Question

8.23 The initial value of the voltage v in the circuit in Fig. 8.1 is zero, and the initial value of the capacitor current, ic(0*), is 45 mA. The expression for the capacitor current is known to be i(t) = Aje-200t + Aze-8001, 1 2 0+, when R is 250 N. Find a) the values of a, wo, L, C, A1, and A2 Hint: u:... dic(0*) _ 1. dt di_(0+) dt dir(ot) dt -v(0) L 1 ic(0+) R C . b) the expression for v(t), t 2 0, c) the expression for ir(t) = 0, d) the expression for iz(t) 2 0. icl + c+V i;1 L311 izl| R Figure 8.1 A A circuit used to illustrate the natural response of a parallel RLC circuit.

Answer 1

ot 502 &d= 200 1800 $ d 500 800-200 3902 a) from Given ort) = Aretot + Ag Boot R-2500 &d= 200 1800 $ d= 1900 - 500 & also av det siste = 800-200 + de 600 * = 3002 odě = 500-20° 4 t0 = V 1600 = levo selo da ARC + c = a verso * TC = Auf tiba Pre ha de a tot 71.5625 A ] V 1600 1400 250 X2x500 1.5625 H A2 Given ic10+) = 45MA & VCO) =O A © Al to Art Ag = {clot) = 45mA ti po we known decto) = -dź (0) at at hoe - tub) = 5685 = oals. deplo) R iclos 45x103 wwb 545A)s 07 C 250 x 4x100 is dicto) = 0-45 = -45 Als, =-200 A1 -800 Aga ry - 200A1 -800 AQ 5-45- Sloving $ A1 =-15MA AQ = COM A Tic = -15 200t +60€ Boot ma boit 304

b) By hypothesis take * Pen VO) = OV & VIE) = Areabot +Ag eBoot R=250 At t=0 Vo) - Art Ag + Art Ag =0 - -= -200 A1 e 2001 - 800 Ag 2800t dt At tzo dvo) = -20041 - 80042 We Knowen duto) = I dt = Ecco) = 45X100 = 11250 dtc AX 106 dt si-200A1 - 800 Ag = 11850 - & @ A1 = 18.750 A4 = -18.75 V. o UL) = 18.75 90018.75 e 800t 207 770 &lt) = V(t) = 18.75 ©2007-18.75 8800 t 250 c) lrlt? 950 -

1 82t) = 758 200t-75 800t MA. do +20] d) { = - Ep-ic =f46230047 458 80047 157 2007- 60 2800t )m 900€ 800€ 900€ 745e + 150 60e ů = -6082002 715 78007 mA 770