As we know, the passive components are resistor, inductor and capacitor. In this page, we find out how they are represented in ac circuits, when a sinusoidal current or voltage is impressed on them. We will also determine how the instantaneous power absorbed by them varies as a function of time. New terms, such as the active power and the reactive power, are introduced in this page.
The relationship between voltage and current in a resistor is shown in Fig. 25. It can be seen that resistance acts just as a scale factor between voltage and current. The slope of the voltage-current characteristic is the resistance.
Let the current through a resistor be expressed by equation (102). The voltage across the resistor can then be expressed, as shown below.
Then the voltage obtained across the resistor can be expressed by equation (103). It is seen that the phase of both the current and the voltage of a resistor is the same. In other words, there is no phase shift or difference in phase between the voltage and the current, as expressed by equation (104). It can be stated that the voltage and the current associated with a resistor vary in phase, and value of a resistor acts as a scale factor, as expressed by equation (105). When the voltage and the current are expressed in the form of phasors, the equation relating voltage and current appears the same as does with dc values. Equations (103) and (105) express the Ohm's law for ac circuits.
The waveforms of current and voltage over two cycles in a 4 W resistor are shown now. It can be seen that voltage and current have the same shape and phase, with the value of resistor being the scale factor. In such a case, the voltage and the current are said to vary in phase.
Given that the voltage and the current associated with a resistor are in phase, they can be expressed, as shown by equations (106) and (107). The instantaneous power absorbed by a resistor is obtained as the product of voltage, and current, as expressed by equation (108). The trigonometric identity expressed by equation (110) can be used used to get the expression for power in equation (109). It can be seen that the expression for power in equation (109) contains a constant part and another part, oscillating at double the source frequency, w. Note that w is the angular frequency, its unit being radians over second. The angular frequency is 2p times the actual frequency. We now have an expression for the instantaneous power. We can get the expression for the average power associated with a resistor.
Power by definition is an average value, and when we use the word, power, we refer to the average power. In ac circuits, we define and use the root mean square values of current and voltage, but when it comes to power, we define only its average value. Power is defined as energy absorbed per second. Note that power absorbed is positive, and power delivered is negative. A resistor absorbs power, and the power absorbed by a resistor can be calculated, as shown by equation (111). The period of a cycle corresponding to a frequency is the reciprocal of frequency, since by definition the product of frequency and the period of a cycle should equal one second. Hence we get equation (112). Equation (113) is obtained from equations (112), (106) and (107). Equation (106) defines the current through the resistor, and equation (107) defines the voltage across a resistor. The expressions for voltage and current, defined by equations (106) and (107), replace terms for voltage and current in equation (112) in order to yield equation (113).
The angular frequency w used in equation (114) is defined in terms of either the frequency, or the cycle period. As expressed by equation (115), cycle period is the reciprocal of frequency. That is, the product of frequency or number of cycle per second and the cycle period equals one second. Using q, as defined by equation (116), we can obtain equation (117), wherein the derivative of time is expressed as derivative of q over w.
In equation (118), we can replace derivative of time by derivative of q over w. Then the upper limit of integral changes to 2p, as shown by equation (119). Using the trigonometric identity defined by equation (120_, we can evaluate the power in a resistor, as shown next.
Equations (121) and (122) show how power absorbed by a resistor is calculated. Power consumed by a resistor is positive, and this forms the basis for passive sign convention. The signs used for voltage, current and power in passive sign convention have been derived based on the behaviour of a resistor. For a resistor, both voltage and current have the same sign. In other words, current enters a resistor through its positive terminal. The power absorbed by a resistor is positive. You need to remember the passive sign convention, to understand this page and the pages to follow.
The plots show the instantaneous power and the average power in a resistor over two cycles. The amplitude of voltage is 40 Volts, and the resistance value is 4 W. It can be seen that the average power is 200 Watts. The oscillating part has twice the source frequency, and its amplitude is 200 Watts. The value of instantaneous power varies from 0 Watt to 400 Watts. It can never, ever be negative. The maximum instantaneous power is 400 W and the minimum instantaneous power is 0 W.
It is the practice to express the power in terms of root mean square values of voltage and current. Given a function, as defined by equation (123), its root mean square value is obtained, as defined by equation (124). To evaluate equation (124), the derivative of time is expressed as derivative of q over w, and the upper limit is changed correspondingly, as indicated by equation (125). Using the trigonometric identity stated in equation (126), we get the root mean square value, as shown next.
We get the root mean square value as the amplitude over square root of 2, and equation (127) presents this information. Using the root mean square values, we can express power in a resistor, according to equation (128). Power in a resistor varies proportionately with the square of root mean square values. The root mean square value is also known as the effective value, and it is explained next, why it is called as the effective value.
In Figure 26, a resistor is shown connected to an ac source on the left side, and a dc side by its side. We can calculate the power delivered, as shown by equations (129) and (130). When these two powers are equal, it is seen that effective voltage is equal to the root mean square voltage. The root mean square voltage is then seen to be equivalent to a dc voltage, represented by the effective voltage. Hence it can be understood why the root mean square value is referred to as the effective voltage.
Within the context of circuit analysis, as described by these pages, an inductor is a linear element. In the case of an inductor, the linearity is between the flux linkage, and current through the inductor, as shown in Fig. 27.
We assume that flux within the core of inductor does not saturate. Flux linkage has the unit of volt seconds, and by definition, voltage is rate of change of flux linkage, as expressed by equation (132). Let inductor current be defined, as expressed by equation (133). Then voltage across inductor is obtained as shown in equation (134). It can be seen that voltage leads the current by 90o.
Equation (135) defines the inductor current, and equation (136) expresses the inductor voltage. Below equation (136), the values of wL, wT, and phase f have been stated. For these values, the waveforms of inductor current and inductor voltage are shown below, where it is seen that the voltage leads the current by 90o.
It can be seen that voltage, indicated by red colour line, leads the current, shown by blue colour line. Next we find the phasor relationship of inductor voltage and inductor current.
Let the current be defined by equation (137). In practice, the current is a sinusoidal function, and the real part of equation (137) represents the cosine function, and the imaginary part represents the sine function. The phasor representation of current is obtained, as shown by equation (138). The inductor voltage is obtained, as expressed by equation (139). Equation (140) expresses the phasor representation of inductor voltage. Since voltage is the product of current and impedance, the impedance due to inductor is jwL. Impedance is a complex value, with real and imaginary parts. The real part of impedance is resistance, and the imaginary part is called the reactance. Since impedance due to inductor has only imaginary part, wL is called the inductive reactance. The unit for reactance is ohm. The sketch in Fig. 28 shows the phasor diagram of an inductor. It can be seen that the voltage in an inductor leads its current by 90o.
In practice, the inductor current is defined as shown by equation (141). Using Euler's identity, it can be expressed as the real part of complex exponential function. The inductor voltage is expressed as shown by equation (142).
The inductor voltage is expressed as shown by equation (142). First, the complex exponential is differentiated, as shown by equation (143). When we multiply complex exponential by j, it gets rotated by 90o in the anti-clockwise direction, as illustrated by equation (144). Taking the real part of the complex exponential after rotating it by 90o, we get equation (145). The operations performed can be summarized as follows
Equations (146) and (147) present a summary of what has been done. The sketch in Fig. 28 shows that the inductor voltage leads its current by 90o. In other words,, the current lags the voltage by 90o in an ideal inductor. You need to remember this.
As shown by equation (148), the reactance of an inductor, called as XL, equals the product of w and inductance L . The reciprocal of impedance is admittance, expressed by Y, and the admittance of an inductor contains only an imaginary component.. This imaginary part is known as susceptance, and it is indicated by letter, BL. The expression for susceptance of an inductor has been presented by equation (149). Note that the susceptance of an inductor is negative, as shown by equation (149). The admittance of an inductor can be expressed, as shown by equation (150) and (151).. The sketch in Fig. 29 shows how an inductor is represented in ac circuits.
Power in an inductor can be determined as shown here. Equation (152) presents expressions for the inductor voltage and the inductor current. Then power is calculated, as shown by equations (153) and (154). Using the trigonometric identity, stated by equation (155), we get that average power in an inductor is zero, as shown by equation (156). The impedance of an inductor is imaginary, and it is logical that the average power consumed is zero, as shown by the sketch shown below.
To obtain the plot shown above, the amplitude of current is taken to be 10 A, the phase of current is taken to be 60o and the product of w and inductance L is taken to be 1 W. For an inductor, we define another term, known as the reactive power, indicated by letter Q. An inductor absorbs reactive power, and the reactive power computed for an inductor has to be positive, since it absorbs reactive power. It is calculated as shown next.
The reactive power, with the unit of VAR, is computed as follows. Express the inductor voltage and the inductor current as phasors, as shown by equations (157) and (158). Here the root mean square values are used as the magnitudes for phasors. Obtain apparent power, defined by letter S, as the product of voltage phasor and the conjugate of current phasor, as shown by equation (159). The symbol for conjugate is the bar over the phasor. The result for apparent power is expressed by equation (160). It is seen that the real part is zero, and the imaginary part is the reactive power of an inductor.
We can express apparent power by the power diagram, and the apparent power is expressed by a complex value. The real part of apparent power equals P, and it is sometimes called as the active power, and the imaginary part of apparent power is reactive power, denoted by letter, Q. A resistor has non-zero, positive active power, and zero reactive power. On the other hand, an inductor has zero active power, and a non-zero, positive reactive power. An inductor absorbs reactive power, and hence it is positive. The definition for apparent power is such that the reactive power for an inductor results as a positive value. Note that the conjugate of current is used for deriving an expression for the apparent power.
A capacitor is a linear element. Its charge and its voltage linearly, with the constant of proportionality being its capacitance.
In this section, we find out how a capacitor can be represented in an ac circuit. In the case of a capacitor, the linearity is between the charge held by the capacitor and its voltage, as shown in Fig. 30, and equation (161) shows how the charge held by a capacitor, its voltage and its capacitance are related with one another. Charge has the unit of Coulomb or the product of amperes and seconds, and by definition, current is rate of change of charge, as expressed by equation (162). Let capacitor voltage be defined, as expressed by equation (163). Then capacitor current is obtained as shown in equation (164). It can be seen that the capacitor voltage lags its current by 90o, or alternatively it can be stated that capacitor current leads its voltage by 90o.
For the voltage and current, expressed in equations (165) and (166), the waveforms obtained for capacitor voltage and capacitor current are shown, and the values used for wC,wT, and phase f, have been stated below equation (166). It is seen that the voltage lags the current by 90o. In a capacitor, the current leads the voltage by 90o , whereas the inductor current lags its voltage by 90o in an inductor. Next we find the phasor relationship of capacitor voltage and capacitor current.
Let the capacitor voltage be defined by equation (167). Then the phasor representation of voltage is obtained, as shown by equation (168). The capacitor current is obtained, as expressed by equation (169). Equation (170) expresses the phasor representation of capacitor current. Since voltage is the product of current and impedance, the impedance due to capacitor is one over (jwC). Impedance is a complex value, with real and imaginary parts, and the impedance of a capacitor has imaginary part only. The imaginary part of the impedance is known as the reactance. The reactance of a capacitor is a negative value. We will see later why the reactance of a capacitor is a negative value.
The phasor diagram for a capacitor shown in Fig. 31 shows that the capacitor current leads its voltage by 90o.
It is clarified here that we get the same result, where the cosine function is expressed as the real part of complex exponential. Equation (171) expresses the capacitor voltage, first as a cosine function and next as the real part of the corresponding exponential function. The capacitor current is expressed as shown by equation (172). First the complex exponential is differentiated. When we multiply complex exponential by j, it gets rotated by 90o, as expressed by equation (173). Taking the real part of the complex exponential after rotating it anti-clockwise by 90o, we get equation (174). We can summarize what has been done, as shown below.
Equation (175) presents a summary of what has been done. The sketch in Fig. 31 shows that the capacitor voltage lags its current by 90o. In a capacitor, the current leads the voltage by 90o. You need to remember this. On the other hand, the current lags the voltage by 90o in an ideal inductor.
The impedance of a capacitor is purely imaginary, as expressed by equation (176). The imaginary part of impedance of a capacitor, called the reactance, is expressed by equation (177). The reactance of a capacitor is called as XC , and it equals the negative of (1/wC), as expressed by equation (177). We can express the impedance in the polar form, as illustrated by equation (178). The reciprocal of impedance is admittance, and it is denoted by letter Y, and the admittance of a capacitor contains only an imaginary component, as shown by equation (179). This imaginary part is known as susceptance, and equation (180) represents susceptance by letter, BC. The expression for susceptance of a capacitor has been presented by equation (181). For a capacitor, its susceptance is a positive value, but its reactance is negative. The sketch in Fig. 32 shows how a capacitor is represented in an ac circuit.
Power in a capacitor can be determined as shown here. Given the capacitor voltage and the capacitor current, as expressed by equation (181), power associated with a capacitor is calculated, as shown by equations (182) and (183). Using the trigonometric identity, stated by equation (184), power in a capacitor is calculated to be zero, as shown by equation (185). The impedance of a capacitor is imaginary, and it is logical that the power consumed is zero, as illustrated by the sketch shown below.
From the plot for power in a capacitor, it can be seen that the average power is zero. The instantaneous power oscillates at twice the source frequency.
As in the case of an inductor, we define reactive power for a capacitor also. Reactive power, indicated by letter Q, is a negative value for capacitor, indicating that a capacitor acts as a source of reactive power. On the other hand, an inductor absorbs reactive power, and reactive power calculated for an inductor has to be positive. Reactive power for a capacitor is calculated as shown next.
The reactive power, with the unit of VAR, is calculated as follows. Express the capacitor voltage and the capacitor current as phasors, as shown by equations (186) and (187). Here the root mean square values are used as the magnitudes for phasors. Obtain apparent power, defined by letter S, as the product of voltage phasor and the conjugate of current phasor, as shown by equation (188). The result for apparent power is expressed by equation (189). It is seen that the real part is zero, and the imaginary part is the reactive power of a capacitor. We can express apparent power by the power diagram, as shown in Fig. 33. The apparent power associated with a capacitor is expressed by a complex value, as shown by equation (189). A capacitor has zero active power, and a non-zero, negative reactive power. A capacitor supplies reactive power, and hence it is negative. The definition for apparent power is such that the reactive power for a capacitor is a negative value.
The table presents the summary on passive elements in ac circuits. For each passive element, the value of its impedance, admittance, resistance, conductance, reactance and susceptance have been presented. Note that admittance Y is the reciprocal of impedance Z. Similarly conductance G is the reciprocal of resistance R, and susceptance B is the reciprocal of reactance X. The unit for resistance, reactance and impedance is Ohm, whereas the unit for conductance, susceptance and admittance is siemens. Note that impedance and admittance are complex values. The real part of impedance is resistance, and its imaginary part is reactance. Similarly real part of admittance is conductance and its imaginary part is susceptance. Inductor has negative susceptance and positive reactance, whereas the reactance of a capacitor is negative and its susceptance is positive.
The next page presents a simple ac circuit, containing a source, a resistor and a capacitor.