When solving for voltage, current, and resistance in a series-parallel circuit, follow the rules which apply to the series part of the circuit, and follow the rules which apply to the parallel part of the circuit. Solving these circuits can be simplified by reducing the circuit to a single equivalent resistance circuit, and redrawing the circuit in simplified form. The circuit is then called an equivalent circuit (Figure 46).

Figure 46 Redrawn Circuit Example

The easiest way to solve these types of circuits is to do it in steps.

Step 1:Find the equivalent resistance of the parallel branch:

Step 2:Find the resistance of the equivalent series circuit:

Step 3:Find total current (I_T):

Step 4:Find I₂ and 13. The voltage across R₁ and R_Z is equal to the applied voltage (V), minus the voltage drop across R_l.

Then, I₂ and 1₃ are calculated.

Y and Delta Network Calculation

Because of its shape, the network shown in Figure 47 is called a T (tee) or Y (wye) network. These are different names for the same network.

Figure 47 T or Y Network

The network shown in Figure 48 is called (pi) or (delta) because the shapes resemble Greek letters and . These are different names for the same network.

Figure 48 (pi) or (delta) Network

In order to analyze the circuits, it may be helpful to convert Y to , or to Y, to simplify the solution. The formulas that will be used for these conversions are derived from Kirchhoff's laws. The resistances in these networks are shown in a three-terminal network. After we use the conversion formulas, one network is equivalent to the other because they have equivalent resistances across any one pair of terminals (Figure 49).

Rule 1: The resistance of any branch of a Y network is equal to the product of the two adjacent sides of a network, divided by the sum of the three resistances.

Figure 49 Y - Equivalent

Rule 2: The resistance of any side of a network is equal to the sum of the Y network resistance, multiplied in pairs, divided by the opposite branch of the Y network.

Let us consider a bridge circuit (Figure 50).

Figure 50 Bridge Circuit

Find R_t at terminals a and d.

Figure 51 Y - A Redrawn Circuit

Step 3:Reduce and simplify the circuit. Note that the 20 and 60 branches are in parallel in Figure 51. Refer to Figures 51 and 52 for redrawing the circuit in each step below.

Figure 52 Steps to Simplify Redrawn Circuit

Summary

The important information in this chapter is summarized below.

DC Circuit Analysis Summary

The current flow at any element in a DC circuit can be determined using loop equations.

The voltage at any point in a DC circuit can be determined using node equations.

The equivalent resistance of series and parallel combinations of elements can be used to simplify DC circuit analysis.