The total resistance of a series-parallel circuit is the sum of the resistances of all the individual resistors in the circuit. In other words, the total resistance is equal to the product of the individual resistances and the sum of the reciprocals of the individual resistances.

The formula for the total resistance of a series-parallel circuit is given by:

$$R_T = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2} + \ldots + \frac{1}{R_n}}$$

where:

* $R_T$ is the total resistance of the circuit * $R_1$, $R_2$, ..., $R_n$ are the resistances of the individual resistors in the circuit

To use this formula, you simply need to add the reciprocals of the individual resistances and then take the reciprocal of the sum. For example, if you have a circuit with three resistors with resistances of 10 ohms, 20 ohms, and 30 ohms, the total resistance of the circuit would be:

$$R_T = \frac{1}{\frac{1}{10} + \frac{1}{20} + \frac{1}{30}} = \frac{60}{11} \approx 5.45 \Omega$$

The total resistance of a series-parallel circuit can be used to calculate the current flowing through the circuit, the voltage drop across the circuit, or the power dissipated by the circuit.

In addition to the total resistance, there are two other important concepts to understand when working with series-parallel circuits:

*

### The equivalent resistance

is the resistance of a single resistor that would have the same effect on the circuit as all of the individual resistors combined. *### The current division ratio

is the ratio of the current flowing through one resistor to the total current flowing through the circuit.The equivalent resistance of a series-parallel circuit can be calculated using the following formula:

$$R_E = \frac{R_1R_2}{R_1 + R_2}$$

where:

* $R_E$ is the equivalent resistance of the circuit * $R_1$ and $R_2$ are the resistances of the two resistors in the circuit

The current division ratio for a series-parallel circuit can be calculated using the following formula:

$$I_1 = \frac{R_2}{R_1 + R_2}I_T$$

where:

* $I_1$ is the current flowing through resistor 1 * $I_2$ is the current flowing through resistor 2 * $I_T$ is the total current flowing through the circuit

Series-parallel circuits can be used to create a variety of different circuit configurations, each with its own unique set of properties. By understanding the total resistance, equivalent resistance, and current division ratio of a series-parallel circuit, you can design circuits that meet your specific needs.

Here are some additional resources that you may find helpful:

* [The Physics Classroom: Series-Parallel Circuits](https://www.physicsclassroom.com/class/circuits/Lesson-1/Series-Parallel-Circuits) * [Khan Academy: Series and Parallel Circuits](https://www.khanacademy.org/science/physics/electricity-and-magnetism/circuits-topic/a/series-and-parallel-circuits) * [SparkFun: Series and Parallel Circuits](https://learn.sparkfun.com/tutorials/series-and-parallel-circuits)

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