Differential Equations and Lumped Element Circuits 8 Introduction Chapter 8 of the text discusses the numerical solution of ordinary differential equations. Differential equations and in particular linear constant coefficient differential equations (LCCDEs), are very important to electrical engineers. Electrical engineering (EE) majors take an entire semester of differential equations. First semester EE majors have had little or no exposure to differential equations. To motivate the future study of differential equations this short overview chapter will describe how LCCDEs appear in the solution of lumped element circuits problems. The Time Domain Response of RLC Circuits A lumped element circuit is composed of an interconnection of resistors, capacitors, and inductors Chapter 8: Introduction 8 1
The voltage current terminal relationships for these elements are shown below Resistor v R = Ri R i R = 1 ---v R R i R v + R - R Inductor v L L di L = ------------- 1 t i L = -- v L L ( λ) dλ i L + v L - L Capacitor i C C dv C = --------------- 1 t v C = --- i C C ( λ) dλ i C + v C - C When an RLC electrical network is configured to have a single input and single output (SISO) voltage (current) xt () RLC yt () Network voltage (current) Chapter 8: The Time Domain Response of RLC Circuits 8 2
Assuming the network is composed of n reactive elements, that is the number of inductors plus the number of capacitors is n, it can be shown that the output yt () and input xt () are related via an nth order LCCDE d n yt () a n -------------- d n 1 yt () n + a n 1 --------------------- n 1 + + a 0 yt () d n xt () b n -------------- d n 1 xt () = n + b n 1 --------------------- n 1 + + b 0 xt () (8.1) Example: An RLC lowpass filter R L v + i it () v - C o Using the terminal relationships defined above we can write that Note that v i Ri L dit ----------- () 1 = + + --- i( λ) dλ C it () C dv o = -------------- t (8.2) (8.3) Chapter 8: The Time Domain Response of RLC Circuits 8 3
Following substitution of (8.3)into (8.2) we obtain or v i RC dv o -------------- LC d2 v o = + ----------------- 2 + v o LCv o ''t () + RCv o ' + v o = v i (8.4) (8.5) To solve (8.5) numerically we can use the MATLAB function lsim(sys,u,t) (control system toolbox, also in student edition) LSIM(SYS,U,T) plots the time response of the LTI system SYS to the input signal described by U and T. The time vector T consists of regularly spaced time samples and U is a matrix with as many columns as inputs and whose i- th row specifies the input value at time T(i). For instance, t = 0:0.01:5; u = sin(t); lsim(sys,u,t) simulates the response of SYS to u(t) = sin(t) during 5 seconds. The input SYS is a system object that can be created to correspond to (8.5) using the MATLAB function tf(b,a) (control system toolbox, also in student edition) The vectors a and b are of the form a = [an an-1 an-2... a0]; b = [bn bn-1 bn-2... b0]; For the RLC circuit considered here a = [LC RC 1]; b = [1]; Chapter 8: The Time Domain Response of RLC Circuits 8 4
The waveform in vector U can be any time varying function represented in sampled form Test inputs can be generated using the MATLAB function GENSIG (control system toolbox, also in student edition) GENSIG Periodic signal generator for time response simulations with LSIM. [U,T] = GENSIG(TYPE,TAU) generates a scalar signal U of class TYPE and period TAU. The following classes are supported: TYPE = 'sin' --- sine wave TYPE = 'square' --- square wave TYPE = 'pulse' --- periodic pulse GENSIG returns a vector T of time samples and the vector U of signal values at these samples. All generated signals have unit amplitude. [U,T] = GENSIG(TYPE,TAU,TF,TS) further specifies the time duration TF of the signal and the spacing TS of the time samples in T. Consider the system response for R = 1 ohm, L = 5 henries, and C = 3 farads We will apply a squarewave with a 200 second period for 500 seconds» [u,t] = gensig('square',200,500,1);» R=1; L=5; C=3; lsim(tf([1],[c*l C*R 1]),u,t)» % A labeled output plot is automatically generated Chapter 8: The Time Domain Response of RLC Circuits 8 5
1.4 1.2 Linear Simulation Results v o R=1; L=5; C=3 1 Amplitude 0.8 0.6 0.4 0.2 0-0.2-0.4 0 50 100 150 200 250 300 350 400 450 500 Time (sec.) By increasing the resistance the overshoot can be reduced, but the risetime is increased Chapter 8: The Time Domain Response of RLC Circuits 8 6
Linear Simulation Results 1.2 v o R=2; L=5; C=3 1 0.8 Amplitude 0.6 0.4 0.2 0-0.2 0 50 100 150 200 250 300 350 400 450 500 Time (sec.) Chapter 8: The Time Domain Response of RLC Circuits 8 7