Lecture 09 Reactive Circuit Elements
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1E6 Electrical Engineering AC Circuit Analysis and Power Lecture 09: Reactive Circuit Elements 9.1 Introduction In the previous analysis of dc circuits all voltages and currents were constant, not varying with time. There was no profile or waveform associated with the source driving an electric circuit. Thus the question of vectors having magnitude and phase did not arise. However, in the case of the analysis of ac circuits, the concept of voltages and currents as vectors is fundamental as the sources driving circuits will essentially be sinusoidal or at least time-varying with periodic properties. Moreover, the circuit elements are not purely resistive, but will include capacitors and inductors which have the property of impedance. This means that they behave in a manner that interacts with the magnitude and phase of the voltages and currents present in a circuit and alter the magnitude and phase of both. It is therefore of interest to characterise the current-voltage relationships of these components in the context of ac electricity. 9.2 Resistive Circuit Elements: In the case of a resistor, under steady-state conditions, if a sinusoidal voltage is placed across it, then the current that flows through the resistor is also sinusoidal and has precisely the same phase as the applied voltage signal. Conversely, if a sinusoidal current is passed through a resistor the resulting voltage which is developed across it is sinusoidal and has the same phase as the source current. The ac voltage and current associated with a resistor are therefore always in phase, as shown in Fig. 1. Imag j V V iR + R I I θ vR t Real -j Fig. 1 Sinusoidal Current and Voltage Associated with a Resistor 1 If the voltage across the resistor is taken as a sinewave so that: vR = Vm Sin ωt Then the current is given as: iR = Vm Sin ωt = I m Sin ωt R The resistance defines the current-voltage relationship as: R= vR Vm Sinωt Vm = = iR I m Sinωt I m If the voltage across the resistor is defined as a general complex vector then: vR = Vm e jωt Then the current will have the same form of complex vector so that: iR = Vm jωt e = I m e jω t R and: vR Vm e jωt Vm R= = = jωt iR I me Im It can be seen that the resistance is a scalar quantity which does not vary with either time or frequency. The value of resistance is shown in Table 1 below for a range of values of voltage and current. Vm Table 1 Values of Resistance for a Range of Voltages and Currents 1V 10V 200V 1V 10V 1V 10V 1V 10V Im 1A 1A 20A 1mA 1mA 1µA 1µA R 1Ω 10Ω 10Ω 1kΩ 100Ω 1MΩ 100kΩ 100kΩ 1MΩ 2 10µA 10µA 9.3 Capacitive Reactance In the case of a capacitor, however, the situation is somewhat different. The voltage developed between the plates of a capacitor is as a result of charge accumulated over time as current flows to supply or remove charge on the plates. This has the characteristic relationship shown in Fig. 2. VC - + iC = C C iC dv C dt Fig. 2 A Capacitor and its Characteristic Equation If the voltage across the capacitor is taken as sine function then: vC = Vm Sin ωt and iC = CVmω Cos ωt or π iC = CVm ω Sin ωt + 2 This shows that in a capacitor with sinusoidal steady-state voltage excitation the current leads the voltage by 90º. This is illustrated in Fig. 3 where the voltage is taken as a sine function and the resulting current becomes a cosine function j VC ω IC VC ω t IC -j Fig. 3 The Relationship Between Current and Voltage in a Capacitor 3 If the voltage across the capacitor is described as a complex vector then: vC = Vm e jωt = Vm (Cos ωt + jSin ωt ) so that: iC = C dvC = −CVmωSinωt + jCVmωCos ωt ) dt iC = ωCVm ( jCos ωt − Sin ωt ) iC = ωCVm ( jCos ωt + j 2 Sinωt ) iC = jωCVm (Cos ωt + jSin ωt ) iC = jωCVm e jωt = jI m e jωt The presence of the j term in the result indicates a phase advancement of 90o in the complex plane. This is consistent with the result obtained above where the exciting voltage was treated as a singular sine function. The relationship between the current and the voltage in a reactive element is defined as the impedance of the element, designated ZC. This is therefore defined for the capacitor as: vC Vm e jωt Vme jωt ZC = = = iC jI m e jωt jωCVme jωt so that: ZC = 1 1 =−j jω C ωC It should be noted that the impedance is a complex parameter and has both magnitude and phase. The j term is associated with the impedance and indicates the fact that current leads voltage by 90o or that voltage lags current by 90o. This is often a point of confusion in Electronic Engineering. 4 The property of reactance is also defined for a capacitor as: XC = vC Vm V 1 = = m = iC I m CVmω ωC Note: The reactance of a capacitor is a scalar quantity and has units of Ohms. It is not complex or imaginary. In addition, the reactance of the capacitor is dependent on the frequency of the sinusoidal excitation. The value of the reactance decreases with increasing frequency as shown in Fig. 4. Consequently, for a given excitation voltage a greater current will flow through the capacitor at higher frequencies. The values of capacitive reactance for ranges of capacitance and frequency are given in Table 2 below. XC ω→0 XC → ∞ ω→∞ XC → 0 ω Fig. 4 Capacitive Reactance as a Function of Frequency. Table 2 Values of Reactance for a Range of Capacitance and Frequency. Frequency Capacitance 1pF 10Hz 50Hz 100Hz 1kHz 10kHz 100kHz 1MHz 10MHz 100MHz 15.9GΩ 3.19GΩ 1.59GΩ 159MΩ 15.9MΩ 1.59MΩ 159kΩ 15.9kΩ 1.59kΩ 10pF 1.59GΩ 319MΩ 159MΩ 15.9MΩ 1.59MΩ 159kΩ 15.9kΩ 1.59kΩ 159Ω 100pF 159MΩ 31.9MΩ 15.9MΩ 1.59MΩ 159kΩ 15.9kΩ 1.59kΩ 159Ω 15.9Ω 1nF 15.9MΩ 3.19MΩ 1.59MΩ 159kΩ 15.9kΩ 1.59kΩ 159Ω 15.9Ω 1.59Ω 10nF 1.59MΩ 319kΩ 159kΩ 15.9kΩ 1.59kΩ 159Ω 15.9Ω 1.59Ω 159mΩ 100nF 159kΩ 31.9kΩ 15.9kΩ 1.59kΩ 159Ω 15.9Ω 1.59Ω 159mΩ 15.9mΩ 1µF 15.9kΩ 3.19kΩ 1.59kΩ 159Ω 15.9Ω 1.59Ω 159mΩ 15.9mΩ 1.59mΩ 10µF 1.59kΩ 319Ω 159Ω 15.9Ω 1.59Ω 159mΩ 15.9mΩ 1.59mΩ 159µΩ 100µF 159Ω 31.9Ω 15.9Ω 1.59Ω 159mΩ 15.9mΩ 1.59mΩ 159µΩ 15.9µΩ 1000µF 15.9Ω 3.19Ω 1.59Ω 159mΩ 15.9mΩ 1.59mΩ 159µΩ 15.9µΩ 1.59µΩ 5 9.3 Inductive Reactance In the case of an inductor, when current flows through it a back emf is developed in a direction which opposes a change in current flow. This back emf is also opposite to the direction of the potential drop across the inductor caused by the external applied electric field which gives rise to the current flow. The characteristic relationship between current and voltage in an inductor is give in Fig. 5 below. VL - + iL vL = L diL dt L Fig. 5 An Inductor and its Characteristic Equation. This time if the current is taken as a negative cosine function then: iL = − I m Cos ωt vL = L I m ω Sin ωt so that: π vL = − L I mω Cos ωt + 2 This shows that in an in an inductor the voltage leads the current by 90º or the current lags the voltage by 90º under steady state sinusoidal excitation conditions. This is shown in Fig. 6 below where the voltage is as a sine function for comparison with the case of the capacitor. j VL VL IL ω ω t IL -j Fig. 6 The Relationship between Current and Voltage in an Inductor. 6 If the current through the inductor is described as a complex vector then: i L = I m e jω t so that: vL = L diL = LI m jωe jωt dt vL = j ωLI m e jωt = jVm e jωt The presence of the j term in the result indicates a phase advancement of 90o in the complex plane. This is consistent with the result obtained above as it shows the voltage to be advanced by 90o in phase compared with the current. Alternatively, the current can be considered to lag behind the voltage by 90o. The relationship between the current and the voltage in the inductive element is again defined as its impedance, designated ZL and given as: vL jVm e jωt jωLI m e jωt ZL = = = jω t iL I me I m e jω t so that: Z L = jω L It is noted again that the impedance is a complex parameter having both magnitude and phase. The j term is associated with the impedance and indicates in this case that current lags the voltage by 90o or that the voltage leads the current by 90o. The property of reactance can also be defined for an inductor as: XL = vL iL = Vm L I m ω = = ωL Im Im The reactance of an inductor is also a scalar quantity and has units of Ohms. It is not complex or imaginary. The reactance of the inductor is also dependent on the frequency of the sinusoidal excitation. In the case of the inductor, however, the magnitude of the reactance increases with frequency so that for a given excitation voltage a smaller current will flow at higher frequencies as shown in 7 Fig. 7. The values of inductive reactance for ranges of inductance and frequency are given in Table 3 below. XL ω→0 XL → 0 ω→∞ XL → ∞ ω Fig. 7 Inductive Reactance as a Function of Frequency. Table 3 Values of Reactance for Ranges of Inductance and Frequency. Frequency Capacitance 1µH 10Hz 50Hz 100Hz 1kHz 10kHz 100kHz 1MHz 10MHz 100MHz 314µΩ 1.57mΩ 3.14mΩ 31.4mΩ 314mΩ 3.14kΩ 31.4kΩ 314kΩ 3.14MΩ 10µH 3.14mΩ 15.7mΩ 31.4mΩ 314mΩ 3.14kΩ 31.4kΩ 314kΩ 3.14MΩ 31.4MΩ 100µH 31.4mΩ 157mΩ 314mΩ 3.14kΩ 31.4kΩ 314kΩ 3.14MΩ 31.4MΩ 314MΩ 1mH 314mΩ 1.57Ω 3.14kΩ 31.4kΩ 314kΩ 3.14MΩ 31.4MΩ 314MΩ 3.14GΩ 10mH 3.14Ω 15.7Ω 31.4kΩ 314kΩ 3.14MΩ 31.4MΩ 314MΩ 3.14GΩ 31.4GΩ 100mH 31.4Ω 157Ω 314kΩ 3.14MΩ 31.4MΩ 314MΩ 3.14GΩ 31.4GΩ 314GΩ 1H 314Ω 1.57kΩ 3.14MΩ 31.4MΩ 314MΩ 3.14GΩ 31.4GΩ 314GΩ 3.14TΩ 10H 3.14kΩ 15.7kΩ 31.4MΩ 314MΩ 3.14GΩ 31.4GΩ 314GΩ 3.14TΩ 31.4TΩ A convenient way to remember the phase relationships between voltage and current in the capacitor and the inductor is given as: C I V I L C capacitor current leads L voltage 8 current lags inductor
