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.

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 *V _{p}*, the peak-to-peak amplitude of the voltage;

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

In this description of *v*(*t*), the frequencies of the sinusoidal components are *f*_{n}=*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

Because *f _{n}*=

Now, when *n*=0, then *f _{n}*=0, so that cos(2π

The integral in Eq. 4-4 is, by definition, the average value of *v*(*t*). Therefore, the notation will be used here to denote that average, rather than *a _{o}*/2. Also, if

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 *φ _{n}* is defined for any

As illustrated in Fig. 4-2, the second term in this definition is a necessary addition to the arctangent, to give *φ _{n}* a full range of 2π radians. That is, when

Given this definition of *φ _{n}* then, it follows that

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

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(*x* − *y*), 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

In this equation,

where *a _{n}* and

Equation 4-8 is a general description of any periodic voltage, in terms of the coefficients *a _{n}* and

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 *V _{p}*,

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:

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 *a _{n}* and

Note also that ω* _{n}*=2π

The Fourier coefficients for the basic voltage waveform *v*_{1}(*t*), which describes *linear rises*, are

and

The Fourier coefficients for the basic voltage waveform *v*_{2}(*t*), which describes *exponential rises*, are

and

where

and

The Fourier coefficients for the basic voltage waveform *v*_{3}(*t*), which describes linear falls, are

and

And, the Fourier coefficients for the basic voltage waveform *v*_{4}(*t*), which describes exponential falls, are

and

where

and

Note, for future reference, that the angle α* _{n}* is associated with exponential rises, and the angle β

Before applying the above results, it is worthwhile to observe some similarities in the expressions obtained. For example, *v*_{1}(*t*) and *v*_{2}(*t*) are both basic waveforms that describe rises. And their Fourier coefficients *a*_{n1}, which is associated with linear rises, and *a*_{n2}, which is associated with exponential rises, are the same except that in *a*_{n2}, *t _{R}* replaces

Similarly, *v*_{3}(*t*) and *v*_{4}(*t*) are both basic waveforms that describe falls. Their Fourier coefficients *a*_{n3}, which is associated with linear falls, and *a*_{n4}, which is associated with exponential falls, are the same except that in *a*_{n4}, *t _{F}* replaces

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

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

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.

Any periodic voltage can be described as a sum of sinusoidal voltages using the Fourier coefficients *a _{n}* and

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 *t _{r}* ≤

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.

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 *t _{r}* ≤

and

Based on these equations, since , it also follows that

where

This last quantity, *D _{n}*, is called the

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

where *D _{n}*=

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

Two specific examples from this voltage category are now examined.

Suppose the values of |*V _{n}*| and φ

Therefore,

can be determined. And |*V _{n}*| can be evaluated either with or with the equation

where

Thus are the values of |*V _{n}*| and φ

The average value of the voltage is

Suppose values of |*V _{n}*| and φ

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

and

**TABLE 4-2**. **Values of | V_{n}| and π_{n} for the Voltage of Fig. 4-7**

And from Eqs. 4-21,

where

Given these expressions, the values of |*V*_{n}| and φ_{n} can again be readily determined. The first 10 values of each are given in Table 4-2 and for plotting purposes,

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*) = *v*_{2}(*t*) + *v*_{4}(*t*). Thus, any one of these voltages has Fourier coefficients of the general form *a*_{n} = *a*_{n2} + *a*_{n4} and *b _{n}* =

and

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, *t*_{R} and *t*_{F} have replaced *t*_{r} and *t*_{f}, and *α*_{n}=arctan(π*f*_{n}*t*_{R}) and *β*_{n}=arctan(π*f*_{n}*t*_{F}) have replaced π*f*_{n}*t*_{r} and π*f*_{n}*t*_{f}

Since based on Eqs. 4-28 it follows that

where

Details of the derivation of this expression for the transition factor *D*_{n} are given in Appendix A. In this case, the variable assignment is *t*_{1}=*t*_{R}, *t*_{2}=*t*_{F}, *x*=*α*_{n}, *y*=*β*_{n}, and *z*=*ω*_{n}*t*_{d} + *β*_{n}. These voltages have an average value of *v*(*t*)=(*V*_{p}/2*T*)(2*t*_{d} − *t*_{R} + *t*_{F}).

Consider the periodic voltage illustrated in Fig. 4-9. The voltage has an exponential rise and an exponential fall, and *t*_{r} = *T*/4, *t*_{d} = *t*_{f} = *T*/2, and *V*_{p} = 10. Therefore, *t*_{R} = *T*/4*e* and *t*_{F} = *T*/2*e*, and from Eq. 4-28 it follows that

and

Also,

where

and *α _{n}*=arctan(

|*V*_{n}| and π_{n} 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 = 5.6.

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*)=*v*_{1}(*t*) + *v*_{4}(*t*) with the Fourier coefficients *a*_{n}=*a*_{n1} + *a*_{n4} and *b*_{n}=*b*_{n1} + *b*_{n4}. Therefore,

where

And, from Eqs. 4-19, the average value of these voltages is *v*(*t*)=(*V*_{p}/2*T*)(2*t*_{d} − *t*_{r} + *t*_{F}).

Consider the voltage waveshape shown in Fig. 4-11. It has a linear rise and an exponential fall, *t*_{r}=*t*_{d}=*T*/10, *t*_{f}=9*T*/10, and *V*_{p}=10. Therefore, *t*_{F}=9*T*/10*e*, and the Fourier coefficients of the voltage are

and

**TABLE 4-4**. **Values of | V_{n}| and π_{n} for the Voltage of Fig. 4-11**

Therefore, the transition factor is

and

Based on these equations, the first 10 values of |*V*_{n}| and π_{n} are those given in Table 4-4. For plotting purposes, .

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*)=*v*_{2}(*t*) + *v*_{3}(*t*). Therefore, *a*_{n}=*a*_{n2} + *a*_{n3} and *b*_{n}=*b*_{n2} + *b*_{n3}, and Eqs. 4-14 and 4-16 give

and

In all of these equations, *α*_{n}=arctan(π*f*_{n}*t*_{R}) and the average value of

Consider the waveshape of Fig. 4-13. Given that *t*_{r}=*T*/4, *t*_{d}=*T*/2, *t*_{f}=*T*/8, and *V*_{p}=10, it follows that *t*_{R}=*T*/4*e* and *α*_{n}=arctan(*n*π/4*e*) from Eqs. 4-28, and that

Also,

where

The first 10 values of |*V*_{n}| and π_{n} are given in of Table 4-5, and s.

**TABLE 4-5**. **Values of | V_{n}| and π_{n} for the Voltage of Fig. 4-13**

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

in which

and

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

First, if the rise is linear, let *t*_{R}=*t*_{r}, and let *A*_{n}=π*f*_{n}*t*_{r}. If the rise is exponential, let *t*_{R}=*t*_{r}/*e* and let *A*_{n}=*α*_{n}=arctan(π*f*_{n}*t*_{R}). If the fall is linear, let *t*_{F}=*t*_{f} and let *B*_{n}=(π*f*_{n}*t*_{f}). If the fall is exponential, let *t*_{F}=*t*_{f}/*e* and let *B*_{n}=*β*_{n}=arctan(π*f*_{n}*t*_{F}). Based on these definitions, the Fourier coefficients of any of the periodic voltage waveforms discussed above can be written as

and

From these expressions, it then follows that

Thus, the transition factor has the general form

and |*V*_{n}| can be more concisely described as

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

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, |*V*_{n}| and π_{n} can be found without first having to find the Fourier coefficients, *a*_{n} and *b*_{n}. The values of *V*_{p}, *t*_{r}, *t*_{f}, *t*_{d}, and *T* can be found, for example, by simply viewing *v*(*t*) on an oscilloscope and measuring them. Then |*V*_{n}| and π_{n} can be found for any value of *n* by calculating *t*_{r}, *t*_{F}, *A*_{n}, *B*_{n}, and *D*_{n} and substituting those values, together with the values of *V*_{p}, *T*, and *t*_{d}, in Eq. 4-38 or in Eqs. 4-39 and 4-40 and in Eq. 4-42.

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 |*V*_{n}| 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 *V*_{p}=10 volts, its total linear rise time is *t*_{r}=*T*/4, its total exponential fall time is *t*_{f}=*T*/2, and the time from the start of *t*_{r} to the start of *t*_{f} is *t*_{d}=*T*/2.

Now, the expression developed in this chapter with which |*V*_{n}| can be evaluated for any given *n* is

In this expression, because the rise is linear, *t*_{R} = *t*_{f} and *A*_{n} = *n*π*ft _{R}*. Also, because the fall is exponential,

and

Substituting these values in the above expression for |*V*_{n}| and solving the resulting equation for *n* = 1 to *n* = 10 yields the following table of values:

In this way, then, the value of |*V _{n}*| can be found for any frequency

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 |*V _{n}*| 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 π

The same values of *A _{n}*,

With these values of |*V _{n}*| and π

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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