Current divider rules and formula

Current divider rules and current divider formula :

A general formula for a current divider is given by

Where,

I_X = Current through any resistor in the parallel circuit =
I_T= Total current of the circuit =
R_T = Equivalent resistance of the parallel circuit
V = Voltage across the parallel circuit = I_TR_T = I_XR_X (as the voltage is same across all the components of the parallel circuit)

In terms of impedance, the formula for a current divider is given by

In terms of admittance, the formula for a current divider is given by

Current Divider Formula for RC Parallel Circuit

Apply current divider rule to the above circuit, the current through the resistor is given by,

Where, Z_C = Impedance of the capacitor

Thus we get,

Current Divider Rule Derivation

Consider a parallel circuit of two resistors R₁and R₂ connected across a supply voltage source of V volts.

Assume that the total current entering the parallel combination of resistors is I_T. The total current I_T divides into two parts I₁and I₂where I₁ is the current flowing through the resistor R₁and I₂ is the current flowing through the resistor R_2.

Therefore, the total current is

I_T = I₁ + I₂…………………………………………………. (1)

I₁ = I_T– I₂ …………………………………………………. (2)

I₂= I_T– I₁ …………………………………………………. (3)

Now, when two resistors are connected in parallel, the equivalent resistor R_eq is given by

R_eq = R₁ parallel to R₂

Now according to the ohm’s law i.e. I = V/R, the current flowing through the resistor R₁ is given by

I₁ = V/R₁

V = I₁R₁ …………………………………………… (5)

Similarly, the current flowing through the resistor R₂ is given by

I₂ = V/R₂

V = I₂R₂ …………………………………………… (6)

compare equation (5) and (6) we get,

V = I₁R₁ = I₂R₂

I₁ = I₂(R₂/R₁)

Put this value of I₁into equation (1) we get,

…………………..(7)

Now put this equation of I₂ into equation (2), we get

………(8)

Thus, from equation (7) and (8) we can say that the current in any branch is equal to the ratio of opposite branch resistance to the total resistance value, multiplied by the total current in the circuit.

In general we can say that,