Practical electric circuits are often constructed using multiple connected resistors.
This section explains how to calculate the equivalent resistance when multiple resistors are connected in parallel.

1.7.1 Parallel-Connected Resistors, Equivalent Resistance, and Equivalent Circuits

A circuit in which resistors R₁ and R₂ [Ω] are connected in parallel, as shown in Figure 1.7-1, is called a parallel circuit.

Figure 1.7-1 Parallel Circuit

In a parallel circuit, when a voltage V [V] is applied between points a and b, the same voltage V [V] is applied across R₁ and R₂.
Therefore, the currents I₁ and I₂ [A] flowing through the respective resistors can be calculated using the following equations.

The total current I [A] in the circuit is calculated using the following equation.

If $\frac{1}{R_1} + \frac{1}{R_2} = \frac{1}{R}\,$, then $I = \frac{V}{R}$. Therefore, the equivalent resistance R is expressed as follows.

Rearranging Equation (1.7-4) gives the following equation.

Equation (1.7-5) applies to a parallel circuit containing two resistors and is commonly described as the product-over-sum formula.

The equivalent resistance of a parallel circuit containing n resistors is expressed by the following equation.

The reciprocal of resistance, $G = \frac{1}{R}$, is called conductance and is measured in siemens (unit symbol: S). Using conductance, the equivalent conductance of a parallel circuit can simply be expressed as the sum of the conductances of the individual branches ($G = G_1 + G_2 + \cdots$).

Figure 1.7-1 can be represented by the equivalent circuit shown in Figure 1.7-2.

Figure 1.7-2 Equivalent Circuit of a Parallel Circuit

1.7.2 Current Division by Resistors

From the equivalent circuit shown in Figure 1.7-2, the voltage V [V] between points a and b can also be calculated using the following equation.

Substituting this into Equations (1.7-1) and (1.7-2) gives the following results.

For two resistors, the product-over-sum formula in Equation (1.7-5) can also be used to obtain the following expressions.

Equations (1.7-8) and (1.7-9) show that, in a parallel circuit, the current divides among the resistors in inverse proportion to their resistance values. This is called the current divider rule and is applied in a shunt resistor (shunt circuit) to extend the measurement range of an ammeter.

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Reference
・Fundamentals of Electricity, Volume I, Corona Publishing Co., Ltd., by Toshio Utsunomiya, Hiroshi Takahashi, and Isao Izumi

Chapter 1: Direct Current (DC) Circuits