CHAPTER 4

FREQUENCY DOMAIN DESCRIPTIONS OF PERIODIC VOLTAGES

4.1 INTRODUCTION

Effective control of unintentional electromagnetic radiations can be greatly facilitated when the radiations are usefully characterized. As previously noted, regulatory limits on unintentional radiations are limits on the amplitudes of their sinusoidal components, and, to verify compliance with the limits, the amplitudes of those sinusoidal components are measured. As a result, useful descriptions associate the amplitudes and frequencies of the sinusoidal components of unintentional radiations with voltage and current parameters that are either well known or easily evaluated.

In addition, electromagnetic radiations are the direct result of time-varying electric currents, and currents are sometimes difficult to observe and evaluate. However, currents are established by voltages, which are usually relatively easy to observe and evaluate with an oscilloscope. Therefore, a good start toward obtaining useful descriptions of unnecessary electromagnetic radiations is to describe time-varying voltages in terms of their sinusoidal components. Then, the currents caused by those voltages and the radiations caused by the currents can be similarly described. Given these frequency-domain descriptions, or sinusoidal component lists, the causes of the radiations can be better understood, and the radiations can be better controlled.

The objective in this chapter, then, is to provide relatively simple methods for describing commonly used periodic voltages as sums of sinusoidal voltages. These frequency-domain descriptions of the voltage waveshapes are then used to describe the currents they cause and the radiations the currents cause.

4.2 MATHEMATICAL FOUNDATIONS

The parameters that enter into the frequency-domain descriptions of periodic voltages given here will be taken from their time-domain descriptions, as observed on an oscilloscope. The parameters used in those descriptions are illustrated in Fig. 4-1. They are Vp, the peak-to-peak amplitude of the voltage; tr, the complete (100%) rise time of the voltage; tf, its complete fall time; td, the time from the start of rise to the start of fall; and T, the time of one full period. Each of these parameters will generally either be known or easily obtainable by viewing a voltage with an oscilloscope.

The mathematical bases for the development that follows are the well-known concepts of the Fourier series, which can be summarized as follows. If a periodic voltage v(t) has a frequency f and a period T=1/f, then it has the Fourier series representation

image

In this description of v(t), the frequencies of the sinusoidal components are fn=nf=n/T, for all integers n ≥ 0, where, as noted above, f=1/T is the frequency of v(t). And, the amplitudes of the sinusoidal components are

image

image

Figure 4-1. The time-domain parameters of a voltage waveshape that are used to obtain a frequency-domain description of the voltage.

and

image

Because fn=nf, for all n ≥ 1, the frequency f is called the fundamental frequency of the Fourier series for v(t). And an and bn are called the Fourier coefficients of v(t).

Now, when n=0, then fn=0, so that cos(2πfnt)=cos(0)=1, and

image

The integral in Eq. 4-4 is, by definition, the average value of v(t). Therefore, the notation image will be used here to denote that average, rather than ao/2. Also, if n=0, sin(2πfnt)=sin(0)=0, and, from Eq. 4-3, it follows that b0=0. Accordingly, bo does not appear in Eq. 4-1.

When n ≥ 1, it will be useful to express the summands in Eq. 4-1 somewhat differently than they are expressed there. To do that, the angle φn is defined for any an and bn to be

image

As illustrated in Fig. 4-2, the second term in this definition is a necessary addition to the arctangent, to give φn a full range of 2π radians. That is, when an is negative, π/2 ≤ φn ≤ 3π/2, and it is necessary to add π to arctan(bn/an), because −π/2 ≤ arctan(bn/an) ≤ π/2.

Given this definition of φn then, it follows that

image

As a result, the summands of Eq. 4-1 can be expressed as

image

image

Figure 4-2. When an is positive, φn=arctan(bn/an), but when an is negative, φn=arctan(bn/an) + π

The latter expression in these equations follows from trigonometric identity A-3 given in Appendix A, which says that cos(x) cos(y) + sin(x) sin(y)=cos(xy), for any x and y.

Thus, any periodic voltage v(t) can be viewed as an infinite sum of sinusoidal voltages that has the general form

image

In this equation,

image

where an and bn are the Fourier coefficients of v(t).

Equation 4-8 is a general description of any periodic voltage, in terms of the coefficients an and bn. Therefore, attention is now turned to the evaluation of |Vn| and φn for periodic voltage waveforms that are commonly used in practice. The practical advantages to be gained by doing this will soon be obvious.

4.3 BASIC VOLTAGE WAVEFORMS

A large number of commonly occurring periodic voltage waveforms can be obtained with the four basic waveforms illustrated in Fig. 4-3. A particular voltage waveform is obtained by adding the graph of the basic waveform that describes its rise to the graph of the basic waveform that describes its fall, and by specifying values for Vp, tr, td, tf, and T. The Fourier coefficients of the waveform are then found simply by adding the known Fourier coefficients of the basic waveforms that were added. Several examples will be considered after the Fourier coefficients of the basic waveforms are obtained.

The basic waveforms of Fig. 4-3 can each be described as a specific function of time, for one full period of time from t = 0, to t = T, as follows:

image

image

Figure 4-3. Four basic voltage waveforms which can be added in pairs to obtain close approximations of numerous periodic voltage waveforms that are used in contemporary electronic devices.

image

image

image

In all of the above expressions, and in subsequent expressions in which it appears, e = 2.71828…, the base of natural logarithms.

With these time-domain expressions for the basic voltage waveforms from t = 0 to t = T, their Fourier coefficients can now be obtained by integration, as indicated in Eqs. 4-2 and 4-3. Details of those integrations are summarized in Appendix B. The results are given below with additional subscripts of 1, 2, 3, and 4 added to an and bn to identify which basic voltage waveform they represent. To simplify these expressions and other expressions to come the following parameters are defined:

image

Note also that ωn=2πf n=2πnf=2πn/T. These expressions will all be used interchangeably.

The Fourier coefficients for the basic voltage waveform v1(t), which describes linear rises, are

image

and

image

The Fourier coefficients for the basic voltage waveform v2(t), which describes exponential rises, are

image

and

image

where

image

and

image

The Fourier coefficients for the basic voltage waveform v3(t), which describes linear falls, are

image

and

image

And, the Fourier coefficients for the basic voltage waveform v4(t), which describes exponential falls, are

image

and

image

where

image

and

image

Note, for future reference, that the angle αn is associated with exponential rises, and the angle βn is associated with exponential falls.

Before applying the above results, it is worthwhile to observe some similarities in the expressions obtained. For example, v1(t) and v2(t) are both basic waveforms that describe rises. And their Fourier coefficients an1, which is associated with linear rises, and an2, which is associated with exponential rises, are the same except that in an2, tR replaces tr, and αn = arctan(πfntR) replaces πfntR. The same is also true for the coefficients bn1 and bn4.

Similarly, v3(t) and v4(t) are both basic waveforms that describe falls. Their Fourier coefficients an3, which is associated with linear falls, and an4, which is associated with exponential falls, are the same except that in an4, tF replaces tf, and βn = arctan(πfntF) replaces πfntf. The same is also true for the coefficients bn3 and bn4.

Finally, the average values of the basic voltage waveforms for rises are

image

And, the average values of the basic voltage waveforms for falls are

image

These values are of relatively little importance here, because average values of voltages do not cause time-varying currents. However, they are essential to fully convert a given time-domain description of a voltage into its equal frequency-domain description.

4.4 COMMONLY OCCURRING PERIODIC VOLTAGE WAVEFORMS

Any periodic voltage can be described as a sum of sinusoidal voltages using the Fourier coefficients an and bn. And a large number of periodic voltages that are commonly used in contemporary electrical engineering practice can be described in the time domain as sums of either v1(t) or v2(t) added to either v3(t) or v4(t). As a result of their similar construction, these voltages also have similar frequency-domain descriptions.

These voltage waveshapes can be divided into four general categories: (1) those that have a linear rise and fall, (2) those that have an exponential rise and fall, (3) those that have a linear rise and an exponential fall, and (4) those that have an exponential rise and a linear fall. Each of these categories is also divisible into four subcategories as a result of the relationships trtd and td + tfT. In other words, either tr = td or tr < td and either td + tf < T or td + tf = T. Thus, there are four different subcategories of the four categories, making a total of 16 basically different kinds of periodic voltage waveshapes to be considered here.

Each of these voltage waveshape categories is now described and examined in detail, and several examples are given. Readers are encouraged to plot the examples on a calculator or computer to verify the results for themselves.

4.5 VOLTAGES WITH A LINEAR RISE AND A LINEAR FALL

Periodic voltage waveforms that have both a linear rise and a linear fall can have any one of the four general waveforms illustrated in Fig. 4-4. The waveforms are different, because of the relationships trtd and td + tfT. However, each of the different voltage waveshapes has the same general description as a sum of sinusoidal voltages in terms of the variables tr, td, tf, T, and Vp. To see this, notice that each of the waveshapes in Fig. 4-4 can be described as v(t) = v1(t) + v3(t). This can be done because v1(t) is the basic waveshape for linear rises and v3(t) is the basic waveshape for linear falls. And because v(t) = v1(t) + v3(t) the Fourier coefficients of v(t) are an = an1 + an3 and bn = bn1 + bn3. Therefore, it follows from Eqs. 4-13 and 4-16 that

image

and

image

Based on these equations, since image, it also follows that

image

where

image

image

Figure 4-4. Periodic voltage waveforms all describable as v(t)=v1(t) + v3(t), each having a linear rise and a linear fall.

This last quantity, Dn, is called the transition factor. The development of this expression for Dn from Eq. 4-20 is detailed in Section A.3 of Appendix A. In that development, for this class of voltages, t1=tr, t2=tf, xf ntr, yfntf, and zfn(2td + tf).

In other words, from the above and from Eq. 4-8, it is seen that all of the voltage waveshapes in Fig. 4-4 have the general frequency-domain representation

image

where Dn=Dn(tr, tf, td,fn) is given in Eqs. 4-21.

The general expression for the average value of these voltages, from Eqs. 4-19, is

image

Two specific examples from this voltage category are now examined.

Example 4-1

Suppose the values of |Vn| and φn are wanted for the trapezoidal waveshape of Fig. 4-5. In that figure, tr = T/10, td = T/2, tf = T/4, and Vp = 10. (Precise units of time and voltage are of no immediate concern.) Substituting these values in Eqs. 4-20 yields

image

image

Figure 4-5. The periodic voltage waveshape of Example 4-1 for which tr = T/10, td = T/2, and tf = T/4.

and

image

Therefore,

image

can be determined. And |Vn| can be evaluated either with image or with the equation

image

where

image

Thus are the values of |Vn| and φn obtained for this voltage waveshape. Their first 10 values are given in Table 4-1, and a plot of the voltage in the frequency domain is given in Fig. 4-6, for the first 10 values of |Vn| and fn. Voltage plots in the frequency domain are sometimes useful to determine how rapidly |Vn| decreases with frequency.

The average value of the voltage is image

Example 4.2

Suppose values of |Vn| and φn are wanted for the triangular voltage waveshape illustrated in Fig. 4-7. As indicated in the figure, Vp=10, tr = td = T/5, and tf = 4T/5. When these values are substituted in Eqs. 4-20, it is seen that

TABLE 4-1. Frequency Domain Parameters for the Voltage of Fig. 4-5

image

image

Figure 4-6. A frequency-domain representation of the trapezoidal voltage waveshape of Example 4-1 and Fig. 4-5.

image

and

image

image

Figure 4-7. The periodic voltage waveshape of Example 4-2 for which tr=td=T/5, and tf=4T/5.

TABLE 4-2. Values of |Vn| and πn for the Voltage of Fig. 4-7

image

And from Eqs. 4-21,

image

where

image

Given these expressions, the values of |Vn| and φn can again be readily determined. The first 10 values of each are given in Table 4-2 and for plotting purposes,image

4.6 VOLTAGES WITH AN EXPONENTIAL RISE AND AN EXPONENTIAL FALL

Periodic voltage waveforms with both an exponential rise and an exponential fall can have any one of the four general forms illustrated in Fig. 4-8. Using the basic waveforms given in Fig. 4-3, any of these voltages can be described as v(t) = v2(t) + v4(t). Thus, any one of these voltages has Fourier coefficients of the general form an = an2 + an4 and bn = bn2 + bn4. Referring back to Eqs. 4-14 and 4-17, it can be seen that

image

and

image

image

Figure 4-8. Periodic voltage waveforms all describable as v(t)=v2(t) + v4(t), each having an exponential rise and an exponential fall.

Notice the similarity in form between these equations and Eqs. 4-20, which describe voltages with a linear rise and fall. The only differences are that, in Eqs. 4-28, tR and tF have replaced tr and tf, and αn=arctan(πfntR) and βn=arctan(πfntF) have replaced πfntr and πfntf

Since image based on Eqs. 4-28 it follows that

image

where

image

Details of the derivation of this expression for the transition factor Dn are given in Appendix A. In this case, the variable assignment is t1=tR, t2=tF, x=αn, y=βn, and z=ωntd + βn. These voltages have an average value of v(t)=(Vp/2T)(2tdtR + tF).

Example 4-3

Consider the periodic voltage illustrated in Fig. 4-9. The voltage has an exponential rise and an exponential fall, and tr = T/4, td = tf = T/2, and Vp = 10. Therefore, tR = T/4e and tF = T/2e, and from Eq. 4-28 it follows that

image

image

Figure 4-9. The periodic voltage waveshape of Example 4-3 for which tR=T/4e, tF=T/2e and td=T/2.

TABLE 4-3. Values of |Vn| and πn for the Voltage of Fig. 4-9

image

and

image

Also,

image

where

image

and αn=arctan(nπ/4e), and βn=arctan(nπ/2e).

|Vn| and πn can now be found for the voltage in Fig. 4-9 using these equations. The values of those parameters for the first 10 values of n are given below in Table 4-3. Also, the average value of the voltage is image = 5.6.

4.7 VOLTAGES WITH A LINEAR RISE AND AN EXPONENTIAL FALL

Periodic voltages that rise linearly and fall exponentially will have one of the four general time-domain waveforms shown in Fig. 4-10. Using the basic waveforms, each of these voltages can be described as v(t)=v1(t) + v4(t) with the Fourier coefficients an=an1 + an4 and bn=bn1 + bn4. Therefore,

image

image

Figure 4-10. Periodic voltage waveforms all describable as v(t)=v1(t) + v4(t), each having a linear rise and an exponential fall.

and

image

where

image

And, from Eqs. 4-19, the average value of these voltages is v(t)=(Vp/2T)(2tdtr + tF).

Example 4-4

Consider the voltage waveshape shown in Fig. 4-11. It has a linear rise and an exponential fall, tr=td=T/10, tf=9T/10, and Vp=10. Therefore, tF=9T/10e, and the Fourier coefficients of the voltage are

image

and

image

image

Figure 4-11. The periodic voltage waveshape of Example 4-4 for which tr=td=T/10, and tF=9T/10e.

TABLE 4-4. Values of |Vn| and πn for the Voltage of Fig. 4-11

image

Therefore, the transition factor is

image

and

image

Based on these equations, the first 10 values of |Vn| and πn are those given in Table 4-4. For plotting purposes, image.

4.8 VOLTAGES WITH AN EXPONENTIAL RISE AND A LINEAR FALL

Periodic voltages that rise exponentially and fall linearly can have any one of the four general forms illustrated in Fig. 4-12. Referring once again to the basic waveforms of Fig. 4-3 and their equations, it is seen that the four voltages of Fig. 4-12 can be described as v(t)=v2(t) + v3(t). Therefore, an=an2 + an3 and bn=bn2 + bn3, and Eqs. 4-14 and 4-16 give

image

and

image

Therefore,

image

Figure 4-12. Periodic voltage waveforms all describable as v(t)=v2(t) + v3(t), each having an exponential rise and a linear fall.

image

In all of these equations, αn=arctan(πfntR) and the average value of image

Example 4-5

Consider the waveshape of Fig. 4-13. Given that tr=T/4, td=T/2, tf=T/8, and Vp=10, it follows that tR=T/4e and αn=arctan(nπ/4e) from Eqs. 4-28, and that

image

Also,

image

where

image

The first 10 values of |Vn| and πn are given in of Table 4-5, and images.

image

Figure 4-13. The periodic voltage waveshape of Example 4-5 for which tR=T/4e, tf=T/8. and td=T/2.

TABLE 4-5. Values of |Vn| and πn for the Voltage of Fig. 4-13

image

4.9 CONSOLIDATION

All of the voltage waveforms considered above have the general frequency domain description

image

in which

image

and

image

where an and bn are the the Fourier coefficients of v(t). For each of the 16 categories of voltage waveforms considered, the Fourier coefficients an are all quite similar to one another, as are the coefficients bn. In fact, with only a few additional generalizations, the similarities can be extended to yield one expression for an and one expression for bn for all of those voltage waveforms. This is done as follows.

First, if the rise is linear, let tR=tr, and let Anfntr. If the rise is exponential, let tR=tr/e and let An=αn=arctan(πfntR). If the fall is linear, let tF=tf and let Bn=(πfntf). If the fall is exponential, let tF=tf/e and let Bn=βn=arctan(πfntF). Based on these definitions, the Fourier coefficients of any of the periodic voltage waveforms discussed above can be written as

image

and

image

From these expressions, it then follows that

image

Thus, the transition factor has the general form

image

and |Vn| can be more concisely described as

image

Similarly, it follows from Eqs. 4-37 and 4-38 that

image

Equations 4-39 and 4-42 are general expressions for the amplitudes and the phases of the frequency-domain components of all of the periodic voltages considered in this chapter. The practical significance of these expressions is that they are functions of only time-domain variables and n. In other words, |Vn| and πn can be found without first having to find the Fourier coefficients, an and bn. The values of Vp, tr, tf, td, and T can be found, for example, by simply viewing v(t) on an oscilloscope and measuring them. Then |Vn| and πn can be found for any value of n by calculating tr, tF, An, Bn, and Dn and substituting those values, together with the values of Vp, T, and td, in Eq. 4-38 or in Eqs. 4-39 and 4-40 and in Eq. 4-42.

4.10 SUMMARY

The purpose of this chapter is to provide a simple, practical method for obtaining frequency-domain descriptions of time-domain voltage waveforms. Sixteen different categories of periodic, time-varying voltages v(t), examples of each of which are shown in Fig. 4-14, were considered here. It was seen that any of those voltages is easily describable as a sum of sinusoidal voltages of different frequencies using only parameters obtained from their time-domain descriptions. The procedure is probably best summarized with a simple example.

Suppose the amplitudes |Vn| of some of the sinusoidal components of the voltage waveform v(t) of Fig. 4-15a are needed. They can be found as follows. The period of that waveform is T=1/f, where f is its frequency. From Fig. 4-15a it is seen that the amplitude of the voltage is Vp=10 volts, its total linear rise time is tr=T/4, its total exponential fall time is tf=T/2, and the time from the start of tr to the start of tf is td=T/2.

Now, the expression developed in this chapter with which |Vn| can be evaluated for any given n is

image

In this expression, because the rise is linear, tR = tf and An = nπftR. Also, because the fall is exponential, tF = tf/e and Bn=arctan(nπftF), where e = 2.718. … Therefore, for this voltage waveform,

image

and

image

Substituting these values in the above expression for |Vn| and solving the resulting equation for n = 1 to n = 10 yields the following table of values:

image

image

Figure 4-14. Sixteen different categories of voltage waveshapes whose sinusoidal components have easily determined amplitudes.

image

Figure 4-15. Time-domain representations of (a) the voltage v(t) and (b) the sum of the first 10 frequency-domain components (—) of v(t) (…).

In this way, then, the value of |Vn| can be found for any frequency fn=nf.

Typically, for purposes of predicting radiations, only the amplitudes of sinusoidal components will be of interest. However, to verify that the values obtained for the amplitudes |Vn| are valid, the frequency-domain description can be plotted in the time domain and compared to the given voltage waveform. To do that the values of πn, the phase angles of the sinusoidal components, must be known. The expression developed here with which πn can be evaluated is

image

The same values of An, Bn, tR, tF, and td as those used to evaluate |Vn| are used here, of course, yielding the following table of radian values for πn:

image

With these values of |Vn| and πn, it can then be graphically verified that

image

A comparison of the given time-domain waveform for v(t) and the sum of the first 10 of its frequency-domain components is shown in Fig. 4-15b.

Based on these results, transitions back and forth between the time domain and the frequency domain are possible with little or no additional mathematics. This will be useful in subsequent discussions here, and it should also be useful for practicing engineers.

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