Examples of Transient RC and RL Circuits The Series RLC Circuit Impulse response of RC Circuit Let s examine the response of the circuit shown on Figure 1 The form of the source voltage Vs is shown on Figure 2 R C Vs vc Figure 1 RC circuit Vs Vp 0 tp t Figure 2 We will investigate the response vc t as a function of the p and Vp The general response is given by t vc t Vp 1 e RC 0 t tp 1 1 If tp RC the capacitor voltage at t tp is equal to Vp Therefore for times t tp the response becomes t tp vc t Vp e RC tp t 6 071 22 071 Spring 2006 Chaniotakis and Cory 1 2 1 A general plot of the response is shown on Figure 3 for RC 1sec tp 6 sec Vp 10Volts Figure 3 If the pulse becomes narrower the value of vc will not reach the maximum value By expanding the exponential in Equation 1 1 we obtain 2 3 t 1 t 1 t vc t Vp 1 1 0 t tp RC 2 RC 6 RC When RC 1 3 t the higher order terms may be neglected resulting in vc t Vp t 0 t tp RC 1 4 At the end of the pulse at t tp the voltage becomes vc t tp 6 071 22 071 Spring 2006 Chaniotakis and Cory Vptp RC 1 5 2 For t tp the response becomes vc t tp Vp tp RC e RC 1 6 The product Vp tp is the area of the pulse and thus the response is proportional to that area As the pulse becomes narrower i e as tp 0 equation 1 6 simplifies to vc t Vp tp RC e RC 1 7 If we constrain the area of the impulse to a constant A Vp tp then as the pulse becomes narrower the amplitude Vp increases resulting in an impulse of strength A Therefore the response of an impulse of strength A is t A RC vc e RC 1 8 Figure 4 Impulse response of RC circuit 6 071 22 071 Spring 2006 Chaniotakis and Cory 3 The spark plug in your car a simplified model Consider the circuit shown on Figure 5 The battery Vb corresponds to the 12 Volt car battery The spark plug is connected actors the inductor and current may flow though it only if the voltage across the gap of the plug exceeds a very large value about 20 kV R Vb vL L spark plug Figure 5 When the switch is closed the current through the inductor reaches a maximum value of Vb R The equation that describes the evolution of the current with the switch closed is i t t Vb L R 1 e R 1 9 And the corresponding voltage across the inductor is given by vL t Vb e t L R 1 10 When the switch is opened the current path is effectively broken and thus the time rate of change of the current becomes arbitrarily large Since the voltage is proportional to di dt the voltage developed across the inductor could become very large As an example let s consider a system with a resistance of 5 a solenoid with an inductance of 10mH connected to a 12 Volt battery How long does it take for the solenoid to reach 99 of its maximum value If the switch is opened in 1 s what is the voltage developed across the solenoid The time constant of the system is L 0 01 0 002sec R 5 The maximum current that can flow in the system is 12 A 2 4 A The time to reach 99 5 of the maximum value is given by 6 071 22 071 Spring 2006 Chaniotakis and Cory 4 t 0 99 1 e 0 002 The voltage across the coil when the switch is opened is v L i 2 4 0 01 24kV 1 10 6 t 6 071 22 071 Spring 2006 Chaniotakis and Cory 5 Response of RC circuit driven by a square wave Let s now consider the RC circuit shown on Figure 6 a driven by a square wave signal of the form shown on Figure 6 b vR R C Vs vc a Vs Vp T t Vp b Figure 6 The response vc t is given by t response final value initial value final value e 1 11 By assuming that the initial value of the voltage across the capacitor is Vp the response during the first half cycle of the square wave is vc t Vp Vp Vp e t Vp 1 2e RC t RC 1 12 During the second half cycle the initial condition is 6 071 22 071 Spring 2006 Chaniotakis and Cory 6 T 2 vc T 2 Vp 1 2e RC 1 13 And the complete response during the second half of the first cycle becomes T 2 t RC RC vc t Vp Vp 1 2e Vp e 1 14 Similarly the response during the first part of the second cycle starts with the value of vc at t T and evolves towards the value Vp If the time constant is small compared to the period of the square wave the response will reach the maximum and minimum values of the square wave as shown on Figure 7 where RC 1 10 4 sec and thus T 2 10RC Figure 7 As the time constant RC increases it takes longer for the response to reach the maximum value Figure 8 shows a plot of the response for T 2 RC Note that the response does not reach the maximum values of the input signal and the average value of the response is equal to the average value of the input signal 6 071 22 071 Spring 2006 Chaniotakis and Cory 7 Figure 8 Figure 9 a and Figure 9 b show the system response for RC 5T 2 for a square wave with a duty factor of 50 that varies between 0 and 5 Volts Notice that the average value is reached within a certain number of oscillations and that there is a variation of the response ripple about the average value The magnitude of this ripple is inversely proportional to the time constant RC This is the first step that one must take when an AC signal is converted to DC Next week when we learn about the diode we will explore this circuit further 6 071 22 071 Spring 2006 Chaniotakis and Cory 8 a b Figure 9 6 071 22 071 Spring 2006 Chaniotakis and Cory 9 Second Order Circuits Series RLC circuit The circuit shown on Figure 10 is called the series RLC circuit We will analyze this circuit in order to determine its transient characteristics once the switch S is closed S vR vL L R Vs C vc Figure 10 The equation that describes the response of the system is obtained by applying KVL around the mesh vR vL vc Vs 1 15 The current flowing …