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PHASORS

INTRODUCTION
MULTIPLICATION AND DIVISION BY j
INTERACTIVE APPLET
PHASORS IN CIRCUIT ANALYSIS
ADDITION OF PHASORS
SUMMARY


INTRODUCTION

Phasors are used to represent sinusoidal signals. Before we take up the description of a phasor, let us look at how a sinusoidal function is defined.

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Equation (63) defines a sinusoidal signal. The sinusoidal signal is completely defined by three parameters, which are its amplitude, its frequency and its phase. The amplitude of the sinusoidal signal, defined by equation (63), is indicated by letter E. The second parameter is its frequency. The frequency is the reciprocal of the cycle period, marked as T. Since frequency is the number of cycles per second, the reciprocal of cycle period yields frequency. The product of frequency and cycle period is one second. The third parameter is the phase.

p02f02

The waveform shown above expresses a sinusoidal signal with zero phase or zero phase angle. That is, f = 0. The signal with zero phase usually represents the reference phasor.

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Using Euler's identity, the sinusoidal signal, represented by equation (63) in time domain, can be represented as shown by equation (64a). Equation (64b) defines the part of sinusoidal signal that can be represented by a phasor. A phasor is completely defined by only two parameters, which are its amplitude and its phase. The frequency of a signal is not explicit from the phasor of a signal. The frequency component is not required for phasor representation, since it has to be the same for all phasors, used within a context. It is necessary to remember that polar form of representation of a phasor is the same as the exponential form. Using the phasor defined in equation (64b), the sinusoidal signal can be represented, as shown by equation (64c). The sketch in Fig. 18 shows how a phasor can be represented. It is similar to a complex value, with a real part and an imaginary part.

F18Phasor1

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MULTIPLICATION AND DIVISION BY j

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Let us see the significance of multiplying a phasor by j. Equation (65) expresses Euler's identity. Using this identity, we get equation (66). We can expand the sine and cosine terms, and obtain equation (67). We know that the square of j is - 1, and hence we get equation (68) from equation (67). We find that j is common to both terms, and equation (68) can be re-presented as equation (69). Using Euler's identity, we get equation (70) from equation (69).

F19Phasor02

The sketch in Fig. 19 shows the effect of multiplying a phasor with angle f by j. Since the resultant is a phasor with an angle equal to sum of f and p/2, multiplying a phasor by j amounts to rotating that phasor by 90o in the counter-clockwise direction, or the positive direction for angle.

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It is possible to express multiplication of a phasor by j as shown by equation (71). We get the same result, but the approach is different. We can represent j as a phasor. Then we multiply two phasors in the exponential form. In such a case, the magnitude of the resultant phasor is the product of the magnitude of multiplicand and the magnitude of multiplier. The angle of the resultant phasor is the algebraic sum of the angle of multiplicand and the angle of multiplier.

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Division of a phasor by j amounts to rotating it by 90o in the clockwise direction. We get equation (72) using Euler's identity. Expand the terms in equation (72) and obtain equation (73). We can multiply equation (73) by j and divide the result by j to get equation (74). Using Euler's identity, the result can be expressed by equation (74). It is seen that division of a phasor by j amounts to rotating it by 90o in the clockwise direction. It is necessary to know the significance of j.

F20Phasor3

The sketch in Fig. 20 shows the effect of dividing a phasor by j. Next an interactive applet is presented to show the relationship between a phasor and its time domain expression.

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INTERACTIVE APPLET

An applet showing the relationship between a phasor and its time domain expression is presented below. The equation at the bottom on the right side of window shows how a phasor, and its time domain expression, are related. Equation (75) defines an expression, in time domain. Using Euler's identity, we can get its real part and and its imaginary part. The plots on the right side of window display the real and the imaginary part of the expression in time domain. On the left side, the phasor is shown in red color. You can change the value of phase angle by entering an integer value into the text box, or by dragging the button on the scroll bar. You need to press the enter button on the keyboard, if you enter a value into the text box.

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PHASORS IN CIRCUIT ANALYSIS

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Information presented at the beginning of this page is repeated here. Equation (75) defines an expression in time domain. The exponential part of signal in equation (75) can be shown to be the product of two exponential functions, as shown by equation (78). The part, containing the magnitude and the phase, can be represented by a phasor, as shown by equation (79). Using this phasor, the time domain can be expressed, as shown by equation 80. It can be seen that when phase angle varies, the time domain signal gets phase-shifted.

The sketch in Fig. 18 shows how a phasor can be represented by a diagram. The positive direction for phase angle is the counter- clockwise direction, and the reference phasor rests on the real axis.

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We apply sinusoidal signals to circuits, and not complex exponential signals. The phase-shifted signal is expressed by equation (82). Using Euler's identity, we can define it, as expressed by equation (83). Hence a cosine signal, expressed by equation (82), is expressed by a phasor, as shown by equation (84). In the context of circuit analysis, a phasor refers only to the real part of complex exponential. A phasor is a shortened representation of a cosine function in circuit analysis. Equation (85) presents the time domain expression for the reference phasor, and it can be expressed, as shown by equation (86). The reference phasor is the phasor with zero phase. The sketch in Fig. 21 shows the relationship between the reference signal and the phase-shifted signal.

F21Sine01

For analysis, we can still deal with the complex exponential signal, instead of the cosine function. After performing the required mathematical tasks, we take the real part of the solution only and discard the imaginary part. This approach is logical and correct for linear circuits. We apply to the circuits the real part of complex exponential, and the real part of the calculated result, in the form of complex exponential, reflects the actual result.

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ADDITION OF PHASORS

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Let us try adding two signals, defined by equations (87) and (88). At first, the solution is obtained without using phasors. Next the solution obtained using phasors is displayed. The plots of two signals, expressed by equations (87) and (88), are shown below by a sketch.

sineAdd

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Without recourse to phasors, we can add the two signals, as shown now. Signal v1 can be expanded into two parts, as shown by equation (89). Signal v2 can be expanded into two parts, as shown by equation (9)0. For expanding the signals into two parts, the trigonometric identity stated by equation (92), can be used. The sum is signal v3, as shown by equation (91). Equation (93) shows the sum of equations (89) and (90). Using the trigonometric identity, stated by equation (92), we can group the two parts in equation (91), and the result obtained is displayed by equation (93). The plot of the sum is shown below.

sumv2

The waveform of the sum of signals, v1 and v2, is shown above. When two or more sinusoidal signals with the same frequency, are added or subtracted from one another, the result is a sinusoidal signal at the same frequency. The magnitude and the phase of the result are dependent on its constituent signals. Remember that the concept of phasors can be applied to signals at the same frequency only. If two signals have two different frequencies, we cannot define them as phasors within a context.

Next we obtain the solution using phasors.

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The phasor representation of signal V1 is shown by equation (94). The expression for phasor V1 is obtained from equation (87). Signal V2 is a sinusoidal signal, represented by a sine function. As the sine function lags the cosine function by 90o, we get the phasor representation of signal v2 as shown by equation (95). The phasor sum of phasors, V1 and V2, is expressed by equation (96) and the corresponding time-domain expression is shown by equation (97). The sketch in Fig. 22 shows how phasors can be added.

F22SinAdd2

F23PhAdd3

The sketch in Fig. 23 shows how the phasors can be added, using a parallelogram. The technique is the same as that used for adding two complex numbers. It is good to use this technique to verify the answers, obtained earlier. The graphical technique is not accurate, but one is not prone to making errors while using the graphical technique.

The same problem is presented now in a different way.

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Let us add signals defined by equations (252) and (253). They are the same signals, defined by equations (89) and (90). To get the phasor representation of v1, we have two options. We can express (252) by the equation (87). The better alternative is to express cosine and sine functions, straightaway as phasors. The cosine part of the signal has zero phase and hence the representation of the cosine part by a phasor is easy. The phasor for the cosine part contains the amplitude of the cosine signal and zero phase. To represent the sinusoidal part, a few steps are needed. The sine signal lags the cosine signal. We can use Euler's identity next and simplify the result, as shown by equation (255). It turns out that phasor representation for sin(wt) is - j. This is so when the cosine signal is the reference phasor. Now we can express equations (252) and (253) in the phasor form, as shown by equations (255) and (256). The sum of two signals is expressed by equation (257), which is the same as equation (96) obtained earlier.

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SUMMARY

This page has described what a phasor is. A phasor is defined by its magnitude and its phase. All the phasors within the same context refer to signals at the same frequency. A phasor is used to represent a sinusoid, unlike a complex value used to represent a two dimensional vector. But phasors can be manipulated just like complex numbers. It is necessary to understand the relationship between and its corresponding time-domain signal, in order to use phasors effectively.

The next page is on the use of passive components in ac circuits.

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