4.4. Combining Resistors

Complex circuit diagrams can be combined by grouping series and parallel resistors. At each step of the process either a group of resistors in series or a group of resistors in parallel can be simplified and represented as one resitor. At the end of the simplificiation, one is left with a circuit diagram that contains one voltage source and one total resistance.

Resistors in Series

The total resistance of multiple resistors wired in series is the sum of the resistance of the individual resistors.

\[ R_t = R_1 + R_2 + R_3 + \ldots + R_n \]

Resistors connected in series all have the same current flowing through them. In the equation below, \(I_1\) is the current flowing through resistor \(R_1\) connected in series.

\[ I_t = I_1 = I_2 = I_3 = \ldots = I_n \]

The voltage drop accross a resistors connected in series sum together to produce the total voltage drop accross the assembly.

\[ V_t = V_1 + V_2 + V_3 + \ldots + V_n \]

Worked Example

GIVEN:

Three resistors are connected in series.

\(R_1 = 10 \ \Omega\), \(R_2 = 20 \ \Omega\), \(R_3 = 30 \ \Omega\)

FIND:

The total resistance of the three resistors connected together, \(R_t\)

SOLUTION:

\[ R_t = R_1 + R_2 + R_3 \]

\[ R_t = 10 \ \Omega + 20 \ \Omega + 30 \ \Omega \]

\[ R_t = 50 \ \Omega \]

Resistors in Parallel

The sum of the reciprocal of the resistance of resistors connected in parallel is the reciprocal of the total resistance.

\[ \frac{1}{R_t} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \ldots + \frac{1}{R_n} \]

The equation above can be re-written to solve for \(R_t\). Note that raising any term to the -1 power is the same as putting that term in the denominator of a fraction.

\[ R_t = \left[\frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \ldots + \frac{1}{R_n}\right]^{-1} \]

Resistors connected in parallel all have the same voltage drop across them.

\[ V_t = V_1 = V_2 = V_3 = \ldots = V_n \]

The current flow through resistors connected in parallel sum together to produce the total current flowing though the assembly.

\[ I_t = I_1 + I_2 + I_3 + \ldots + I_n \]

Worked Example

GIVEN:

Three resistors are connected in parallel.

\(R_1 = 10 \ \Omega\), \(R_2 = 20 \ \Omega\), \(R_3 = 30 \ \Omega\)

FIND:

The total resistance of the three resistors connected together, \(R_t\)

SOLUTION:

\[ \frac{1}{R_t} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \]

\[ \frac{1}{R_t} = \frac{1}{10 \ \Omega} + \frac{1}{20 \ \Omega} + \frac{1}{30 \ \Omega} \]

\[ \frac{1}{R_t} = \frac{6}{60 \ \Omega} + \frac{3}{60 \ \Omega} + \frac{2}{60 \ \Omega} = \frac{11}{60 \ \Omega} \]

\[ R_t = \frac{60}{11} \ \Omega \approx 5.45 \ \Omega \]

Complex Circuit Diagrams

Complex circuit digrams containing many resistors can be simplied into a circuit with just one resistor. Simplification is accomplished by combining resistors in series and in parellel squentially until there is only one resistor which respresents the total resistance of the entire assembly.