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ch05 - Attia John Okyere Transient Analysis Electronics and...

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Attia, John Okyere. “Transient Analysis.” Electronics and Circuit Analysis using MATLAB. Ed. John Okyere Attia Boca Raton: CRC Press LLC, 1999 © 1999 by CRC PRESS LLC
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CHAPTER FIVE TRANSIENT ANALYSIS 5.1 RC NETWORK Considering the RC Network shown in Figure 5.1 , we can use KCL to write Equation (5.1). R C V o (t) Figure 5.1 Source-free RC Network C dv t dt v t R o o ( ) ( ) + = 0 (5.1) i.e., dv t dt v t CR o o ( ) ( ) + = 0 If V m is the initial voltage across the capacitor, then the solution to Equation (5.1) is v t V e m t CR 0 ( ) = (5.2) where CR is the time constant Equation (5.2) represents the voltage across a discharging capacitor. To obtain the voltage across a charging capacitor, let us consider Figure 5.2 . © 1999 CRC Press LLC
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V o (t) R C V s Figure 5.2 Charging of a Capacitor Using KCL, we get C dv t dt v t V R o o s ( ) ( ) + = 0 (5.3) If the capacitor is initially uncharged, that is v t 0 ( ) = 0 at t = 0, the solution to Equation (5.3) is given as v t V e S t CR 0 1 ( ) = (5.4) Examples 5.1 and 5.2 illustrate the use of MATLAB for solving problems related to RC Network. Example 5.1 Assume that for Figure 5.2 C = 10 µ F, use MATLAB to plot the voltage across the capacitor if R is equal to (a) 1.0 k , (b) 10 k and (c) 0.1 k . Solution MATLAB Script % Charging of an RC circuit % c = 10e-6; r1 = 1e3; © 1999 CRC Press LLC
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tau1 = c*r1; t = 0:0.002:0.05; v1 = 10*(1-exp(-t/tau1)); r2 = 10e3; tau2 = c*r2; v2 = 10*(1-exp(-t/tau2)); r3 = .1e3; tau3 = c*r3; v3 = 10*(1-exp(-t/tau3)); plot(t,v1,'+',t,v2,'o', t,v3,'*') axis([0 0.06 0 12]) title('Charging of a capacitor with three time constants') xlabel('Time, s') ylabel('Voltage across capacitor') text(0.03, 5.0, '+ for R = 1 Kilohms') text(0.03, 6.0, 'o for R = 10 Kilohms') text(0.03, 7.0, '* for R = 0.1 Kilohms') Figure 5.3 shows the charging curves. Figure 5.3 Charging of Capacitor © 1999 CRC Press LLC
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From Figure 5.3 , it can be seen that as the time constant is small, it takes a short time for the capacitor to charge up. Example 5.2 For Figure 5.2 , the input voltage is a rectangular pulse with an amplitude of 5V and a width of 0.5s. If C = 10 µ F, plot the output voltage, v t 0 ( ) , for resistance R equal to (a) 1000 , and (b) 10,000 . The plots should start from zero seconds and end at 1.5 seconds. Solution MATLAB Script % The problem will be solved using a function program rceval function [v, t] = rceval(r, c) % rceval is a function program for calculating % the output voltage given the values of % resistance and capacitance. % usage [v, t] = rceval(r, c) % r is the resistance value(ohms) % c is the capacitance value(Farads) % v is the output voltage % t is the time corresponding to voltage v tau = r*c; for i=1:50 t(i) = i/100; v(i) = 5*(1-exp(-t(i)/tau)); end vmax = v(50); for i = 51:100 t(i) = i/100; v(i) = vmax*exp(-t(i-50)/tau); end end % The problem will be solved using function program % rceval % The output is obtained for the various resistances c = 10.0e-6; r1 = 2500; © 1999 CRC Press LLC
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