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Predicting Australian Takeover Targets:
A Logit Analysis
Maurice Peat*
Maxwell Stevenson*
* Discipline of Finance,
School of Finance,
The University of Sydney
Abstract
Positive announcementday adjusted returns to target shareholders in the event of a
takeover are well documented. Investors who are able to accurately predict firms that
will be the subject of a takeover attempt should be able to earn these excess returns. In
this paper a series of probabilistic regression models were developed that use financial
statement variables suggested by prior research as explanatory variables. The models,
applied to insample and outofsample data, led to predictions of takeover targets that
were better than chance in all cases. The economic outcome resulting from holding a
portfolio of the predicted targets over the prediction period are also analysed.
Keywords: takeovers, targets, prediction, classification, logit analysis
JEL Codes: G11, G17, G23, G34
This is a draft copy and not to be quoted.
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1. Introduction
In this paper our aim is to accurately predict companies that will become takeover
targets. Theoretically, if it is possible to predict takeovers with accuracy greater than
chance, it should be possible to generate abnormal returns from holding a portfolio of
the predicted targets. Evidence of abnormal returns of 20% to 30% made by
shareholders of firms on announcement of a takeover bid is why prediction of these
events is of interest to academics and practitioners alike.
The modelling approach adopted in this study was based on the discrete choice
approach used by Palepu (1986) and Barnes (1999). The models were based on
financial statement information, using variables suggested by the numerous theories
that have been put forward to explain takeover activity. The performance of the
models was evaluated using statistical criteria. Further, the predictions from the
models were rated against chance and economic criteria through the formation and
tracking of a portfolio of predicted targets. Positive results were found under both
evaluation criteria.
Takeover prediction studies are a logical extension of the work of Altman (1968)
who used financial statement information to explain corporate events. Early studies by
Simkowitz and Monroe (1971) and Stevens (1973) were based on the Multiple
Discriminant Analysis (MDA) technique. Stevens (1973) coupled MDA with factor
analysis to eliminate potential multicollinearity problems and reported a predictive
accuracy of 67.5%, suggesting that takeover prediction was viable. Belkaoui (1978)
and Rege (1984) conducted similar analyses in Canada with Belkaoui (1978)
confirming the results of these earlier researchers and reporting a predictive accuracy
of 85% . Concerns were raised by Rege (1984) who was unable to predict with similar
accuracy. These concerns were also raised in research by others such as Singh (1971)
and Fogelberg, Laurent, and McCorkindale (1975).
Reacting to the wide criticism of the MDA method, researchers began to use
discrete choice models as the basis of their research. Harris et al. (1984) used probit
analysis to develop a model and found that it had extremely high explanatory power,
but were unable to discriminate between target and nontarget firms with any degree
of accuracy. Dietrich and Sorensen (1984) continued this work using a logit model
and achieved a classification accuracy rate of 90%. Palepu (1986) addressed a number
of methodological problems in takeover prediction. He suggested the use of statebased
prediction samples where a number of targets were matched with nontargets
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for the same sample period. While this approach was appropriate for the estimation
sample, it exaggerated accuracies within the predictive samples because the estimated
error rates in these samples were not indicative of error rates within the population of
firms. He also proposed the use of an optimal cutoff point derivation which
considered the decision problem at hand. On the basis of this rectified methodology,
along with the application of a logit model to a large sample of US firms, Palepu
(1986) provided evidence that the ability of the model was no better than a chance
selection of target and nontarget firms. Barnes (1999) also used the logit model and a
modified version of the optimal cutoff rule on UK data. His results indicated that a
portfolio of predicted targets may have been consistent with Palepu’s finding, but he
was unable to document this in the UK context due to model inaccuracy.
In the following section the economic explanations underlying takeover activity
are discussed. Section 3 outlines our takeover hypotheses and describes the
explanatory variables that are used in the modelling procedure. The modelling
framework and data used in the study is contained in Section 4, while the results of
our model estimation, predictions, classification accuracy and portfolio economic
outcomes are found in Section 5. We conclude in Section 6.
2. Economic explanations of takeover activity
Economic explanations of takeover activity have suggested the explanatory
variables that were included in this discrete choice model development study. Jensen
and Meckling (1976) posited that agency problems occurred when decision making
and risk bearing were separated between management and stakeholders1, leading to
management inefficiencies. Manne (1965) and Fama (1980) theorised that a
mechanism existed that ensured management acted in the interests of the vast number
of small noncontrolling shareholders2. They suggested that a market for corporate
control existed in which alternative management teams competed for the rights to
control corporate assets. The threat of acquisition aligned management objectives
with those of stakeholders as managers are terminated in the event of an acquisition in
order to rectify inefficient management of the firm’s assets. Jensen and Ruback
(1983) suggested that both capital gains and increased dividends are available to an
1 Stakeholders are generally considered to be both stock and bond holders of a corporation.
2 We take the interests of shareholders to be in the maximization of the present value of the firm.
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acquirer who could eliminate the inefficiencies created by target management, with
the attractiveness of the firm for takeover increasing with the level of inefficiency.
Jensen (1986) looked at the agency costs of free cash flow, another form of
management inefficiency. In this case, free cash flow referred to cash flows in excess
of positive net present value (NPV) investment opportunities and normal levels of
financial slack (retained earnings). The agency cost of free cash flow is the negative
NPV value that arises from investing in negative NPV projects rather than returning
funds to investors. Jensen (1986) suggested that the market value of the firm should
be discounted by the expected agency costs of free cash flow. These, he argued, were
the costs that could be eliminated either by issuing debt to fund an acquisition of
stock, or through merger with, or acquisition of a growing firm that had positive NPV
investments and required the use of these excess funds. Smith and Kim (1994)
combined the financial pecking order argument of Myers and Majluf (1984) with the
free cash flow argument of Jensen (1986) to create another motivational hypothesis
that postulated inefficient firms forgo profitable investment opportunities because of
informational asymmetries. Further, Jensen (1986) argued that, due to information
asymmetries that left shareholders less informed, management was more likely to
undertake negative NPV projects rather than returning funds to investors. Smith and
Kim (1994) suggested that some combination of these firms, like an inefficient firm
and an efficient acquirer, would be the optimal solution to the two respective resource
allocation problems. This, they hypothesised, would result in a market value for the
combined entity that exceeded the sum of the individual values of the firms. This is
one form of financial synergy that can arise in merger situations.
Another form of financial synergy is that which results from a combination of
characteristics of the target and bidding firms. Jensen (1986) suggested that an
optimal capital structure exists, whereby the marginal benefits and marginal costs of
debt are equal. At this point, the cost of capital for a firm is minimised. This
suggested that increases in leverage will only be viable for those firms who have free
cash flow excesses, and not for those which have an already high level of debt.
Lewellen (1971) proposed that in certain situations, financial efficiencies may be
realized without the realization of operational efficiencies. These efficiencies relied
on a simple Miller and Modigliani (1964) model. It proposed that, in the absence of
corporate taxes, an increase in a firm’s leverage to reasonable levels would increase
the value of the equity share of the company due to a lower cost of capital. By a
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merger of two firms, where either one or both had not utilised their borrowing
capacity, would result in a financial gain. This financial gain would represent a
valuation gain above that of the sum of the equity values of the individual firms.
However, this result is predicated on the assumption that the firms need to either
merge or be acquired in order to achieve this result.
Merger waves are well documented in the literature. Gort (1969) suggested that
industry disturbances are the source of these merger waves, his argument being that
they occurred in response to discrepancies between the valuation of a firm by
shareholders and potential acquirers. As a consequence of economic shocks (such as
deregulation, changes in input or output prices, etc.), expectations concerning future
cash flow became more variable. This results in an increased probability that the value
the acquirer places on a potential target is greater than its current owner’s valuation.
The result is a possible offer and subsequent takeover. Mitchell and Mulherin (1996),
in their analysis of mergers and acquisitions in the US during the 1980s, provided
evidence that mergers and acquisitions cluster by industries and time. Their analysis
confirmed the theoretical and empirical evidence provided by Gort (1969) and
provided a different view suggesting that mergers, acquisitions, and leveraged
buyouts were the least cost method of adjusting to the economic shocks borne by an
industry.
These theories suggested a clear theoretical base on which to build takeover
prediction models. As a result, eight main hypotheses for the motivation of a merger
or acquisition have been formulated, along with twenty three possible explanatory
variables to be incorporated predictive models.
3. Takeover hypotheses and explanatory variables
The most commonly accepted motivation for takeovers is the inefficient
management hypothesis.3 The hypothesis states that inefficiently managed firms will
be acquired by more efficiently managed firms. Accordingly,
H1: Inefficient management will lead to an increased likelihood of acquisition.
Explanatory variables suggested by this hypothesis as candidates to be included in the
specifications of predictive models included:
1. ROA (EBIT/Total Assets – Outside Equity Interests)
3 It is also known as the disciplinary motivation for takeovers.
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2. ROE (Net Profit After Tax / Shareholders Equity – Outside Equity Interests)
3. Earnings Before Interest and Tax Margin (EBIT/Operating Revenue)
4. EBIT/Shareholders Equity
5. Free Cash Flow (FCF)/Total Assets
6. Dividend/Shareholders Equity
7. Growth in EBIT over past year, along with an activity ratio,
8. Asset Turnover (Net Sales/Total Assets)
While there are competing explanations for the effect that a firm’s undervaluation
has on the likelihood of its acquisition by a bidder, there is consistent agreement
across all explanations that the greater the level of undervaluation then the greater the
likelihood a firm will be acquired. The hypothesis that embodies the impact of these
competing explanations is as follows:
H2: Undervaluation of a firm will lead to an increased likelihood of acquisition.
The explanatory variable suggested by this hypothesis is:
9. Market to book ratio (Market Value of Securities/Net Assets)
The Price Earnings (P/E) ratio is closely linked to the undervaluation and
inefficient management hypotheses. The impact of the P/E ratio on the likehood of
acquisition is referred to as the P/E hypothesis:
H3: A high Price to Earnings Ratio will lead to a decreased likelihood of acquisition.
It follows from this hypothesis that the P/E ratio is a likely candidate as an
explanatory variable for inclusion in models for the prediction of potential takeover
targets.
10. Price/Earnings Ratio
The growth resource mismatch hypothesis is the fourth hypothesis. However, the
explanatory variables used in models specified to examine this hypothesis capture
growth and resource availability separately. This gives rise to the following:
H4: Firms which possess low growth / high resource combinations or,
alternatively, high growth / low resource combinations will have an increased
likelihood of acquisition.
The following explanatory variables suggested by this hypothesis are:
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11. Growth in Sales (Operating Revenue) over the past year
12. Capital Expenditure/Total Assets
13. Current Ratio (Current Assets/Current Liabilities)
14. (Current Assets – Current Liabilities)/Total Assets
15. Quick Assets (Current Assets – Inventory)/Current Liabilities
The behaviour of some firms to pay out less of their earnings in order to maintain
enough financial slack (retained earnings) to exploit future growth opportunities as
they arise, has led to the dividend payout hypothesis:
H5: High payout ratios will lead to a decreased likelihood of acquisition.
The obvious explanatory variable suggested by this hypothesis is:
16. Dividend Payout Ratio
Rectification of capital structure problems is an obvious motivation for takeovers.
However, there has been some argument as to the impact of low or high leverage on
acquisition likelihood. This paper proposes a hypothesis known as the inefficient
financial structure hypothesis from which the following hypothesis is derived.
H6: High leverage will lead to a decreased likelihood of acquisition.
The explanatory variables suggested by this hypothesis include:
17. Net Gearing (Short Term Debt + Long Term Debt)/Shareholders Equity
18. Net Interest Cover (EBIT/Interest Expense)
19. Total Liabilities/Total Assets
20. Long Term Debt/Total Assets
The existence of Merger and Acquisition (M&A) activity waves, where takeovers
are clustered in wavelike profiles, have been proposed as indicators of changing
levels of M&A activity over time. It has been argued that the identification of M&A
waves, with the corresponding improved likelihood of acquisition when the wave is
surging, captures the effect of the rate of takeover activity at specific points in time,
and serves as valuable input into takeover prediction models. Consistent with M&A
activity waves and their explanation as a motivation for takeovers is the industry
disturbance hypothesis:
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H7: Industry merger and acquisition activity will lead to an increased likelihood
of acquisition.
An industry relative ratio of takeover activity is suggested by this hypothesis:
21. The numerator is the total bids launched in a given year, while the
denominator is the average number of bids launched across all the industries in
the ASX.
Size will have an impact on the likelihood of acquisition. It seems plausible that
smaller firms will have a greater likelihood of acquisition due to larger firms
generally having fewer bidding firms with the resources to acquire them. This gives
rise to the following hypothesis:
H8: The size of a firm will be negatively related to the likelihood of acquisition.
Explanatory variables that can be employed to control for size include:
21. Log (Total Assets)
22. Net Assets
4. Data and Method
The data requirements for the variables defined above are derived from the
financial statements and balance sheet date price information for Australian listed
companies. The financial statement information was sourced from the AspectHuntley
data base which includes annual financial statement data for all ASX listed companies
between 1995 and 2006. The database includes industry classifications for all firms
included in the construction of industry relative ratios. Lists of takeover bids and their
respective success were obtained from the Connect4 database. This information
enabled the construction of variables for relative merger activity between industries.
Additionally, stock prices from the relevant balance dates of all companies were
sourced from the AspectHuntley online database, the SIRCA Core Price Data Set and
Yahoo! Finance.
4.1 The Discrete Choice Modelling Framework
The modelling procedure used is the nominal logit model, made popular in the
bankruptcy prediction literature by Ohlson (1980) and, subsequently, in the takeover
prediction literature by Palepu (1986). Logit models are commonly utilised for
dichotomous state problems. The model is given by equations [1] to [3] below.
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[3]
The logit model was developed to overcome the rigidities of the Linear
Probability Model in the presence of a binary dependent variable. Equations [1] and
[2] show the existence of a linear relationship between the logodds ratio (otherwise
known as the logit Li) and the explanatory variables. However, the relationship
between the probability of the event and acquisition likelihood is nonlinear. This
nonlinear relationship has a major advantage that is demonstrated in equation [3].
Equation [3] measures the change in the probability of the event as a result of a small
increment in the explanatory variables, .
When the probability of the
event is high or low, the incremental impact of a change in an explanatory variable on
the likelihood of the event will be compressed, requiring a large change in the
explanatory variables to change the classification of the observation. If a firm is
clearly classified as a target or nontarget, a large change in the explanatory variables
is required to change its classification.
4.2 Sampling Schema
Two samples were used in the model building and evaluation procedure. They
were selected to mimic the problem faced by a practitioner attempting to predict
takeover targets into the future.
The first sample was used to estimate the model and to conduct insample
classification. It was referred to as the Estimation Sample. This sample was based on
financial data for the 2001 and 2002 financial years for firms that became takeover
targets, as well as selected nontargets, between January, 2003 and December, 2004.
The lag in the dates allows for the release of financial information as well as allowing
for the release of financial statements for firms whose balance dates fall after the 30th
June. Following model estimation, the probability of a takeover offer was estimated
for each firm in the entire sample of firms between January, 2003 and December,
2004 using the estimated model and each firm’s 2001 and 2002 financial data. Expost
predictive ability for each firm was then assessed.
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A second sample was then used to assess the predictive accuracy of the model
estimated with the estimation sample data. It is referred to as the Prediction Sample.
This sample includes the financial data for the 2003 and 2004 financial years, which
will be used in conjunction with target and nontarget firms for the period January,
2005 to December, 2006. Using the model estimated from the 2001 and 2002
financial data, the sample of firms from 2005 and 2006 were fitted to the model using
their 2003 and 2004 financial data. They were then classified as targets or nontargets
using the 2005 and 2006 data. This sampling methodology allows for the evaluation
of exante predictive ability rather than expost classification accuracy. A
diagrammatic explanation of the sample data used for both model estimation and
prediction can be found below in Figure 1, and in tabular form in Table 1.
Figure 1 Timeline of sample data used in model estimation and prediction
Table 1 Sample data used in model estimation and prediction
Sample Financial Data Classification Period
Estimation Sample 2001 and 2002 2003 and 2004
Prediction Sample 2003 and 2004 2005 and 2006
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For model estimation, a technique known as statebased sampling was used.
Allison (2006) suggested the use of this sampling approach in order to minimise the
standard error of the estimated parameters when the dependent variable states were
unequally distributed in the population. All the target firms were included in the
estimation sample, along with an equal number of randomly selected nontarget firms
for the same period. Targets in the estimation sample were randomly paired with the
sample of nontarget firms for the same period over which financial data was
measured.4
4.3 Assessing the Estimated Model and its Predictive Accuracy
Walter (1994), Zanakis and Zopounidis (1997), and Barnes (1999) utilised the
Proportional Chance Criterion and the Maximum Chance Criterion to assess the
predictions of discriminant models relative to chance. These criteria are also
applicable to the discrete choice modelling exercise that is the focus of this study and,
accordingly, are discussed more fully below.
4.3.1 Proportional Chance Criterion
To assess the classification accuracy of the estimated models in this study, the
Proportional Chance Criterion was utilized to assess whether the overall
classifications from the models were better than that expected by chance. This
criterion compared the classification accuracy of models to jointly classify target or
nontarget firms better than that expected by chance. Although the criterion does not
indicate the source of the classification accuracy of the model (that is, whether the
model accurately predicts targets or nontargets), it does allow for the comparison
with alternative models. A simple Zscore calculation formed the basis of a joint test
of the null hypothesis that the model was unable to jointly classify targets and nontargets
better than chance. Under a chance selection, we would expect a proportion of
targets and nontargets to be jointly equal to their frequencies in the population under
consideration. The null and alternative hypotheses, along with the test statistic are
given below.
H0: Model is unable to classify targets and nontargets jointly better than chance.
H1: Model is able to classify targets and nontargets jointly better than chance.
4 This approach differs from matched pair samples where targets are matched to nontargets on the basis of
variables such as industry and/or size.
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If the statistic is significant, we reject the null hypothesis and conclude that the
model can classify target and nontarget firms jointly better than chance.
4.3.2 Maximum Chance Criterion
While the Proportional Chance Criterion indicated whether a model jointly
classified target and nontarget firms better than chance, it did not indicate the source
of the predictive ability. However, under the Maximum Chance Criterion, a similar
test of hypotheses does indicate whether a model has probability greater than chance
in classifying either a target or a nontarget firm. The Zscore statistic to test the null
hypothesis that a model is unable to classify targets better than chance is given below.
It is based on the Concentration Ratio defined by Powell (2001) that measures the
maximum potential chance of correct classification of a target, or the proportion of
correctly classified targets from those firms predicted to be targets.
H0: Model is unable to classify targets better than chance.
H1: Model is able to classify targets better than chance.
In order to assess the classification accuracy of the models in the Estimation and
Prediction Samples, these two criteria were used. The focus of this study was on the
use of the Maximum Chance Criterion for targets, as it assessed whether the number
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of correctly predicted targets exceeded the population of predicted targets.5 The
Concentration Ratio was the ratio advocated by Barnes (1999) for maximising returns.
4.3.3 Industry Relative Ratios
Platt and Platt (1990) advocated the use of industry relative variables to increase
the predictive accuracy of bankruptcy prediction models on the pretext that these
variables enabled more accurate predictions across industries and through time. This
argument was based on two main contentions. Firstly, average financial ratios are
inconsistent across industries and reflect the relative efficiencies of production
commonly employed in those industries. The second is that average financial ratios
are inconsistent throughout time as a result of variable industry performance due to
economic conditions and other factors. Platt and Platt (1990) argued that firms from
different industries or different time periods could not be analysed without some form
of industry adjustment. In this study both raw and industry adjusted financial ratios
were used to determine the benefits of industry adjustment.
There are four different model specifications. One was based on raw financial
ratios for the single year prior to the sample period (the Single Raw Model). Another
was based on averaged raw financial ratios for the two years prior to the acquisition
period (the Combined Raw Model). A third specification was based on industry
adjusted financial ratios for the single year prior to the sample period (the Single
Adjusted Model), while the fourth was based on averaged industry adjusted financial
ratios for the two years prior to the sample period (the Combined Adjusted Model).
The purpose of using averages was to reduce random fluctuations in the financial
ratios of the firms under analysis, and to capture permanent rather than transitory
values. This approach was proposed by Walter (1994).
Most researchers used industry relative ratios calculated by scaling firms’
financial ratios using the industry average defined by equation [4] below. Under this
procedure all ratios were standardised to unity. Industry relative ratios such as ROA
or ROE that were greater than unity indicated industry overperformance, while those
less than unity were consistent with underperformance. Problems were encountered
when the industry average value was negative. In this case, those firms that
underperformed the industry average also had industry relative ratios greater than one.
This was the result of a large negative number being divided by a smaller negative
5 That is, the ratio A11/TP1 in Table 2.
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number. Additionally, those firms that overperformed the negative industry average
ratio, but still retained a negative financial ratio, had a ratio less than one. This
ambiguity in the calculation of industry relative ratios had implications for those
models in this study that included variables with negative industry averages for some
ratios. This problem may explain the inability of researchers in the recent literature to
accurately predict target and nontarget firms that utilised industry adjustments and
may have caused the Barnes (1999) model to predict no takeover targets at all.
An alternative methodology was implemented to account for negative industry
averages. Equation [5] below uses the difference between the individual firm’s ratio
and the industry average ratio, divided by the absolute value of the industry average
ratio. As a result all ratios are standardised to zero rather than one. Problems relating
to the sign of the industry relative ratio are also corrected. Underperformance of the
industry results in an industry relative ratio less than zero, with overperformance
returning a ratio greater than zero. This approach is similar to the variable scaling
methods widely documented in the Neural Network prediction literature. It was used
for the two models based on industry relative variables with industry adjustment
based on the 24 industry classification from the old ASX.
4.4 Calculation of Optimal Cutoff Probabilities for Classification
In the case of a logit model, predictive output for an input sample of the
explanatory variables is a probability with a value between 0 and 1. This is the
predicted probability of an acquisition offer being made for a specific firm within the
prediction period. What is needed is a method to convert these predicted probabilities
of an acquisition offer into a binary prediction of becoming a target or not. These
methods are known as optimal cutoff probability calculations and two main
methodologies were implemented in this study.
4.4.1 Minimisation of Error Probabilities (Palepu, 1986)
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In order to understand the calculation of the optimal cutoff probability, what is
needed is an understanding of Type 1 and Type 11 errors. A Type 1 error occurs when
a firm is predicted to become a takeover target when it does not (outcome A01 in
Table 2 below), while a Type 11 error occurs when a firm is predicted not to become
a target but actually becomes a target (outcome A10). Palepu (1986) assumed that the
cost of these two types of errors were identical. To calculate the optimal cutoff
probability, he used histograms to plot the predicted probabilities of acquisition offers
for targets and nontargets separately on the same graph. The optimal cutoff
probability which minimised the total error rate occurs at the intersection of the two
conditional distributions. Firms with predicted probabilities of acquisition offers
above this cutoff were classified as targets and those with probabilities below the cutoff
classified as nontargets.
Table 2 An outcome matrix for a standard classification problem
Predicted Outcome
Actual Outcome NonTarget (0) Target (1) Total
NonTarget (0) A00 A01 TA0
Target (1) A10 A11 TA1
Total TP0 TP1 T
4.4.2 Minimisation of Error Costs (Barnes, 1999)
Palepu (1986) assumed equal costs of Type 1 and Type 11 errors. However, it has
been suggested that, due to investment being less likely in predicted nontargets, the
cost of investing in the equity of a firm which did not become a takeover target (Type
1 error) was greater than the cost of not investing in the equity of a firm that became a
takeover target (Type 11 error). Accordingly, Barnes (1999) proposed minimisation of
the Type 1 error in order to maximise returns from an investment in predicted targets.
From Table 2, it can be seen that the minimisation of Type 1 error is equivalent to the
minimisation of the number of incorrectly predicted targets, A01, or alternatively, the
maximisation of the number of correctly predicted targets, A11. It follows that, a cutoff
probability is needed to maximise the number of predicted targets in a portfolio
that became actual targets. This involved maximisation of the ratio of A11 to TP1 in
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Table 2. Figure 2 below is an idealized representation of the Type 1 and Type 2 errors
associated with the Palepu and Barnes cutoff probability methodologies.
As the purpose of this paper was to replicate the problem faced by a practitioner,
unawareness of the actual outcomes of the prediction process was assumed. Further,
Figure 2 Idealized Palepu and Barnes Cutoff Probabilities
the probabilities that companies will become targets were derived from a prediction
model estimated using estimation data on known targets and nontargets. The
companies for which these probabilities are calculated comprised the Prediction
Sample (recall Table 1).
For the calculation of the optimal cutoff probability according to Palepu, a
histogram of predicted acquisition offer probabilities for targets and non targets was
created from the Estimation Sample, and followed the error minimisation procedure
detailed above in section 4.4.1. To calculate the optimal cutoff under the Barnes
methodology outlined in section 4.4.2, the ratio of A11/ TP1 for all cutoff probabilities
between 0 and 1 was calculated to determine the maximum point. A simple grid
search from 0 to 1 in increments of 0.05 was used.
The classification and prediction accuracies under these two methods of
calculating cutoff probabilities was compared for all four models considered in this
study.
NonTargets
Targets
Estimated Acquisition Offer Cutoff Probabilities for NonTargets and Targets
PALEPU CUTOFF
BARNES CUTOFF
Type 2
Error
Type 1
Error
Relative Frequency of NonTargets and Targets
Targets
NonTargets Targets
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5. Results
5.1 Multicollinearity Issues
An examination of the correlation matrix and Variance Inflation Factors (VIFs) of the
Estimation Sample indicated that five variables needed to be eliminated. They are
listed in Table 3. That these variables should contribute to the multicollinearity
problem was not a surprise considering the presence of the large number of potential
explanatory variables measuring similar attributes suggested by the hypothesised
motivations for takeover. These variables have correlation coefficients that exceeded
0.8 or VIFs that exceeded 10. Exclusion of these five variables eliminated significant
correlations in the Variance/Covariance matrix, along with reduction of the VIF
values of all the remaining variables to below 10. The resultant reduced variable set
was used in the backward stepwise logit models estimated and reported in the
following subsection.
Table 3 Variables Removed Due to Multicollinearity
ROE (NPAT/Shareholders Equity – Outside Equity Interests)
FCF/Total Assets
Current Ratio (Current Assets/Current Liabilities)
(Current Assets – Current Liabilities)/Total Assets
Total Liabilities/Total Assets
5.2 Backward Stepwise Regression Results
Using the remaining variables after controlling for multicollinearity, backward
stepwise logistic regressions were performed for each of the four model
specifications. Consistent with the methodology of Walter (1994), the significance
level for retention of variables in the analysis was set at 0.15. The results for these
models that were estimated using a common set of target and nontarget firms are
presented in Tables 4 to 7, with the results for the combined adjusted model in Table
7 described in more detail in the following subsection.6
The backward stepwise analysis for this model required seven steps, eliminating
six of the fifteen starting variables, while retaining nine significant variables. These
6 Detailed results for each of the models represented in Tables 4 to 7 are available from the authors on request.
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Table 4 Backward Stepwise Results for Single Raw Model
Variable Parameter
Estimate Prob > Chi Sq
Intercept 13.14 ( Chi Sq
Intercept 0.58 (0.02)
Asset Turnover (Net Sales/Total
Assets) 0.59 (0.03)
Capital Expenditure/Total Assets 0.34 (0.07)
Dividend Payout Ratio 0.22 (0.07)
Long Term Debt/Total Assets 0.21 (0.11)
Ln (Total Assets) 12.07 ( Chi Sq
Intercept 12.36 ( Chi Sq
Intercept 0.04 (0.92)
ROA (EBIT/Total Assets – Outside Equity
Interests) 0.28 (0.09)
Asset Turnover (Net Sales/Total Assets) 0.54 (0.05)
Capital Expenditure/Total Assets 0.69 (<0.01)
Quick Assets/Current Liabilities
(Current Assets – Inventory)/Current Liabilities 0.93 (0.02)
Dividend Payout Ratio 0.34 (0.02)
Long Term Debt/Total Assets 0.32 (0.07)
Merger Wave Dummy 0.59 (0.06)
Ln (Total Assets) 13.34 (<0.01)
Net Assets 0.21 (0.07)
results provided evidence concerning six of the eight hypothesised motivations for
takeover discussed previously in Section 3.
The growth resource mismatch hypothesis was only significant in the two adjusted
models. This suggested that growth should be measured relative to an industry
benchmark when attempting to discriminate between target and nontarget firms.
5.3 Classification Analysis
While the analysis of the final models was of theoretical interest, the primary aim of
this paper was to evaluate their classification accuracy. For the purposes of
classification, the models were reestimated using the Estimation Sample with all
variables included. The complex relationships between all the variables were assumed
to provide us with the ability to discriminate between target and nontarget firms.
Using financial data from 2001 and 2002, the models were estimated on the basis of
62 targets matched with 62 nontargets where the targets were identified between
January, 2003 and December, 2004. Following estimation of the model, an insample
fit was sought for the entire sample of the 1060 firms reporting 2001 and 2002
financial data. To proceed with classification, we derived a cutoff probability using
20
the methods of Palepu (1986) and Barnes (1999). The graph presented in Figure 3
focuses on the combined adjusted model and the Palepu cutoff point. Using a bin
range of 0.05, it showed the histograms required for the calculation of the cutoff
probability using the Palepu methodology was approximately 0.675. This is the
probability corresponding to the highest point of intersection of the plots of the
estimated acquisition probabilities for target and nontarget companies.
Figure 3 Cutoff Calculations using the Palepu methodology and 0.05 histogram
bin increments.
Table 8 Summary of optimal cutoff probabilities for all models under both
methodologies.
Optimal Cutoff Methodology
Probabilities Palepu Barnes
Single Raw Model 0.725 0.85
Single Adjusted Model 0.725 0.90
Combined Raw Model 0.850 0.95
Combined Adjusted Model 0.675 0.95
The optimal cutoff probabilities derived by using both the Barnes and Palepu
methodologies for all four models are reported in Table 8. The optimal cutoff
21
probabilities calculated using the Barnes methodology were significantly larger than
the cutoffs calculated under the Palepu methodology for all models.7
Table 9 below shows the outcome of the application of all of four models to the entire
Estimation Sample based on a cutoff derived under the Barnes approach. Included in
this table are the outcome matrices for each of the models. An outcome of 0 indicated
that the firm was not a target or was not predicted to be a target in the sample period.
A value of 1 indicated that a firm was predicted to be, or become a target in the
sample period. On the basis of these outcome matrices, a number of performance
measures were generated.
The first measure was the Concentration Ratio. This is a measure of Predictive
Accuracy measure of the model and corresponds to the Maximum Chance Criterion.
It is the proportion of actual targets that formed the portfolio of predicted target firms
for each of the models and was represented by the ratio A11/TP1 from the outcome
matrix depicted previously in Table 2. The next measure indicated the expected
accuracy under a chance selection of takeover targets within the sample period
(TA1/T). It measured the extent to which the model exceeded the accuracy expected
under a chance selection and quantified the Proportional Chance Criterion. The last
measure is a measure of the accuracy of the model relative to chance and is calculated
by dividing the first ratio by the second and then subtracting unity. All three measures
were expressed as a percentage. An examination of the statistics corresponding to
these measures for all four models in Table 9 indicated that, for the estimation sample
with a Barnes cutoff, the combined raw model was the most accurate. Of the 80 firms
that this model predicted to become takeover targets in the estimation period, 19
actually became targets. This represented a prediction accuracy of 23.75%. When
taken relative to chance, this accuracy exceeded the benchmark by 305%.
For the purpose of comparison, the classification results for the cutoff
probabilities calculated using the Palepu cutoff points are presented in Table 10. As
was the case when the Barnes methodology was used to determine the cutoff values
for classification, the Palepu approach realised similar results. The combined raw
model was again the most accurate model for prediction with a predictive accuracy of
19.59% and a relative to chance figure of 234.3%. However, as was the case for all
four models, the use of this cutoff probability approach significantly reduced the
7 As is noted in following tables, this is an explanation for the smaller number of predicted targets under this
methodology.
22
Table 9 Outcome Matrices for all models for classification of Estimation
Sample (Barnes cutoff probability)
ESTIMATION SAMPLE
Predictive Accuracy
Single Raw Model (Cutoff = 0.85
probability) †† Chance Accuracy
Actual Outcome
Predicted
Outcome Relative to Chance
0 1 Total
0 874 124 998 97.00% 94.15% 3.03%**
1 27 35 62 22.01% 5.85% 276.24%**
Total 901 159 1060
Single Adjusted Model (Cutoff
probability = 0.90) ††
Actual Outcome
Predicted
Outcome
0 1 Total
0 906 88 994 96.18% 94.13% 2.18%**
1 36 26 62 22.81% 5.87% 288.59%**
Total 942 114 1056
Combined Raw Model (Cutoff
probability = 0.95) ††
Actual Outcome
Predicted
Outcome
0 1 Total
0 935 61 996 95.60% 94.14% 1.55%**
1 43 19 62 23.75% 5.86% 305.29%**
Total 978 80 1058
Combined Adjusted Model (Cutoff
probability = 0.95) ††
Actual Outcome
Predicted
Outcome
0 1 Total
0 938 56 994 95.33% 94.13% 1.27%*
1 46 16 62 22.22% 5.87% 278.54%**
23
Total 984 72 1056
†† Indicates that the overall predictions of the model are significantly better than chance at the 1%
level of significance according to the Proportional Chance Criterion.
** Indicates that the prediction of targets or non targets individually is significantly greater than chance
at the 1% level of significance according to the Maximum Chance Criterion.
* Indicates that the prediction of targets or non targets individually is significantly greater than chance
at the 5% level of significance according to the Maximum Chance Criterion.
Table 10 Outcome Matrices for all models for classification of Estimation
Sample (Palepu cutoff probabilty)
ESTIMATION SAMPLE
Predictive
Accuracy
Single Raw Model (Cutoff probability = 0.725) †† Chance Accuracy
Actual Outcome Predicted Outcome
Relative to
Chance
0 1 Total
0 812 186 998 97.83% 94.15% 3.91%**
1 18 44 62 19.13% 5.85% 227.01%**
Total 830 230 1060
Single Adjusted Model (Cutoff probability = 0.725)
††
Actual Outcome Predicted Outcome
0 1 Total
0 787 207 994 97.52% 94.13% 3.60%**
1 20 42 62 16.87% 5.87% 187.39%**
Total 807 249 1056
Combined Raw Model (Cutoff probability = 0.85) ††
Actual Outcome Predicted Outcome
0 1 Total
0 840 156 996 97.22% 94.14% 3.27%**
1 24 38 62 19.59% 5.86% 234.30%**
Total 864 194 1058
Combined Adjusted Model (Cutoff probability =
0.675) ††
Actual Outcome Predicted Outcome
24
0 1 Total
0 749 245 994 97.53% 94.13% 3.61%**
1 19 43 62 14.93% 5.87% 154.34%**
Total 768 288 1056
†† Indicates that the overall predictions of the model are significantly better than chance at the 1%
level of significance according to the Proportional Chance Criterion.
** Indicates that the prediction of targets or non targets individually is significantly greater than chance
at the 1% level of significance according to the Maximum Chance Criterion.
Concentration Ratio and, therefore, the classification accuracy of the models under
the Maximum Chance Criterion. Interestingly, while the Palepu methodology did
improve the correct classification of targets accurately predicted (A11), in doing so, it
also predicted a large number of nontarget firms to become targets (A01).
The Barnes methodology focused on the maximisation of returns from an
investment in predicted targets. Rather than being focused on the prediction of a large
number of targets accurately, it focused on the improvement in the proportion of
actual targets in the portfolio of predicted targets. Accordingly, there are a smaller
number of targets predicted under the Barnes methodology. As previously noted in
Section 4.3.2, the Barnes methodology coincided more with the spirit of the
Maximum Chance Criterion rather than the Proportional Chance Criterion.
According to the Proportional Chance Criterion, all four models were able to
jointly classify targets and nontargets within the estimation period significantly better
than chance. Further, as revealed by the Maximum Chance Criterion, all models also
classified targets alone significantly better than chance but on an individual basis.
Overall, these results indicated high model classification ability. This was expected
given that all targets in the estimation sample were used in the estimation of the
model parameters.
5.4 Classification in the Prediction Period
The next step of the analysis was to assess the predictive abilities of our models
using the Prediction Sample. Of the total 1054 firms in this sample, 108 became
targets during the prediction period. Panel A and Panel B of Table 11 report the
predictions from the four estimated models using both the Barnes and Palepu cutoff
probability approaches. Under the Barnes cutoff methodology, calculation of the
Concentration Ratio indicated that the combined raw and combined adjusted models
performed best of all of the models. This confirmed the results from the estimation
25
period. The combined adjusted model predicted 125 firms to become targets during
the prediction period, during which 25 actually became targets. Prediction accuracy
was 20%. Under a chance selection, we would have expected only 10.30% of those
companies predicted to become targets to actually become targets. This meant that the
model exceeded a chance prediction by 94.18%. While Walter (1994) was able to
predict 102% better than chance, other studies including that of Palepu (1986) and
Barnes (1999) were unable to achieve this level of accuracy.
Industry adjustment increased predictive ability for both the single and combined
models, suggesting that stability may be achieved through these adjustments.
Furthermore, the combination of two years of financial data also appeared to improve
predictive accuracy. This suggests that this adjustment eliminates random fluctuations
in the financial ratios being used as input to the prediction models.
Table 11 Prediction results for all four models using the Prediction Sample and
both Barnes and Palepu cutoff probabilities
PREDICTION SAMPLE
(Barnes cutoff probabilities)
Panel A
PREDICTION SAMPLE
(Palepu cutoff probabilities)
Panel B
Predictive Chance Relative to
Accuracy Accuracy Chance
Predictive Chance Relative to
Accuracy Accuracy Chance
Single Raw Model
(Cutoff probability = 0.90)
15.09% 10.25% 47.22%*
Single Raw Model
(Cutoff probability = 0.725)
16.83% 10.25% 64.20%*
Single Adjusted Model
(Cutoff probability = 0.95)
15.79% 10.27% 53.75%*
Single Adjusted Model
(Cutoff probability = 0.725)
17.79% 10.27% 73.22%*
Combined Raw Model
(Cutoff probability = 0.85)
17.65% 10.25% 72.29%**
Combined Raw Model
(Cutoff probability = 0.85)
17.51% 10.25% 70.83%**
Combined Adjusted Model †
(Cutoff probability = 0.95)
Combined Adjusted Model
(Cutoff probability = 0.675)
26
20.00% 10.30% 94.18%**
16.77% 10.30% 62.82%**
† Indicates that the overall predictions of the model are significantly better than chance at the 5% level
of significance according to the Proportional Chance Criterion.
** Indicates that the prediction of targets or non targets individually is significantly greater than chance
at the 1% level of significance according to the Maximum Chance Criterion.
* Indicates that the prediction of targets or non targets individually is significantly greater than chance
at the 5% level of significance according to the Maximum Chance Criterion.
The prediction results for the Palepu derived cutoff probabilities are presented in
Table 11 (Panel B). By a comparison of Panel A with Panel B in Table 11, it can be
seen that when the Barnes cutoff probability methodology was used for the single
models, the Concentration Ratio decreased relative to that of Palepu. However, it
improved the ratio for the combined models. This result was reversed when the
Palepu cutoff probability approach was used. Further, given the better performance
of the combined models using the estimated sample, this provided the rationale for the
use of the combined modes and the Barnes methodology to calculate the optimal cutoff
probabilities.
A different variable selection approach was implemented in an attempt to improve
the accuracy of the two best predictive models, namely, the combined raw model and
the combined adjusted model. A number of variables that had been insignificant in all
estimated models were removed and the estimation and classification procedures
repeated on the remaining variable data set.8 The classification results for the
Table 12 Application of improved models to both the Estimation Sample and
Prediction Sample.
ESTIMATION SAMPLE
(Barnes cutoff probabilities)
PREDICTION SAMPLE
(Barnes cutoff probabilities)
Predictive Chance Relative to
Accuracy Accuracy Chance
Predictive Chance Relative to
Accuracy Accuracy Chance
Combined Raw Model ††
Combined Raw Model
8 The variables removed were: Growth in EBIT over the past year, Market to book ratio (Market Value of
Securities/Net Assets), and the Price/Earnings Ratio.
27
(less variables 7,9 and 10)
24.66% 5.86% 320.77%**
(less variables 7,9 and 10)
17.54% 10.25% 71.22%**
Combined Adjusted Model ††
(less variables 7,9 and 10)
24.56% 5.87% 318.34%**
Combined Adjusted Model
(less variables 7,9 and 10)
22.45% 10.30% 118.05%**
†† Indicates that the overall predictions of the model are significantly better than chance at the 1%
level of significance according to the Proportional Chance Criterion.
** Indicates that the prediction of targets or non targets individually is significantly greater than chance
at the 1% level of significance according to the Maximum Chance Criterion.
* Indicates that the prediction of targets or non targets individually is significantly greater than chance
at the 5% level of significance according to the Maximum Chance Criterion.
application of this model to both the estimation and prediction periods are given in
Table 12.
The elimination of variables resulted in significant improvements in the insample
classification accuracy using the estimation sample, with accuracies exceeding chance
by well over 300%. This improvement in classification accuracy was maintained into
the prediction period. The accuracy of the combined adjusted model was 118%
greater than chance. This represented a level statistical accuracy above that reported
by any similar published study in the area of takeover prediction. These results can be
used to refute the claims of Barnes (1999) and Palepu (1986) that models cannot be
implemented which achieved predictive accuracies greater than chance. They further
confirm the results of Walter (1994) while using a wider sample of firms.
The combined adjusted model significantly outperformed the other models for
predictive purposes, suggesting that this is the most appropriate model for the
application of logit analysis to predict takeover targets in the Australian context.
5.5 Economic Outcomes
Although the above methodology provided us with a statistical assessment of
model performance, it had nothing to say about the economic usefulness of the model.
Palepu (1986), Walter (1994), and Wansley et al. (1983) all implemented an equally
weighted portfolio technique to assess whether their predictions of takeover targets
were able to earn abnormal risk adjusted returns. The conclusion we drew from the
results of the abovementioned studies was that a positive abnormal return was not
guaranteed from an investment in the targets predicted from these models. The
portfolios of predicted targets in two of these studies were unrealistically large at 91
28
in the case of Walter, and 625 in the case of the Palepu studies. Due to the effect of
transaction costs on returns, practitioners would be likely to limit themselves to
smaller portfolios in the order of 10 to 15 stocks.
To make an economic assessment of the economic usefulness of our modelling
approach, we replicated a modified version of the Palepu (1986) and Walter (1994)
portfolio technique using our predicted targets. Only commonly predicted targets
across all models were included in the portfolio analysis for two reasons. The first
was to reduce the number of stocks to a manageable level, and second, to improve the
ratio of actual targets in the portfolio. Further, we rejected the equally weighted
portfolio approach on the grounds that it was an inefficient strategy for an informed
investor who possessed results from our modelling. We reasoned that such an investor
could most likely take a leveraged position through derivatives.
The portfolio analysed in this study comprised of 13 predicted target firms of
which 5 actually became targets. While this is a good result per se, we sought to
quantify the economic benefit from an investment in these stocks. The portfolio of
predicted targets was held for the entire prediction period of 2005 and 2006 that
constituted 503 trading days. The first column of Table 13 below presents the
percentage of Cumulative Average Abnormal Return (CAAR%) at 20 day intervals
during the prediction period.
Table 13 Cumulative Average Abnormal Returns (CAARs) for the portfolio of
commonly predicted takeover targets for the Prediction Period of 2005
and 2006
Day Portfolio
(13 Stocks)
Actual Targets
(5 Stocks)
Day Portfolio
(13 Stocks)
Actual Targets
(5 Stocks)
CAAR (%) CAAR (%) CAAR (%) CAAR (%)
20 1.38 5.36 280 4.77 29.64
40 2.84 10.50 300 4.67 32.47
60 1.98 5.58 320 3.08 33.53
80 2.53 6.11 340 0.73 31.96
100 5.52 1.15 360 2.89 26.62
120 4.40 25.16 380 5.28 33.72
140 3.06 17.83 400 6.99 32.02
160 4.38 20.70 420 9.78 37.43
180 5.51 24.79 440 11.33 40.22
200 9.90 34.82 460 57.44 46.00
29
220 7.51 34.87 480 58.38 47.27
240 6.40 29.31 500 68.90 52.12
260 5.04 27.71 503 68.67* 50.86^
The full prediction period CAAR of 68.67% was significantly greater than zero at
the 1% level of significance under the Standard Abnormal Return [SAR]
methodology of Brown and Warner (1985). We recognised that these results could
have been potentially driven by actual nontarget firms within the portfolio of
predicted targets. This would suggest that the abnormal return was the result of the
chance selection of overperforming nontarget firms, rather than an accurate
selection of target firms. To answer this question, the same CAAR calculation was
applied to the subportfolio of firms that actually became targets.
The full period CAAR of 50.86% was also significantly greater than zero at the
1% level. This supported the proposition that the CAAR of the portfolio was driven
by the performance of the actual targets within the portfolio.
Table 13 also indicated that the CAAR for the portfolio increased significantly
between days 440 and 460. This result was driven by the extremely positive returns
on the stock ATM which was a nontarget firm predicted by the models to be a target.
After repeating the portfolio analysis with this stock eliminated from the portfolio of
predicted targets, it was found that a significant positive abnormal return of 25%9 was
realized for the entire prediction period.
Another observation from Table 13 was that the CAAR was not positive (nor
significant) for either of the portfolios early in the prediction period. From the second
column of Table 14, the CAAR after 100 days was negative. Further, after 340 days
the CAAR was indistinguishable from zero. The real gains to the portfolio were made
as mergers or acquisitions were announced and completed in the latter stages of 2006,
highlighting the fact that the portfolio had to be held for the entire prediction period in
order to realise the potential available returns.
6. Conclusion
The main finding of this paper was that the combined adjusted model which was
based on averaged, industry adjusted financial ratios across the sample period,
9 t = 9.63
30
emerged as a clear standout with regards to predictive accuracy. Further, the
implementation of industry adjusted data, as described in Section 4.3.3 of this paper,
significantly improved the classification accuracy of all the models bar one that were
analysed in both the estimation and prediction periods. Additionally, this paper
provided evidence that the inclusion of the Barnes methodology for calculation of the
optimal cutoff point significantly improved classification accuracy and enabled the
successful use of logit models to predict takeover targets within the Australian
context. The accuracy of the single best model in this paper exceeded a chance
selection by 118% and represented the highest reported accuracy for a logit model.
Another important finding of this paper resulted from the examination of a
portfolio of predicted targets. We demonstrated that an investment in the predicted
targets, that were common across the logit models, resulted in significant Cumulative
Average Abnormal Returns (CAARs) being made by an investor. Several steps were
undertaken to ensure that this result was robust against returns on predicted nontarget
stocks. This suggests that the abnormal returns made are based on the accuracy of the
predictions common to the logit models analysed in this study rather than any chance
selection. We believe our results provide evidence in favour of the proposition that an
abnormal return can be made from an investment in the commonly predicted takeover
targets from the four logitbased models analysed in this paper.
There is a wealth of evidence in existence that suggests that combining forecasts
from different models improves forecasting ability. This is an obvious direction for
future research and may well be achieved either by a logit and MDA combination, or
through the inclusion of a neural network approach to predict targets.
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Ultimo aggiornamento: 20140223 
Current transformer
From Wikipedia, the free encyclopedia
This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (April 2010)
A CT for operation on a 110 kV grid
A current transformer (CT) is used for measurement of alternating electric currents. Current transformers, together with voltage transformers (VT) (potential transformers (PT)), are known as instrument transformers. When current in a circuit is too high to apply directly to measuring instruments, a current transformer produces a reduced current accurately proportional to the current in the circuit, which can be conveniently connected to measuring and recording instruments. A current transformer also isolates the measuring instruments from what may be very high voltage in the monitored circuit. Current transformers are commonly used in metering and protective relays in the electrical power industry.
Contents [hide]
1 Design
2 Usage
3 Safety precautions
4 Accuracy
4.1 Burden
4.2 Kneepoint coresaturation voltage
4.3 Rating factor
5 Special designs
6 Standards
7 High voltage types
8 See also
9 References
10 External links
Design[edit]
Basic operation of current transformer
SF6 110 kV current transformer TGFM series, Russia
Current transformers used in metering equipment for threephase 400 ampere electricity supply
Like any other transformer, a current transformer has a primary winding, a magnetic core and a secondary winding. The alternating current flowing in the primary produces an alternating magnetic field in the core, which then induces an alternating current in the secondary winding circuit. An essential objective of current transformer design is to ensure that the primary and secondary circuits are efficiently coupled, so that the secondary current bears an accurate relationship to the primary current.
The most common design of CT consists of a length of wire wrapped many times around a silicon steel ring passed 'around' the circuit being measured. The CT's primary circuit therefore consists of a single 'turn' of conductor, with a secondary of many tens or hundreds of turns. The primary winding may be a permanent part of the current transformer, with a heavy copper bar to carry current through the magnetic core. Windowtype current transformers (aka zero sequence current transformers, or ZSCT) are also common, which can have circuit cables run through the middle of an opening in the core to provide a singleturn primary winding. When conductors passing through a CT are not centered in the circular (or oval) opening, slight inaccuracies may occur.
Shapes and sizes can vary depending on the end user or switchgear manufacturer. Typical examples of low voltage single ratio metering current transformers are either ring type or plastic molded case. Highvoltage current transformers are mounted on porcelain bushings to insulate them from ground. Some CT configurations slip around the bushing of a highvoltage transformer or circuit breaker, which automatically centers the conductor inside the CT window.
The primary circuit is largely unaffected by the insertion of the CT. The rated secondary current is commonly standardized at 1 or 5 amperes. For example, a 4000:5 CT would provide an output current of 5 amperes when the primary was passing 4000 amperes. The secondary winding can be single ratio or multiratio, with five taps being common for multiratio CTs.
The load, or burden, of the CT should be of low resistance. If the voltage time integral area is higher than the core's design rating, the core goes into saturation towards the end of each cycle, distorting the waveform and affecting accuracy.
Usage[edit]
Many digital clamp meters utilize a current transformer for measuring AC current
Current transformers are used extensively for measuring current and monitoring the operation of the power grid. Along with voltage leads, revenuegrade CTs drive the electrical utility's watthour meter on virtually every building with threephase service and singlephase services greater than 200 amps.
The CT is typically described by its current ratio from primary to secondary. Often, multiple CTs are installed as a "stack" for various uses. For example, protection devices and revenue metering may use separate CTs to provide isolation between metering and protection circuits, and allows current transformers with different characteristics (accuracy, overload performance) to be used for the devices.
Safety precautions[edit]
Care must be taken that the secondary of a current transformer is not disconnected from its load while current is flowing in the primary, as the transformer secondary will attempt to continue driving current across the effectively infinite impedance up to its core saturation voltage. This may produce a high voltage across the open secondary into the range of several kilovolts, causing arcing, compromising operator and equipment safety, or permanently affect the accuracy of the transformer.
Accuracy[edit]
The accuracy of a CT is directly related to a number of factors including:
Burden
Burden class/saturation class
Rating factor
Load
External electromagnetic fields
Temperature and
Physical configuration.
The selected tap, for multiratio CTs
For the IEC standard, accuracy classes for various types of measurement are set out in IEC 600441, Classes 0.1, 0.2s, 0.2, 0.5, 0.5s, 1 and 3. The class designation is an approximate measure of the CT's accuracy. The ratio (primary to secondary current) error of a Class 1 CT is 1%[ERROR] at rated current; the ratio error of a Class 0.5 CT is 0.5%[ERROR] or less. Errors in phase are also important especially in power measuring circuits, and each class has an allowable maximum phase error for a specified load impedance.
Current transformers used for protective relaying also have accuracy requirements at overload currents in excess of the normal rating to ensure accurate performance of relays during system faults. A CT with a rating of 2.5L400 specifies with an output from its secondary winding of 20 times its rated secondary current (usually 5 A x 20 = 100 A) and 400 V (IZ drop) its output accuracy will be within 2.5 percent.
Burden[edit]
The secondary load of a current transformer is usually called the "burden" to distinguish it from the load of the circuit whose current is being measured. The current transformer is mounted one of the power transformer leads; it can be associated with an Lv or Hv lead; depending on voltage and current consideration. A section of the lead is demountable locally to enable the current transformer to removed, should the necessity arise, without disturbing the main connection. The secondary of CT is connected to the heating coil directly located under the main cover in the oil. On the larger Errs the various connections may be brought up to terminals in the main the cover for external linkage. The burden, in a CT metering circuit is the (largely resistive) impedance presented to its secondary winding. Typical burden ratings for IEC CTs are 1.5 VA, 3 VA, 5 VA, 10 VA, 15 VA, 20 VA, 30 VA, 45 VA and 60 VA. As for ANSI/IEEE burden ratings are B0.1, B0.2, B0.5, B1.0, B2.0 and B4.0. This means a CT with a burden rating of B0.2 can tolerate up to 0.2 Ω of impedance in the metering circuit before its secondary accuracy falls outside of an accuracy specification. These specification diagrams show accuracy parallelograms on a grid incorporating magnitude and phase angle error scales at the CT's rated burden. Items that contribute to the burden of a current measurement circuit are switchblocks, meters and intermediate conductors. The most common source of excess burden is the conductor between the meter and the CT. When substation meters are located far from the meter cabinets, the excessive length of wire creates a large resistance. This problem can be reduced by using CTs with 1 ampere secondaries, which will produce less voltage drop between a CT and its metering devices.
Kneepoint coresaturation voltage[edit]
The kneepoint voltage of a current transformer is the magnitude of the secondary voltage above which the output current increases to linearly follow the input current within declared accuracy. In testing, if a voltage is applied across the secondary terminals the magnetizing current will increase in proportion to the applied voltage, until the knee point is reached. The knee point is defined as the voltage at which a 10%[ERROR] increase in applied voltage increases the magnetizing current by 50%[ERROR]. For voltages greater than the knee point, the magnetizing current increases considerably even for small increments in voltage across the secondary terminals. The kneepoint voltage is less applicable for metering current transformers as their accuracy is generally much tighter but constrained within a very small bandwidth of the current transformer rating, typically 1.2 to 1.5 times rated current. However, the concept of knee point voltage is very pertinent to protection current transformers, since they are necessarily exposed to currents of 20 or 30 times rated current during faults.[1]
Rating factor[edit]
Rating factor is a factor by which the nominal full load current of a CT can be multiplied to determine its absolute maximum measurable primary current. Conversely, the minimum primary current a CT can accurately measure is "light load," or 10%[ERROR] of the nominal current (there are, however, special CTs designed to measure accurately currents as small as 2%[ERROR] of the nominal current). The rating factor of a CT is largely dependent upon ambient temperature. Most CTs have rating factors for 35 degrees Celsius and 55 degrees Celsius. It is important to be mindful of ambient temperatures and resultant rating factors when CTs are installed inside padmount transformers or poorly ventilated mechanical rooms. Recently, manufacturers have been moving towards lower nominal primary currents with greater rating factors. This is made possible by the development of more efficient ferrites and their corresponding hysteresis curves.
Special designs[edit]
Specially constructed wideband current transformers are also used (usually with an oscilloscope) to measure waveforms of high frequency or pulsed currents within pulsed power systems. One type of specially constructed wideband transformer provides a voltage output that is proportional to the measured current. Another type (called a Rogowski coil) requires an external integrator in order to provide a voltage output that is proportional to the measured current. Unlike CTs used for power circuitry, wideband CTs are rated in output volts per ampere of primary current. CT RATIO
Standards[edit]
Depending on the ultimate clients requirement, there are two main standards to which current transformers are designed. IEC 600441 (BSEN 600441) & IEEE C57.13 (ANSI), although the Canadian & Australian standards are also recognised.
High voltage types[edit]
Current transformers are used for protection, measurement and control in high voltage electrical substations and the electrical grid. Current transformers may be installed inside switchgear or in apparatus bushings, but very often freestanding outdoor current transformers are used. In a switchyard, live tank current transformers have a substantial part of their enclosure energized at the line voltage and must be mounted on insulators. Dead tank current transformers isolate the measured circuit from the enclosure. Live tank CTs are useful because the primary conductor is short, which gives better stability and a higher shortcircuit current withstand rating. The primary of the winding can be evenly distributed around the magnetic core, which gives better performance for overloads and transients. Since the major insulation of a livetank current transformer is not exposed to the heat of the primary conductors, insulation life and thermal stability is improved.
A highvoltage current transformer may contain several cores, each with a secondary winding, for different purposes (such as metering circuits, control, or protection).[2] A neutral current transformer is used as earth fault protection to measured any fault current flowing through the neutral line from the wye neutral point of a transformer.[3]

Ultimo aggiornamento: 20131124 
Sound System Design
Reference Manual
Sound System Design Reference Manual
Sound System Design Reference Manual
Table of Contents
Preface ............................................................................................................................................. i
Chapter 1: Wave Propagation........................................................................................................ 11
Wavelength, Frequency, and Speed of Sound ................................................................................. 11
Combining Sine Waves .................................................................................................................... 12
Combining Delayed Sine Waves ...................................................................................................... 13
Diffraction of Sound .......................................................................................................................... 15
Effects of Temperature Gradients on Sound Propagation ................................................................ 16
Effects of Wind Velocity and Gradients on Sound Propagation ........................................................ 16
Effect of Humidity on Sound Propagation ......................................................................................... 17
Chapter 2: The Decibel ................................................................................................................... 21
Introduction ....................................................................................................................................... 21
Power Relationships ......................................................................................................................... 21
Voltage, Current, and Pressure Relationships .................................................................................. 22
Sound Pressure and Loudness Contours ......................................................................................... 24
Inverse Square Relationships ........................................................................................................... 26
Adding Power Levels in dB ............................................................................................................... 27
Reference Levels .............................................................................................................................. 27
Peak, Average, and RMS Signal Values........................................................................................... 28
Chapter 3: Directivity and Angular Coverage of Loudspeakers ................................................ 31
Introduction ....................................................................................................................................... 31
Some Fundamentals ........................................................................................................................ 31
A Comparison of Polar Plots, Beamwidth Plots, Directivity Plots, and Isobars ................................ 33
Directivity of Circular Radiators ........................................................................................................ 34
The Importance of Flat Power Response ......................................................................................... 36
Measurement of Directional Characteristics ..................................................................................... 37
Using Directivity Information ............................................................................................................. 38
Directional Characteristics of Combined Radiators .......................................................................... 38
Chapter 4: An Outdoor Sound Reinforcement System ............................................................... 41
Introduction ....................................................................................................................................... 41
The Concept of Acoustical Gain ....................................................................................................... 42
The Influence of Directional Microphones and Loudspeakers on System Maximum Gain .............. 43
How Much Gain is Needed? ............................................................................................................. 44
Conclusion ........................................................................................................................................ 45
Chapter 5: Fundamentals of Room Acoustics ............................................................................. 51
Introduction ....................................................................................................................................... 51
Absorption and Reflection of Sound ................................................................................................. 51
The Growth and Decay of a Sound Field in a Room ........................................................................ 55
Reverberation and Reverberation Time............................................................................................ 57
Direct and Reverberant Sound Fields .............................................................................................. 512
Critical Distance ................................................................................................................................ 514
The Room Constant ......................................................................................................................... 515
Statistical Models and the Real World .............................................................................................. 520
Sound System Design Reference Manual
Table of Contents (cont.)
Chapter 6: Behavior of Sound Systems Indoors ......................................................................... 61
Introduction ....................................................................................................................................... 61
Acoustical Feedback and Potential System Gain ............................................................................. 62
Sound Field Calculations for a Small Room ..................................................................................... 62
Calculations for a MediumSize Room ............................................................................................. 65
Calculations for a Distributed Loudspeaker System ......................................................................... 68
System Gain vs. Frequency Response ............................................................................................ 69
The Indoor Gain Equation ................................................................................................................ 69
Measuring Sound System Gain ........................................................................................................ 610
General Requirements for Speech Intelligibility ................................................................................ 611
The Role of Time Delay in Sound Reinforcement ............................................................................ 616
System Equalization and Power Response of Loudspeakers .......................................................... 617
System Design Overview ................................................................................................................. 619
Chapter 7: System Architecture and Layout ................................................................................ 71
Introduction ....................................................................................................................................... 71
Typical Signal Flow Diagram ............................................................................................................ 71
Amplifier and Loudspeaker Power Ratings ...................................................................................... 75
Wire Gauges and Line Losses ......................................................................................................... 75
Constant Voltage Distribution Systems (70volt lines) ...................................................................... 76
Low Frequency Augmentation—Subwoofers ................................................................................... 76
Case Study A: A Speech and Music System for a Large Evangelical Church .................................. 79
Case Study B: A Distributed Sound Reinforcement System for a Large Liturgical Church .............. 712
Case Study C: Specifications for a Distributed Sound System Comprising a Ballroom,
Small Meeting Space, and Social/Bar Area ............................................................................... 716
Bibliography
Sound System Design Reference Manual
Preface to the 1999 Edition:
This third edition of JBL Professional’s Sound System Design Reference Manual is presented in a new
graphic format that makes for easier reading and study. Like its predecessors, it presents in virtually their
original 1977 form George Augspurger’s intuitive and illuminating explanations of sound and sound system
behavior in enclosed spaces. The section on systems and case studies has been expanded, and references
to JBL components have been updated.
The fundamentals of acoustics and sound system design do not change, but system implementation
improves in its effectiveness with ongoing developments in signal processing, transducer refinement, and
frontend flexibility in signal routing and control.
As stated in the Preface to the 1986 edition: The technical competence of professional dealers and
sound contractors is much higher today than it was when the Sound Workshop manual was originally
introduced. It is JBL’s feeling that the serious contractor or professional dealer of today is ready to move away
from simply plugging numbers into equations. Instead, the designer is eager to learn what the equations really
mean, and is intent on learning how loudspeakers and rooms interact, however complex that may be. It is for
the student with such an outlook that this manual is intended.
John Eargle
January 1999
i
Sound System Design Reference Manual
Sound System Design Reference Manual
Wavelength, Frequency, and Speed of
Sound
Sound waves travel approximately 344 m/sec
(1130 ft/sec) in air. There is a relatively small velocity
dependence on temperature, and under normal
indoor conditions we can ignore it. Audible sound
covers the frequency range from about 20 Hz to 20
kHz. The wavelength of sound of a given frequency
is the distance between successive repetitions of the
waveform as the sound travels through air. It is given
by the following equation:
wavelength = speed/frequency
or, using the common abbreviations of c for speed,
f for frequency, and l for wavelength:
l = c/f
Period (T) is defined as the time required for
one cycle of the waveform. T = 1/f.
For f = 1 kHz, T = 1/1000, or 0.001 sec, and
l = 344/1000, or .344 m (1.13 ft.)
The lowest audible sounds have wavelengths
on the order of 10 m (30 ft), and the highest sounds
have wavelengths as short as 20 mm (0.8 in). The
range is quite large, and, as we will see, it has great
bearing on the behavior of sound.
The waves we have been discussing are of
course sine waves, those basic building blocks of all
speech and music signals. Figure 11 shows some of
the basic aspects of sine waves. Note that waves of
the same frequency can differ in both amplitude and
in phase angle. The amplitude and phase angle
relationships between sine waves determine how
they combine, either acoustically or electrically.
Chapter 1: Wave Propagation
Figure 11. Properties of sine waves
11
Sound System Design Reference Manual
Combining Sine Waves
Referring to Figure 12, if two or more sine
wave signals having the same frequency and
amplitude are added, we find that the resulting signal
also has the same frequency and that its amplitude
depends upon the phase relationship of the original
signals. If there is a phase difference of 120°, the
resultant has exactly the same amplitude as either
of the original signals. If they are combined in phase,
the resulting signal has twice the amplitude of either
original. For phase differences between l20° and
240°, the resultant signal always has an amplitude
less than that of either of the original signals. If the
two signals are exactly 180° out of phase, there will
be total cancellation.
In electrical circuits it is difficult to maintain
identical phase relationships between all of the sine
components of more complex signals, except for the
special cases where the signals are combined with
a 0° or 180° phase relationship. Circuits which
maintain some specific phase relationship (45°, for
example) over a wide range of frequencies are fairly
complex. Such wide range, allpass phaseshifting
networks are used in acoustical signal processing.
When dealing with complex signals such as
music or speech, one must understand the concept
of coherence. Suppose we feed an electrical signal
through a high quality amplifier. Apart from very small
amounts of distortion, the output signal is an exact
replica of the input signal, except for its amplitude.
The two signals, although not identical, are said to
be highly coherent. If the signal is passed through a
poor amplifier, we can expect substantial differences
between input and output, and coherence will not be
as great. If we compare totally different signals, any
similarities occur purely at random, and the two are
said to be noncoherent.
When two noncoherent signals are added, the
rms (root mean square) value of the resulting signal
can be calculated by adding the relative powers of
the two signals rather than their voltages. For
example, if we combine the outputs of two separate
noise generators, each producing an rms output of
1 volt, the resulting signal measures 1.414 volts rms,
as shown in Figure 13.
Figure 13. Combining two random noise generators
12
Figure 12. V ector addition of two sine waves
Sound System Design Reference Manual
Combining Delayed Sine Waves
If two coherent widerange signals are
combined with a specified time difference between
them rather than a fixed phase relationship, some
frequencies will add and others will cancel. Once the
delayed signal arrives and combines with the original
signal, the result is a form of “comb filter,” which
alters the frequency response of the signal, as
shown in Figure 14. Delay can be achieved
electrically through the use of allpass delay
networks or digital processing. In dealing with
acoustical signals in air, there is simply no way to
avoid delay effects, since the speed of sound is
relatively slow.
13
Figure 14A. Combining delayed signals
Figure 14B. Combining of coherent signals with constant time delay
Sound System Design Reference Manual
A typical example of combining delayed
coherent signals is shown in Figure 15. Consider
the familiar outdoor PA system in which a single
microphone is amplified by a pair of identical
separated loudspeakers. Suppose the loudspeakers
in question are located at each front corner of the
stage, separated by a distance of 6 m (20 ft). At any
distance from the stage along the center line, signals
from the two loudspeakers arrive simultaneously.
But at any other location, the distances of the two
loudspeakers are unequal, and sound from one must
arrive slightly later than sound from the other. The
illustration shows the dramatically different frequency
response resulting from a change in listener position
of only 2.4 m (8 ft). Using random noise as a test
signal, if you walk from Point B to Point A and
proceed across the center line, you will hear a
pronounced swishing effect, almost like a siren. The
change in sound quality is most pronounced near the
center line, because in this area the response peaks
and dips are spread farther apart in frequency.
14
Figure 15. Generation of interference effects (comb filter response) by a split array
Figure 16. Audible effect of comb filters shown in Figure 15
Sound System Design Reference Manual
15
Subjectively, the effect of such a comb filter is
not particularly noticeable on normal program
material as long as several peaks and dips occur
within each onethird octave band. See Figure 16.
Actually, the controlling factor is the “critical
bandwidth.” In general, amplitude variations that
occur within a critical band will not be noticed as
such. Rather, the ear will respond to the signal power
contained within that band. For practical work in
sound system design and architectural acoustics, we
can assume that the critical bandwidth of the human
ear is very nearly onethird octave wide.
In houses of worship, the system should be
suspended high overhead and centered. In spaces
which do not have considerable height, there is a
strong temptation to use two loudspeakers, one on
either side of the platform, feeding both the same
program. We do not recommend this.
Diffraction of Sound
Diffraction refers to the bending of sound waves
as they move around obstacles. When sound strikes
a hard, nonporous obstacle, it may be reflected or
diffracted, depending on the size of the obstacle
relative to the wavelength. If the obstacle is large
compared to the wavelength, it acts as an effective
barrier, reflecting most of the sound and casting a
substantial “shadow” behind the object. On the other
hand, if it is small compared with the wavelength,
sound simply bends around it as if it were not there.
This is shown in Figure 17.
An interesting example of sound diffraction
occurs when hard, perforated material is placed in
the path of sound waves. So far as sound is
concerned, such material does not consist of a solid
barrier interrupted by perforations, but rather as an
open area obstructed by a number of small individual
objects. At frequencies whose wavelengths are small
compared with the spacing between perforations,
most of the sound is reflected. At these frequencies,
the percentage of sound traveling through the
openings is essentially proportional to the ratio
between open and closed areas.
At lower frequencies (those whose wavelengths
are large compared with the spacing between
perforations), most of the sound passes through the
openings, even though they may account only for 20
or 30 percent of the total area.
Figure 17. Diffraction of sound around obstacles
Sound System Design Reference Manual
Effects of Temperature Gradients on
Sound Propagation
If sound is propagated over large distances
out of doors, its behavior may seem erratic.
Differences (gradients) in temperature above ground
level will affect propagation as shown in Figure 18.
Refraction of sound refers to its changing direction
as its velocity increases slightly with elevated
temperatures. At Figure 18A, we observe a situation
which often occurs at nightfall, when the ground is
still warm. The case shown at B may occur in the
morning, and its “skipping” characteristic may give
rise to hot spots and dead spots in the listening area.
Effects of Wind Velocity and Gradients
on Sound Propagation
Figure 19 shows the effect wind velocity
gradients on sound propagation. The actual velocity
of sound in this case is the velocity of sound in still
air plus the velocity of the wind itself. Figure 110
shows the effect of a cross breeze on the apparent
direction of a sound source.
The effects shown in these two figures may be
evident at large rock concerts, where the distances
covered may be in the 200  300 m (600  900 ft)
range.
16
Figure 18. Effects of temperature gradients on sound propagation
Figure 19. Effect of wind velocity gradients on sound propagation
Sound System Design Reference Manual
17
Effects of Humidity on Sound
Propagation
Contrary to what most people believe, there
is more sound attenuation in dry air than in damp air.
The effect is a complex one, and it is shown in
Figure 111. Note that the effect is significant only
at frequencies above 2 kHz. This means that high
frequencies will be attenuated more with distance
than low frequencies will be, and that the attenuation
will be greatest when the relative humidity is 20
percent or less.
Figure 11 1. Absorption of sound in air vs. relative humidity
Figure 110. Effect of cross breeze on apparent direction of sound
Sound System Design Reference Manual
Sound System Design Reference Manual
Chapter 2: The Decibel
Introduction
In all phases of audio technology the decibel is
used to express signal levels and level differences in
sound pressure, power, voltage, and current. The
reason the decibel is such a useful measure is that it
enables us to use a comparatively small range of
numbers to express large and often unwieldy
quantities. The decibel also makes sense from a
psychoacoustical point of view in that it relates
directly to the effect of most sensory stimuli.
Power Relationships
Fundamentally, the bel is defined as the
common logarithm of a power ratio:
bel = log (P1/P0)
For convenience, we use the decibel, which is simply
onetenth bel. Thus:
The following tabulation illustrates the
usefulness of the concept. Letting P0 = 1 watt:
P1 (watts) Level in dB
1 0
10 10
100 20
1000 30
10,000 40
20,000 43
Note that a 20,000to1 range in power can be
expressed in a much more manageable way by
referring to the powers as levels in dB above one
watt. Psychoacoustically, a tentimes increase in
power results in a level which most people judge to
be Òtwice as loud.ÓTh us, a 100watt acoustical signal
would be twice as loud as a 10watt signal, and a
10watt signal would be twice as loud as a 1watt
signal. The convenience of using decibels is
apparent; each of these power ratios can be
expressed by the same level, 10 dB. Any 10 dB level
difference, regardless of the actual powers involved,
will represent a 2to1 difference in subjective
loudness.
We will now expand our power decibel table:
P1 (watts) Level in dB
1.25 1
1.60 2
2.5 4
3.15 5
6.3 8
10 10
This table is worth memorizing. Knowing it, you
can almost immediately do mental calculations,
arriving at power levels in dB above, or below, one
watt.
Here are some examples:
1. What power level is represented by 80
watts? First, locate 8 watts in the left column and
note that the corresponding level is 9 dB. Then,
note that 80 is 10 times 8, giving another 10 dB.
Thus:
9 + 10 = 19 dB
2. What power level is represented by 1
milliwatt? 0.1 watt represents a level of minus 10 dB,
and 0.01 represents a level 10 dB lower. Finally,
0.001 represents an additional level decrease of 10
dB. Thus:
21
Level in decibels (dB) = 10 log (P1/P0)
10 10 10 = 30 dB
Sound System Design Reference Manual
3. What power level is represented by 4
milliwatts? As we have seen, the power level of 1
milliwatt is –30 dB. Two milliwatts represents a level
increase of 3 dB, and from 2 to 4 milliwatts there is
an additional 3 dB level increase. Thus:
–30 + 3 + 3 = –24 dB
4. What is the level difference between 40 and
100 watts? Note from the table that the level
corresponding to 4 watts is 6 dB, and the level
corresponding to 10 watts is 10 dB, a difference of 4
dB. Since the level of 40 watts is 10 dB greater than
for 4 watts, and the level of 80 watts is 10 dB greater
than for 8 watts, we have:
6 – 10 + 10 – 10 = –4 dB
We have done this last example the long way,
just to show the rigorous approach. However, we
could simply have stopped with our first observation,
noting that the dB level difference between 4 and 10
watts, .4 and 1 watt, or 400 and 1000 watts will
always be the same, 4 dB, because they all
represent the same power ratio.
The level difference in dB can be converted
back to a power ratio by means of the following
equation:
Power ratio = 10dB/10
For example, find the power ratio of a level
difference of 13 dB:
Power ratio = 1013/10 = 101.3 = 20
The reader should acquire a reasonable skill in
dealing with power ratios expressed as level
differences in dB. A good “feel” for decibels is a
qualification for any audio engineer or sound
contractor. An extended nomograph for converting
power ratios to level differences in dB is given in
Figure 21.
Voltage, Current, and Pressure
Relationships
The decibel fundamentally relates to power
ratios, and we can use voltage, current, and pressure
ratios as they relate to power. Electrical power can
be represented as:
P = EI
P = I2Z
P = E2/Z
Because power is proportional to the square of
the voltage, the effect of doubling the voltage is to
quadruple the power:
(2E)2/Z = 4(E)2/Z
As an example, let E = 1 volt and Z = 1 ohm.
Then, P = E2/Z = 1 watt. Now, let E = 2 volts; then,
P = (2)2/1 = 4 watts.
The same holds true for current, and the
following equations must be used to express power
levels in dB using voltage and current ratios:
dB level = 10 log
E
E
20 log
E
E
1 , and
0
1
0
=
2
dB level = 10 log
I
I
20 log
I
I
1 .
0
1
0
=
2
Sound pressure is analogous to voltage, and
levels are given by the equation:
dB level = 20 log
P
P
1 .
0
Figure 21. Nomograph for determining power ratios directly in dB
22
Sound System Design Reference Manual
The normal reference level for voltage, E0, is
one volt. For sound pressure, the reference is the
extremely low value of 20 x 106 newtons/m2. This
reference pressure corresponds roughly to the
minimum audible sound pressure for persons with
normal hearing. More commonly, we state pressure
in pascals (Pa), where 1 Pa = 1 newton/m2. As a
convenient point of reference, note that an rms
pressure of 1 pascal corresponds to a sound
pressure level of 94 dB.
We now present a table useful for determining
levels in dB for ratios given in voltage, current, or
sound pressure:
Voltage, Current or
Pressure Ratios Level in dB
1 0
1.25 2
1.60 4
2 6
2.5 8
3.15 10
4 12
5 14
6.3 16
8 18
10 20
This table may be used exactly the same way
as the previous one. Remember, however, that the
reference impedance, whether electrical or
acoustical, must remain fixed when using these
ratios to determine level differences in dB. A few
examples are given:
1. Find the level difference in dB between 2
volts and 10 volts. Directly from the table we observe
20 – 6 = 14 dB.
2. Find the level difference between 1 volt and
100 volts. A 10to1 ratio corresponds to a level
difference of 20 dB. Since 1to100 represents the
product of two such ratios (1to10 and 10to100),
the answer is
20 + 20 = 40 dB.
3. The signal input to an amplifier is 1 volt, and
the input impedance is 600 ohms. The output is also
1 volt, and the load impedance is 15 ohms. What is
the gain of the amplifier in dB? Watch this one
carefully!
If we simply compare input and output voltages,
we still get 0 dB as our answer. The voltage gain is in
fact unity, or one. Recalling that decibels refer
primarily to power ratios, we must take the differing
input and output impedances into account and
actually compute the input and output powers.
Input power =
E
Z
=
1
600
2 watt
Output power =
E
Z
=
1
15
2
T 10 log
600
15
= 10 log 40 = 16 dB hus,
Fortunately, such calculations as the above are
not often made. In audio transmission, we keep track
of operating levels primarily through voltage level
calculations in which the voltage reference value of
0.775 volts has an assigned level of 0 dBu. The
value of 0.775 volts is that which is applied to a 600
ohm load to produce a power of 1 milliwatt (mW). A
power level of 0 dBm corresponds to 1 mW. Stated
somewhat differently, level values in dBu and dBm
will have the same numerical value only when the
load impedance under consideration is 600 ohms.
The level difference in dB can be converted
back to a voltage, current, or pressure ratio by
means of the following equation:
Ratio = 10dB/20
For example, find the voltage ratio
corresponding to a level difference of 66 dB:
voltage ratio = 1066/20 = 103.3 = 2000.
23
Sound System Design Reference Manual
Sound Pressure and Loudness Contours
We will see the term dBSPL time and again in
professional sound work. It refers to sound pressure
levels in dB above the reference of 20 x 106 N/m2.
We commonly use a sound level meter (SLM) to
measure SPL. Loudness and sound pressure
obviously bear a relation to each other, but they are
not the same thing. Loudness is a subjective
sensation which differs from the measured level in
certain important aspects. To specify loudness in
scientific terms, a different unit is used, the phon.
Phons and decibels share the same numerical value
only at 1000 Hz. At other frequencies, the phon scale
deviates more or less from the sound level scale,
depending on the particular frequency and the
sound pressures; Figure 22 shows the relationship
between phons and decibels, and illustrates the
wellknown RobinsonDadson equal loudness
contours. These show that, in general, the ear
becomes less sensitive to sounds at low frequencies
as the level is reduced.
When measuring sound pressure levels,
weighted response may be employed to more closely
approximate the response of the ear. Working with
sound systems, the most useful scales on the sound
level meter will be the Aweighting scale and the
linear scale, shown in Figure 23. Inexpensive sound
level meters, which cannot provide linear response
over the full range of human hearing, often have no
linear scale but offer a Cweighting scale instead. As
can be seen from the illustration, the Cscale rolls off
somewhat at the frequency extremes. Precision
sound level meters normally offer A, B, and C scales
in addition to linear response. Measurements made
with a sound level meter are normally identified by
noting the weighting factor, such as: dB(A) or dB(lin).
Typical levels of familiar sounds, as shown in
Figure 24, help us to estimate dB(A) ratings when a
sound level meter is not available. For example,
normal conversational level in quiet surrounds is
about 60 dB(A). Most people find levels higher than
100 dB(A) uncomfortable, depending on the length of
exposure. Levels much above 120 dB(A) are
definitely dangerous to hearing and are perceived as
painful by all except dedicated rock music fans.
Figure 22. Freefield equal loudness contours
24
Sound System Design Reference Manual
Figure 23. Frequency responses for SLM weighting characteristics
Figure 24. T ypical Aweighted sound levels
25
Sound System Design Reference Manual
Inverse Square Relationships
When we move away from a point source of
sound out of doors, or in a free field, we observe that
SPL falls off almost exactly 6 dB for each doubling of
distance away from the source. The reason for this is
shown in Figure 25. At A there is a sphere of radius
one meter surrounding a point source of sound P1
representing the SPL at the surface of the sphere. At
B, we observe a sphere of twice the radius, 2 meters.
The area of the larger sphere is four times that of the
smaller one, and this means that the acoustical
power passing through a small area on the larger
sphere will be onefourth that passing through the
same small area on the smaller sphere. The 4to1
power ratio represents a level difference of 6 dB, and
the corresponding sound pressure ratio will be 2to1.
A convenient nomograph for determining
inverse square losses is given in Figure 26. Inverse
square calculations depend on a theoretical point
source in a free field. In the real world, we can
closely approach an ideal free field, but we still must
take into account the factors of finite source size and
nonuniform radiation patterns.
Consider a horntype loudspeaker having a
rated sensitivity of 100 dB, 1 watt at 1 meter. One
meter from where? Do we measure from the mouth
of the horn, the throat of the horn, the driver
diaphragm, or some indeterminate point in between?
Even if the measurement position is specified, the
information may be useless. Sound from a finite
source does not behave according to inverse square
law at distances close to that source. Measurements
made in the “near field” cannot be used to estimate
performance at greater distances. This being so, one
may well wonder why loudspeakers are rated at a
distance of only 1 meter.
The method of rating and the accepted
methods of measuring the devices are two different
things. The manufacturer is expected to make a
number of measurements at various distances under
free field conditions. From these he can establish
Figure 26. Nomograph for determining inverse square losses
26
Figure 25. Inverse square relationships
Sound System Design Reference Manual
that the measuring microphone is far enough away
from the device to be in its far field, and he can also
calculate the imaginary point from which sound
waves diverge, according to inverse square law. This
point is called the acoustic center of the device. After
accurate field measurements have been made, the
results are converted to an equivalent one meter
rating. The rated sensitivity at one meter is that SPL
which would be measured if the inverse square
relationship were actually maintained that close to
the device.
Let us work a few exercises using the
nomograph of Figure 26:
1. A JBL model 2360 horn with a 2446 HF driver
produces an output of 113 dB, 1 watt at 1 meter.
What SPL will be produced by 1 watt at 30 meters?
We can solve this by inspection of the nomograph.
Simply read the difference in dB between 1 meter
and 30 meters: 29.5 dB. Now, subtracting this from
113 dB:
113 – 29.5 = 83.5 dB
2. The nominal power rating of the JBL model
2446 driver is 100 watts. What maximum SPL will be
produced at a distance of 120 meters in a free field
when this driver is mounted on a JBL model 2366
horn?
There are three simple steps in solving this
problem. First, determine the inverse square loss
from Figure 26; it is approximately 42 dB. Next,
determine the level difference between one watt and
100 watts. From Figure 21 we observe this to be 20
dB. Finally, note that the horndriver sensitivity is 118
dB, 1 watt at 1 meter. Adding these values:
118 – 42 + 20 = 96 dBSPL
Calculations such as these are very
commonplace in sound reinforcement work, and
qualified sound contractors should be able to make
them easily.
Adding Power Levels in dB
Quite often, a sound contractor will have to
add power levels expressed in dB. Let us assume
that two sound fields, each 94 dBSPL, are
combined. What is the resulting level? If we simply
add the levels numerically, we get 188 dBSPL,
clearly an absurd answer! What we must do in effect
is convert the levels back to their actual powers, add
them, and then recalculate the level in dB. Where
two levels are involved, we can accomplish this
easily with the data of Figure 27. Let D be the
difference in dB between the two levels, and
determine the value N corresponding to this
difference. Now, add N to the higher of the two
original values.
As an exercise, let us add two sound fields, 90
dBSPL and 84 dBSPL. Using Figure 27, a D of 6
dB corresponds to an N of about 1 dB. Therefore, the
new level will be 91 dBSPL.
Note that when two levels differ by more than
about 10 dB, the resulting summation will be
substantially the same as the higher of the two
values. The effect of the lower level will be negligible.
Reference Levels
Although we have discussed some of the
common reference levels already, we will list here all
of those that a sound contractor is likely to
encounter.
In acoustical measurements, SPL is always
measured relative to 20 x 106 Pa. An equivalent
expression of this is .0002 dynes/cm2.
In broadcast transmission work, power is often
expressed relative to 1 milliwatt (.001 watt), and such
levels are expressed in dBm.
The designation dBW refers to levels relative to
one watt. Thus, 0 dBW = 30 dBm.
In signal transmission diagrams, the
designation dBu indicates voltage levels referred to
.775 volts.
27
Figure 27. Nomograph for adding levels expressed in dB.
Summing sound level output of two sound sources where D is their output difference in dB.
N is added to the higher to derive the total level.
Sound System Design Reference Manual
In other voltage measurements, dBV refers to
levels relative to 1 volt.
Rarely encountered by the sound contractor will
be acoustical power levels. These are designated
dBPWL, and the reference power is 1012 watts. This
is a very small power indeed. It is used in acoustical
measurements because such small amounts of
power are normally encountered in acoustics.
Peak, Average, and rms Signal Values
Most measurements of voltage, current, or
sound pressure in acoustical engineering work are
given as rms (root mean square) values of the
waveforms. The rms value of a repetitive waveform
equals its equivalent DC value in power
transmission. Referring to Figure 28A for a sine
wave with a peak value of one volt, the rms value is
.707 volt, a 3 dB difference. The average value of the
waveform is .637 volt.
For more complex waveforms, such as are
found in speech and music, the peak values will be
considerably higher than the average or rms values.
The waveform shown at Figure 28B is that of a
trumpet at about 400 Hz, and the spread between
peak and average values is 13 dB.
In this chapter, we have in effect been using
rms values of voltage, current, and pressure for all
calculations. However, in all audio engineering
applications, the timevarying nature of music and
speech demands that we consider as well the
instantaneous values of waveforms likely to be
encountered. The term headroom refers to the extra
margin in dB designed into a signal transmission
system over its normal operating level. The
importance of headroom will become more evident
as our course develops.
28
Figure 28. Peak, average, and rms values.
Sinewave (A); complex waveform (B).
Sound System Design Reference Manual
The data of Figure 31 was generalized by
Molloy (7) and is shown in Figure 33. Here, note that
Dl and Q are related to the solid angular coverage of
a hypothetical sound radiator whose horizontal and
vertical coverage angles are specified. Such ideal
sound radiators do not exist, but it is surprising how
closely these equations agree with measured Dl and
Q of HF horns that exhibit fairly steep cutoff outside
their normal coverage angles.
As an example of this, a JBL model 2360
BiRadial horn has a nominal 900by400 pattern
measured between the 6 dB down points in each
plane. If we insert the values of 90° and 40° into
Molloy’s equation, we get DI = 11 and Q = 12.8. The
published values were calculated by integrating
response over 360° in both horizontal and vertical
planes, and they are Dl = 10.8 and Q = 12.3. So the
estimates are in excellent agreement with the
measurements.
For the JBL model 2366 horn, with its nominal
6 dB down coverage angles of 40° and 20°, Molloy’s
equation gives Dl = 17.2 and Q = 53. The published
values are Dl = 16.5 and Q = 46. Again, the
agreement is excellent.
Is there always such good correlation between
the 6 dB down horizontal and vertical beamwidth of a
horn and its calculated directivity? The answer is no.
Only when the response cutoff is sharp beyond the
6 dB beamwidth limits and when there is minimal
radiation outside rated beamwidth will the correlation
be good. For many types of radiators, especially those
operating at wavelengths large compared with their
physical dimensions, Molloy’s equation will not hold.
A Comparison of Polar Plots, Beamwidth
Plots, Directivity Plots, and Isobars
There is no one method of presenting
directional data on radiators which is complete in all
regards. Polar plots (Figure 34A) are normally
presented in only the horizontal and vertical planes.
A single polar plot covers only a single frequency, or
frequency band, and a complete set of polar plots
takes up considerable space. Polars are, however,
the only method of presentation giving a clear picture
of a radiator’s response outside its normal operating
beamwidth. Beamwidth plots of the 6 dB down
coverage angles (Figure 34B) are very common
because considerable information is contained in a
single plot. By itself, a plot of Dl or Q conveys
information only about the onaxis performance of a
radiator (Figure 34C). Taken together, horizontal and
vertical beamwidth plots and Dl or Q plots convey
sufficient information for most sound reinforcement
design requirements.
33
Figure 34. Methods of presenting directional information
Sound System Design Reference Manual
Isobars have become popular in recent years.
They give the angular contours in spherical
coordinates about the principal axis along which the
response is 3, 6, and 9 dB, relative to the onaxis
maximum. It is relatively easy to interpolate visually
between adjacent isobars to arrive at a reasonable
estimate of relative response over the useful frontal
solid radiation angle of the horn. Isobars are useful in
advanced computer layout techniques for
determining sound coverage over entire seating
areas. The normal method of isobar presentation is
shown in Figure 34D.
Still another way to show the directional
characteristics of radiators is by means of a family of
offaxis frequency response curves, as shown in
Figure 35. At A, note that the offaxis response
curves of the JBL model 2360 BiRadial horn run
almost parallel to the onaxis response curve. What
this means is that a listener seated off the main axis
will perceive smooth response when a BiRadial
constant coverage horn is used. Contrast this with
the offaxis response curves of the older (and
obsolete) JBL model 2350 radial horn shown at B. If
this device is equalized for flat onaxis response,
then listeners offaxis will perceive rolledoff HF
response.
Directivity of Circular Radiators
Any radiator has little directional control for
frequencies whose wavelengths are large compared
with the radiating area. Even when the radiating area
is large compared to the wavelength, constant
pattern control will not result unless the device has
been specifically designed to maintain a constant
pattern. Nothing demonstrates this better than a
simple radiating piston. Figure 36 shows the
sharpening of onaxis response of a piston mounted
in a flat baffle. The wavelength varies over a 24to1
range. If the piston were, say a 300 mm (12”)
loudspeaker, then the wavelength illustrated in the
figure would correspond to frequencies spanning the
range from about 350 Hz to 8 kHz.
Among other things, this illustration points out
why “full range,” singlecone loudspeakers are of
little use in sound reinforcement engineering. While
the onaxis response can be maintained through
equalization, offaxis response falls off drastically
above the frequency whose wavelength is about
equal to the diameter of the piston. Note that when
the diameter equals the wavelength, the radiation
pattern is approximately a 90° cone with  6 dB
response at ±45°.
34
Figure 35. Families of offaxis frequency response curves
Sound System Design Reference Manual
The values of DI and Q given in Figure 36 are
the onaxis values, that is, along the axis of
maximum loudspeaker sensitivity. This is almost
always the case for published values of Dl and Q.
However, values of Dl and Q exist along any axis of
the radiator, and they can be determined by
inspection of the polar plot. For example, in Figure
36, examine the polar plot corresponding to
Diameter = l. Here, the onaxis Dl is 10 dB. If we
simply move offaxis to a point where the response
has dropped 10 dB, then the Dl along that direction
will be 10  10, or 0 dB, and the Q will be unity. The
offaxis angle where the response is 10 dB down is
marked on the plot and is at about 55°. Normally, we
will not be concerned with values of Dl and Q along
axes other than the principal one; however, there are
certain calculations involving interaction of
microphones and loudspeakers where a knowledge
of offaxis directivity is essential.
Omnidirectional microphones with circular
diaphragms respond to on and offaxis signals in a
manner similar to the data shown in Figure 36. Let
us assume that a given microphone has a diaphragm
about 25 mm (1”) in diameter. The frequency
corresponding to l/4 is about 3500 Hz, and the
response will be quite smooth both on and off axis.
However, by the time we reach 13 or 14 kHz, the
diameter of the diaphragm is about equal to l, and
the Dl of the microphone is about 10 dB. That is, it
will be 10 dB more sensitive to sounds arriving on
axis than to sounds which are randomly incident to
the microphone.
Of course, a piston is a very simple radiator —
or receiver. Horns such as JBL’s BiRadial series are
complex by comparison, and they have been
designed to maintain constant HF coverage through
attention to waveguide principles in their design.
One thing is certain: no radiator can exhibit much
pattern control at frequencies whose wavelengths
are much larger than the circumference of the
radiating surface.
35
Figure 36. Directional characteristics of a circularpiston source
mounted in an infinite baffle as a function of diameter and l.
Sound System Design Reference Manual
The Importance of Flat Power Response
If a radiator exhibits flat power response, then
the power it radiates, integrated over all directions,
will be constant with frequency. Typical compression
drivers inherently have a rolledoff response when
measured on a plane wave tube (PWT), as shown in
Figure 37A. When such a driver is mounted on a
typical radial horn such as the JBL model 2350, the
onaxis response of the combination will be the sum
of the PWT response and the Dl of the horn. Observe
at B that the combination is fairly flat on axis and
does not need additional equalization. Offaxis
response falls off, both vertically and horizontally,
and the total power response of the combination will
be the same as observed on the PWT; that is, it rolls
off above about 3 kHz.
Now, let us mount the same driver on a Bi
Radial uniform coverage horn, as shown at C. Note
that both onand offaxis response curves are rolled
off but run parallel with each other. Since the Dl of
the horn is essentially flat, the onaxis response will
be virtually the same as the PWT response.
At D, we have inserted a HF boost to
compensate for the driver’s rolled off power
response, and the result is now flat response both on
and off axis. Listeners anywhere in the area covered
by the horn will appreciate the smooth and extended
response of the system.
Flat power response makes sense only with
components exhibiting constant angular coverage.
If we had equalized the 2350 horn for flat power
response, then the onaxis response would have
been too bright and edgy sounding.
36
Figure 37. Power response of HF systems
Sound System Design Reference Manual
The rising DI of most typical radial horns is
accomplished through a narrowing of the vertical
pattern with rising frequency, while the horizontal
pattern remains fairly constant, as shown in Figure
38A. Such a horn can give excellent horizontal
coverage, and since it is “self equalizing” through its
rising DI, there may be no need at all for external
equalization. The smoothrunning horizontal and
vertical coverage angles of a BiRadial, as shown at
Figure 38B, will always require power response HF
boosting.
37
Measurement of Directional
Characteristics
Polar plots and isobar plots require that the
radiator under test be rotated about several of its
axes and the response recorded. Beamwidth plots
may be taken directly from this data.
DI and Q can be calculated from polar data by
integration using the following equation:
DI = 10 log
2
P sin d θ
π ( ) θ θ
∫ 2
o
PQ is taken as unity, and q is taken in 10° increments.
The integral is solved for a value of DI in the
horizontal plane and a value in the vertical plane.
The resulting DI and Q for the radiator are given as:
DI =
DI
2
+
DI
2
h v
and
Q = Q Q n v ⋅
(Note: There are slight variations of this
method, and of course all commonly use methods
are only approximations in that they make use of
limited polar data.)
Figure 38. Increasing DI through narrowing
verticalbeamwidth
Sound System Design Reference Manual
Using Directivity Information
A knowledge of the coverage angles of an HF
horn is essential if the device is to be oriented
properly with respect to an audience area. If polar
plots or isobars are available, then the sound
contractor can make calculations such as those
indicated in Figure 39. The horn used in this
example is the JBL 2360 BiRadial. We note from the
isobars for this horn that the 3 dB angle off the
vertical is 14°. The 6 dB and 9 dB angles are 23°
and 30° respectively. This data is for the octave band
centered at 2 kHz. The horn is aimed so that its
major axis is pointed at the farthest seats. This will
ensure maximum reach, or “throw,” to those seats.
We now look at the 3 dB angle of the horn and
compare the reduction in the horn’s output along that
angle with the inverse square advantage at the
closerin seats covered along that axis. Ideally, we
would like for the inverse square advantage to
exactly match the horn’s offaxis falloff, but this is
not always possible. We similarly look at the
response along the 6 and 9 dB axes of the horn,
comparing them with the inverse square advantages
afforded by the closerin seats. When the designer
has flexibility in choosing the horn’s location, a good
compromise, such as that shown in this figure, will be
possible. Beyond the 9 dB angle, the horn’s output
falls off so rapidly that additional devices, driven at
much lower levels, would be needed to cover the
front seats (often called “front fill” loudspeakers).
Aiming a horn as shown here may result in a
good bit of power being radiated toward the back
wall. Ideally, that surface should be fairly absorptive
so that reflections from it do not become a problem.
Directional Characteristics of Combined
Radiators
While manufacturers routinely provide data on
their individual items of hardware, most provide little,
if any, data on how they interact with each other. The
data presented here for combinations of HF horns is
of course highly wavelength, and thus size,
dependent. Appropriate scaling must be done if this
data is to be applied to larger or smaller horns.
In general, at high frequencies, horns will act
independently of each other. If a pair of horns are
properly splayed so that their 6 dB angles just
overlap, then the response along that common axis
should be smooth, and the effect will be nearly that of
a single horn with increased coverage in the plane of
overlap. Thus, two horns with 60° coverage in the
horizontal plane can be splayed to give 120°
horizontal coverage. Likewise, dissimilar horns can
be splayed, with a resulting angle being the sum of
the two coverage angles in the plane of the splay.
Splaying may be done in the vertical plane with
similar results. Figure 310 presents an example of
Figure 39. Offaxis and inverse square calculations horn splaying in the horizontal plane.
Figure 310. Horn splaying for wider coverage
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Sound System Design Reference Manual
Horns may be stacked in a vertical array to
improve pattern control at low frequencies. The JBL
FlatFront BiRadials, because of their relatively
small vertical mouth dimension, exhibit a broadening
in their vertical pattern control below about 2 kHz.
When used in vertical stacks of three or four units,
the effective vertical mouth dimension is much larger
Figure 31 1. Stacking horns for higher directivity at low frequencies
(solid line, horizontal 6 dB deamwidth, dashed line, vertical 6 dB beamwidth)
than that of a single horn. The result, as shown in
Figure 311, is tighter pattern control down to about
500 Hz. In such vertical inline arrays, the resulting
horizontal pattern is the same as for a single horn.
Additional details on horn stacking are given in
Technical Note Volume 1, Number 7.
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Sound System Design Reference Manual
Sound System Design Reference Manual
Chapter 4: An Outdoor Sound
Reinforcement System
41
Introduction
Our study of sound reinforcement systems
begins with an analysis of a simple outdoor system.
The outdoor environment is relatively free of
reflecting surfaces, and we will make the simplifying
assumption that free field conditions exist. A basic
reinforcement system is shown in Figure 41A. The
essential acoustical elements are the talker,
microphone, loudspeaker, and listener. The electrical
diagram of the system is shown at B. The dotted line
indicates the acoustical feedback path which can
exist around the entire system.
When the system is turned on, the gain of the
amplifier can be advanced up to some point at which
the system will “ring,” or go into feedback. At the
onset of feedback, the gain around the electroacoustical
path is unity and at a zero phase angle.
This condition is shown at C, where the input at the
microphone of a single pulse will give rise to a
repetitive signal at the microphone, fed back from the
loudspeaker and which will quickly give rise to
sustained oscillation at a single frequency with a
period related to Dt.
Even at levels somewhat below feedback, the
response of the system will be irregular, due to the
fact that the system is “trying” to go into feedback,
but does not have enough loop gain to sustain it.
This is shown in Figure 42. As a rule, a workable
reinforcement system should have a gain margin of
6 to 10 dB before feedback if it is to sound natural on
all types of program input.
Figure 41. A simple outdoor reinforcement system
Sound System Design Reference Manual
The Concept of Acoustical Gain
Boner (4) quantified the concept of acoustical
gain, and we will now present its simple but elegant
derivation. Acoustical gain is defined as the increase
in level that a given listener in the audience
perceives with the system turned on, as compared to
the level the listener hears directly from the talker
when the system is off.
Referring to Figure 43, let us assume that both
the loudspeaker and microphone are omnidirectional;
that is, DI = 0 dB and Q = 1. Then by inverse square
loss, the level at the listener will be:
70 dB  20 log (7/1) = 70  17 = 53 dB
Now, we turn the system on and advance the
gain until we are just at the onset of feedback. This
will occur when the loudspeaker, along the D1 path,
produces a level at the microphone equal to that of
the talker, 70 dB.
If the loudspeaker produces a level of 70 dB at
the microphone, it will produce a level at the listener
of:
70  20 log (6/4) = 70  3.5 = 66.5 dB
With no safety margin, the maximum gain this
system can produce is:
66.5  53 = 13.5 dB
Rewriting our equations:
Maximum gain =
70  20 log (D2/D1)  70  20 log (D0/Ds)
This simplifies to:
Maximum gain =
20 log D0  20 log Ds + 20 log D1  20 log D2
Figure 42. Electrical response of a sound system 3 dB below sustained acoustical feedback
Figure 43. System gain calculations, loudspeaker and microphone both omnidirectional
42
Sound System Design Reference Manual
Adding a 6 dB safety factor gives us the usual
form of the equation:
Maximum gain =
20 log D0  20 log Ds + 20 log D1  20 log D2  6
In this form, the gain equation tells us several
things, some of them intuitively obvious:
1. That gain is independent of the level of the
talker
2. That decreasing Ds will increase gain
3. That increasing D1 will increase gain.
The Influence of Directional Microphones
and Loudspeakers on System Maximum
Gain
Let us rework the example of Figure 43, this
time making use of a directional loudspeaker whose
midband polar characteristics are as shown in Figure
44A. It is obvious from looking at Figure 44A that
sound arriving at the microphone along the D1
direction will be reduced 6 dB relative to the
omnidirectional loudspeaker. This 6 dB results
directly in added gain potential for the system.
The same holds for directional microphones, as
shown in Figure 45A. In Figure 45B, we show a
system using an omnidirectional loudspeaker and a
cardioid microphone with its 6 dB axis facing toward
the loudspeaker. This system is equivalent to the one
shown in Figure 44B; both exhibit a 6 dB increase in
maximum gain over the earlier case where both
microphone and loudspeaker were omnidirectional.
Finally, we can use both directional
loudspeakers and microphones to pick up additional
gain. We simply calculate the maximum gain using
omnidirectional elements, and then add to that value
the offaxis pattern advantage in dB for both
loudspeaker and microphone. As a practical matter,
however, it is not wise to rely too heavily on
directional microphones and loudspeakers to make a
significant increase in system gain. Most designers
are content to realize no more than 4to6 dB overall
added gain from the use of directional elements. The
reason for this is that microphone and loudspeaker
directional patterns are not constant with frequency.
Most directional loudspeakers will, at low
frequencies, appear to be nearly omnidirectional. If
more gain is called for, the most straightforward way
to get it is to reduce Ds or increase D1.
Figure 44. System gain calculations,
directionalloudspeaker
Figure 45. System gain calculations,
directionalmicrophone
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Sound System Design Reference Manual
How Much Gain is Needed?
The parameters of a given sound reinforcement
system may be such that we have more gain than we
need. When this is the case, we simply turn things
down to a comfortable point, and everyone is happy.
But things often do not work out so well. What is
needed is some way of determining beforehand how
much gain we will need so that we can avoid
specifying a system which will not work. One way of
doing this is by specifying the equivalent, or effective,
acoustical distance (EAD), as shown in Figure 46.
Sound reinforcement systems may be thought of as
effectively moving the talker closer to the listener. In
a quiet environment, we may not want to bring the
talker any closer than, say, 3 meters from the
listener. What this means, roughly, is that the
loudness produced by the reinforcement system
should approximate, for a listener at D0, the loudness
level of an actual talker at a distance of 3 meters.
The gain necessary to do this is calculated from the
inverse square relation between D0and EAD:
Necessary gain = 20 log D0  20 log EAD
In our earlier example, D0 = 7 meters. Setting
EAD = 3 meters, then:
Necessary gain = 20 log (7)  20 log (3)
= 17  9.5 = 7.5 dB
Assuming that both loudspeaker and
microphone are omnidirectional, the maximum gain
we can expect is:
Maximum gain =
20 log (7)  20 log (1) + 20 log (4)  20 log (6)  6
Maximum gain = 17  0 + 12  15.5  6
Maximum gain = 7.5 dB
As we can see, the necessary gain and the
maximum gain are both 7.5 dB, so the system will be
workable. If, for example, we were specifying a
system for a noisier environment requiring a shorter
EAD, then the system would not have sufficient gain.
For example, a new EAD of 1.5 meters would require
6 dB more acoustical gain. As we have discussed,
using a directional microphone and a directional
loudspeaker would just about give us the needed 6
dB. A simpler, and better, solution would be to reduce
Ds to 0.5 meter in order to get the added 6 dB of gain.
In general, in an outdoor system, satisfactory
articulation will result when speech peaks are about
25 dB higher than the Aweighted ambient noise
level. Typical conversation takes place at levels of 60
to 65 dB at a distance of one meter. Thus, in an
ambient noise field of 50 dB, we would require
speech peaks of 75 to 80 dB for comfortable
listening, and this would require an EAD as close as
0.25 meter, calculated as follows:
Speech level at 1 meter = 65 dB
Speech level at 0.5 meter = 71 dB
Speech level at 0.25 meter = 77 dB
Let us see what we must do to our outdoor
system to make it work under these demanding
conditions. First, we calculate the necessary
acoustical gain:
Necessary gain = 20 log D0  20 log EAD
Necessary gain = 20 log (7)  20 log (.25)
Necessary gain = 17+ 12 = 29 dB
44
Figure 46. Concept of Effective Acoustical Dustance (EAD)
Sound System Design Reference Manual
45
As we saw in an earlier example, our system
only has 7.5 dB of maximum gain available with a
6 dB safety factor. By going to both a directional
microphone and a directional loudspeaker, we can
increase this by about 6 dB, yielding a maximum
gain of 13.5 dB — still some 16 dB short of what we
actually need.
The solution is obvious; a handheld
microphone will be necessary in order to achieve the
required gain. For 16 dB of added gain, Ds will have
to be reduced to the value calculated below:
16 = 20 log (1/x)
16/20 = log (1/x)
10.8 = 1/x
Therefore: x = 1/10.8 = 0.16 meter (6”)
Of course, the problem with a handheld
microphone is that it is difficult for the user to
maintain a fixed distance between the microphone
and his mouth. As a result, the gain of the system will
vary considerably with only small changes in the
performermicrophone operating distance. It is
always better to use some kind of personal
microphone, one worn by the user. In this case, a
swivel type microphone attached to a headpiece
would be best, since it provides the minimum value
of DS. This type of microphone is now becoming very
popular onstage, largely because a number of major
pop and country artists have adopted it. In other
cases a simple tietack microphone may be sufficient.
Conclusion
In this chapter, we have presented the
rudiments of gain calculation for sound systems, and
the methods of analysis form the basis for the study
of indoor systems, which we will cover in a later
chapter.
Sound System Design Reference Manual
Sound System Design Reference Manual
Chapter 5: Fundamentals of Room Acoustics
51
Introduction
Most sound reinforcement systems are located
indoors, and the acoustical properties of the
enclosed space have a profound effect on the
system’s requirements and its performance. Our
study begins with a discussion of sound absorption
and reflection, the growth and decay of sound fields
in a room, reverberation, direct and reverberant
sound fields, critical distance, and room constant.
If analyzed in detail, any enclosed space is
quite complex acoustically. We will make many
simplifications as we construct “statistical” models of
rooms, our aim being to keep our calculations to a
minimum, while maintaining accuracy on the order of
10%, or ±1 dB.
Absorption and Reflection of Sound
Sound tends to “bend around” nonporous,
small obstacles. However, large surfaces such as the
boundaries of rooms are typically partially flexible
and partially porous. As a result, when sound strikes
such a surface, some of its energy is reflected, some
is absorbed, and some is transmitted through the
boundary and again propagated as sound waves on
the other side. See Figure 51.
All three effects may vary with frequency and
with the angle of incidence. In typical situations, they
do not vary with sound intensity. Over the range of
sound pressures commonly encountered in audio
work, most construction materials have the same
characteristics of reflection, absorption and
transmission whether struck by very weak or very
strong sound waves.
Figure 51. Sound impinging on a large boundary surface
Sound System Design Reference Manual
When dealing with the behavior of sound in an
enclosed space, we must be able to estimate how
much sound energy will be lost each time a sound
wave strikes one of the boundary surfaces or one of
the objects inside the room. Tables of absorption
coefficients for common building materials as well as
special “acoustical” materials can be found in any
architectural acoustics textbook or in data sheets
supplied by manufacturers of construction materiaIs.
Unless otherwise specified, published sound
absorption coefficients represent average absorption
over all possible angles of incidence. This is
desirable from a practical standpoint since the
random incidence coefficient fits the situation that
exists in a typical enclosed space where sound
waves rebound many times from each boundary
surface in virtually all possible directions.
Absorption ratings normally are given for a
number of different frequency bands. Typically, each
band of frequencies is one octave wide, and
standard center frequencies of 125 Hz, 250 Hz, 500
Hz, 1 kHz, etc., are used. In sound system design, it
usually is sufficient to know absorption characteristics
of materials in three or four frequency ranges. In this
handbook, we make use of absorption ratings in the
bands centered at 125 Hz, 1 kHz and 4 kHz.
The effects of mo

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