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Analysis of resistive circuits

Two electric circuits are said to be equivalent with respect to a pair of terminals if the voltages across the terminals and currents through the terminals are identical for both networks.

If <math>V_1=V_2<math> and <math>I_1=I_2<math>, then with respect to terminals ab and xy, circuit 1 and circuit 2 are equivalent.

Resistors in series and in parallel

Resistors in series: <math>R_\mathrm{eq} = R_1 + R_2 + \,\cdots\, + R_n.<math>

Resistors in parallel: <math>\frac{1}{R_\mathrm{eq}} = \left( \frac{1}{R_1} \right) + \left( \frac{1}{R_2} \right) + \,\cdots\, + \left( \frac{1}{R_n} \right).<math>

Special case: Two resistors in parallel: <math>R_\mathrm{eq} = \left( \frac{R_1R_2}{R_1 + R_2} \right).<math>

Delta-wye transformation

The transformation is used to establish equivalence for networks with 3 terminals.

For equivalence, the resistance between any pair of terminals must be the same for both networks.

Delta-to-wye transformation equations

<math>R_1 = \left( \frac{R_aR_b}{R_a + R_b + R_c} \right)<math>

<math>R_2 = \left( \frac{R_bR_c}{R_a + R_b + R_c} \right)<math>

<math>R_3 = \left( \frac{R_cR_a}{R_a + R_b + R_c} \right)<math>

Wye-to-delta transformation equations

<math>R_a = \left( \frac{R_1R_2 + R_2R_3 + R_3R_1}{R_2} \right)<math>

<math>R_b = \left( \frac{R_1R_2 + R_2R_3 + R_3R_1}{R_3} \right)<math>

<math>R_c = \left( \frac{R_1R_2 + R_2R_3 + R_3R_1}{R_1} \right)<math>

Voltage and current division

If the two networks are equivalent with respect to terminals ab, then V and I must be identical for both networks. Thus

<math>V = RI_s<math> or <math>I_s = \left( \frac{V_s}{R} \right)<math>

Nodal analysis

General procedure

1. Label all nodes in the circuit. Arbitrarily select any node as reference.

2. Define a voltage variable from every remaining node to the reference. These voltage variables must be defined as voltage rises with respect to the reference node.

3. Write a KCL equation for every node except the reference.

4. Solve the resulting system of equations.

Mesh analysis

General procedure

Mesh - a loop that does not contain an inner loop.

1. Count the number of ?window panes? in the circuit. Assign a mesh current to each window pane.

2. Write a KVL equation for every mesh whose current is unknown.

3. Solve the resulting equations.

Choice of method

Given the choice, which method should be used? Nodal analysis or mesh analysis?

Y-delta transform

• Kirchhoff's circuit laws