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circuits

# circuits - Kirchoffs Laws Direct KCL KVL Ohms Law V = IR VG...

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Kirchoff’s Laws Direct: KCL, KVL, Ohm’s Law V IR VG I = = G R = 1 Ohm’s Law: V I I 1 1 2 3 6 = = V I 2 3 12 = (always get 1 equation/Resistor) KCL: A: - + B: 5 0 5 0 1 2 3 1 2 3 I I I I I I + = + = eq. are dependent (in general, get n -1 indep. for nodes) KVL: + = V V 1 2 24 0 write 1 loop equation for each loop with a voltage not in the current set of equations. Eliminate either V 1 or I using Ohm’s Law eq: A: + + = 5 3 6 12 0 1 1 2 V V V = + 1 3 1 6 1 12 1 2 1 1 5 24 V V KVL: + = V V 1 2 24 0 Cramer’s Rule: A X B [ ] = v v X Det A B A Det A i i i N = [ ] [ ] + 0 1 1 .. .. 5 24 V + - 6 3 V 1 V 2 + I 1 I 2 - + 12 I 3 A B

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We can always write in terms of only V , or I variables using ohm’s law: KCL: A : + + = I I I 1 2 3 0 KVL: M 1 : + + = 12 2 4 0 1 2 I I M 2 : + = 4 6 8 0 2 3 1 I I I = 1 1 1 2 4 0 8 4 6 0 12 0 1 2 3 I I I Note: if V IR = R G = 1 V I G R + B 2 6 4 12 V V 1 V 3 8 1 I A M 1 M 2 V 2 + + + - - + I 3 I 2 I 1 Current Dependent Voltage Source: +
Rules for nodes: 1) Convert all voltage to current sources 1) Determine a reference node and identify unknown relative voltages 2) Use KCL at each unknown node: at node A: G V G V G V I AA A AB B AN N A = .... B: - G V G V G V I AB A BB B BN N B + = .... N: - G V G V G V I AN A BN B NN N N + = .... where: G ii = Σ all conductances connected to node i G ji = Σ all conductances between node i and node j I i = Σ all current sources connected to node i Define N equations in N unknowns all other voltages and currents by Ohm’s Law from V and S . 2 2 2 2 1 1 1 R V 10 5V 1 1 R V R + - I 1 2 A 10 1 Ω = . eg: 4 2 5 4 10 20 + - 9 A 12 V Eq: 3 A 1 4 Ref 1 10 1 20 1 4 1 2 1 5 9 A A B C

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A: 0 25 5 25 0 5 0 25 3 9 . . . . . + + ( ) = V V V A B C ( A ) B: 0 5 . V A + + + ( ) = 0 5 0 1 0 2 02 0 . . . . V V B C C: .25 V A + + + ( ) = 0 2 0 2 0 25 0 05 9 . . . . V V B C 1 0 0 5 2 5 5 8 2 25 2 5 6 0 9 . . .
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circuits - Kirchoffs Laws Direct KCL KVL Ohms Law V = IR VG...

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