Two Equivalent Representations S 1 R S closes at t 0 10 V C 2 1 vs t vs 10 0 R C 2 vC vC vs 0 0 vs 0 10 t S 1 R 5K S closes at t 0 10 V C 2 F 2 vC Find vC 0 Replace C by battery with voltage vC 0 then calculate vC 0 R 5K 1 Step 1 vC 0 5 10 V 2 vC 0 vC 0 5 Find vC t Replace C by open circuit then calculate vC vC t R 5K 1 Step 2 10 V 2 Step 3 Calculate the time constant RC vC vC t 10 V Fundamental Behavior of 1st order Circuits The voltage vjk t between any pair of nodes j and k and the current io t at any terminal of any linear resistive circuit is always an exponential waveform except for the trivial circuit consisting of a current source resp voltage source connected across a capacitor resp inductor whose solution is a linear ramp function Problem Calculate iR t under the same setting S 1 5K S closes at t 0 iC t 10 V C 2 F 2 Step 1 iR t vC Calculate iR 0 Replace C by a 5 V battery i 0 5 K 1 R vC 0 5 10 V 2 Calculate Step 2 iR 0 vC 0 vC 0 5V 5 10 1 mA 5K Calculate iR t Replace C by open circuit 1 5K iR t iC t 0 10 V 2 Calculate iR t 0 Step 3 Calculate time constant R C 5 103 2 10 6 10 ms iR 10 ms 10 20 30 40 0 0 5 1 0 63 iR 50 60 t ms t t e 10 ms t 0 t e 10 ms t 0 Verification of solution iC t iC iR t 1 0 5 0 iC t 0 63 10 20 30 40 t 1 t 10 10 3 vC t vC 0 e C 0 50 dt t t 1 10 10 3 5 e 3 0 2 10 t 3 5 5 e10 10 5 t 3 10 5e10 10 t 0 dt 60 t ms Step 3 Calculate time constant R C 5 103 2 10 6 10 ms iR 10 ms 10 20 30 40 50 60 0 0 5 1 0 63 iR t t 1i e 10 ms Verification Let us calculate iC t iR t e vC t vC 0 t 10 ms 1 t iC d 0 C t t 1 10 10 3 5 e 3 0 2 10 t 3 5 5 e10 10 5 10 5e t t 0 dt t ms t 0 Finding Solutions by Inspection S R 1 S closes at t 0 10 V C vC 2 Problem Assume vC 0 5 V where t 0 denotes the time just before switch S made contact with resistor R Sketch vC t for t 0 where t 0 denotes the instant switch S made contact with resistor R Solution vC 0 vC 0 5 V vC t 10 V R C 5 103 2 10 6 10 ms vC 10 ms 10 vC 5 0 Calculate 10 vC 20 30 t t 10 5e10 40 50 60 t t 10 5e10 t 0 t ms How to Find iL t How to Find vC t 1 iC 0 1 N vC 2 vC vC t 1 N 2 Step 1 Open capacitor Step 2 Calculate vC Open capacitor Then vC t vC Open capacitor 1 vL 0 N iL 2 Step 1 iL iL t N 2 Short capacitor Step 2 Calculate iL Short inductor Then i L t i L Short inductor How to Find vc to How to Find iL to 1 vo 1 circuit condition changed at t to vo 2 Step 1 2 Step 1 Calculate vC to Calculate iL to Step 2 vC t o circuit condition changed at t to Step 2 vC t o i L to i L to L How to Find G L R How to Find R C 1 1 Req N C voc 2 Req N L voc 2 Step 1 Thevenin equivalent circuit Find Thevenin equivalent circuit of N Step 2 Req C Step 1 Thevenin equivalent circuit Find Thevenin or Norton equivalent circuit of N Step 2 L R eq Substitution Theorem Let N be a circuit made of a nonlinear resistive one port NR terminated in an arbitrary one port NL as shown in Fig a 1 If N has a unique solution v v t for all t then NL may be substituted by a voltage source v t without affecting the branch voltage and branch current solution inside NR provided the substituted circuit Nv in Fig b has a unique solution for all t 2 If N has a unique solution i i t for all t then NL may be substituted by a current source i t without affecting the branch voltage and branch current solution inside NR provided the substituted circuit Ni in Fig c has a unique solution for all t Circuit N
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