20.8

Circuits Wired Partially in Series and Partially in Parallel

Often an electric circuit is wired partially in series and partially in parallel. The key to determining the current, voltage, and power in such a case is to deal with the circuit in parts, with the resistances in each part being either in series or in parallel with each other. Example 12 shows how such an analysis is carried out.

| Example 12 | | | A Four-Resistor Circuit

Figure 20-24 shows a circuit composed of a 24-V battery and four resistors, whose resistances are 110, 180, 220, and . Find (a) the total current supplied by the battery and (b) the voltage between points A and B in the circuit. Reasoning The total current that is supplied by the battery can be obtained from Ohm’s law, I = V/R, where R is the equivalent resistance of the four resistors. The equivalent resistance can be calculated by dealing with the circuit in parts. The voltage VAB between the points A and B is also given by Ohm’s law, VAB = IRAB, where I is the current and RAB is the equivalent resistance between the two points.
Solution

a. | The resistor and the resistor are in series, so they are equivalent to a single resistor whose resistance is (see Figure 20-24a). The resistor is in parallel with the resistor. Their equivalent resistance can be obtained from Equation 20.17:

The circuit is now equivalent to a circuit containing a resistor in series with a resistor (see Figure 20-24b). This combination behaves like a single resistor whose resistance is (see Figure 20-24c). The total current from the battery is, then,

b. | The current I = 0.10 A passes through the resistance between points A and B. Therefore, Ohm’s law indicates that the voltage across the resistor between points A and B is

Figure 20-24 The circuits shown in this picture are equivalent.

Check Your Understanding 3

You have three identical resistors. By connecting all three together in various ways, how many different values of the total resistance can you obtain, and what are those values? Express your answers in terms of R, the resistance of one of the resistors. Note that some of the values are greater than R and some are less than R. (The answers are given at the end of the book.)
Background: The concepts of series and parallel wiring hold the key to this question. A given combination of resistors may be part series and part parallel.
For similar questions (including calculational counterparts), consult Self-Assessment Test 20.2, which is described at the end of Section 20.13.

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