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# Electrical Quantity Division Principles

In this chapter, let us discuss about the following two division principles of electrical quantities.

- Current Division Principle
- Voltage Division Principle

## Current Division Principle

When two or more passive elements are connected in parallel, the amount of current that flows through each element gets **divided** (shared) among themselves from the current that is entering the node.

Consider the following **circuit diagram**.

The above circuit diagram consists of an input current source **I _{S}** in parallel with two resistors

**R**and

_{1}**R**. The voltage across each element is

_{2}**V**. The currents flowing through the resistors

_{S}**R**and

_{1}**R**are

_{2}**I**and

_{1}**I**respectively.

_{2}The **KCL equation** at node **P** will be

$$I_S = I_1 + I_2$$

Substitute $I_1 = \frac{V_S}{R_1}$ and $I_2 = \frac{V_S}{R_2}$ in the above equation.

$$I_S = \frac{V_S}{R_1} + \frac{V_S}{R_2} = V_S \lgroup \frac {R_2 + R_1 }{R_1 R_2} \rgroup$$

$$\Rightarrow V_S = I_S \lgroup \frac{R_1R_2}{R_1 + R_2} \rgroup$$

Substitute the value of

*V*in $I_1 = \frac{V_S}{R_1}$._{S}

$$I_1 = \frac{I_S}{R_1}\lgroup \frac{R_1 R_2}{R_1 + R_2} \rgroup$$

$$\Rightarrow I_1 = I_S\lgroup \frac{R_2}{R_1 + R_2} \rgroup$$

Substitute the value of

*V*in $I_2 = \frac{V_S}{R_2}$._{S}

$$I_2 = \frac{I_S}{R_2} \lgroup \frac{R_1 R_2}{R_1 + R_2} \rgroup$$

$$\Rightarrow I_2 = I_S \lgroup \frac{R_1}{R_1 + R_2} \rgroup$$

From equations of *I _{1}* and

*I*, we can generalize that the current flowing through any passive element can be found by using the following formula.

_{2}$$I_N = I_S \lgroup \frac{Z_1\rVert Z_2 \rVert...\rVert Z_{N-1}}{Z_1 + Z_2 + ... + Z_N}\rgroup$$

This is known as **current division principle** and it is applicable, when two or more passive elements are connected in parallel and only one current enters the node.

Where,

*I*is the current flowing through the passive element of N_{N}^{th}branch.*I*is the input current, which enters the node._{S}*Z*are the impedances of 1_{1}, Z_{2}, …,Z_{N}^{st}branch, 2^{nd}branch, …, N^{th}branch respectively.

## Voltage Division Principle

When two or more passive elements are connected in series, the amount of voltage present across each element gets **divided** (shared) among themselves from the voltage that is available across that entire combination.

Consider the following **circuit diagram**.

The above circuit diagram consists of a voltage source, V_{S} in series with two resistors R_{1} and R_{2}. The current flowing through these elements is I_{S}. The voltage drops across the resistors R_{1} and R_{2} are V_{1} and V_{2} respectively.

The **KVL equation** around the loop will be

$$V_S = V_1 + V_2$$

Substitute

*V*and_{1}= I_{S}R_{1}*V*in the above equation_{2}= I_{S}R_{2}

$$V_S = I_S R_1 + I_S R_2 = I_S(R_1 + R_2)$$

$$I_S = \frac{V_S}{R_1 + R_2}$$

Substitute the value of

*I*in_{S}*V*=_{1}*I*._{S}R_{1}

$$V_1 = \lgroup \frac {V_S}{R_1 + R_2} \rgroup R_1$$

$$\Rightarrow V_1 = V_S \lgroup \frac {R_1}{R_1 + R_2} \rgroup$$

Substitute the value of

*I*in_{S}*V*=_{2}*I*._{S}R_{2}

$$V_2 = \lgroup \frac {V_S}{R_1 + R_2} \rgroup R_2$$

$$\Rightarrow V_2 = V_S \lgroup \frac {R_2}{R_1 + R_2} \rgroup$$

From equations of *V _{1}* and

*V*, we can generalize that the voltage across any passive element can be found by using the following formula.

_{2}$$V_N = V_S \lgroup \frac {Z_N}{Z_1 + Z_2 +....+ Z_N}\rgroup$$

This is known as **voltage division principle** and it is applicable, when two or more passive elements are connected in series and only one voltage available across the entire combination.

Where,

*V*is the voltage across N_{N}^{th}passive element.*V*is the input voltage, which is present across the entire combination of series passive elements._{S}*Z*,_{1}*Z*, …,_{2}*Z*are the impedances of 1_{3}^{st}passive element, 2^{nd}passive element, …, N^{th}passive element respectively.