Voltage and Current Dividers

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Contents

Voltage Division

When we have a voltage across a string of resistors connected in series, the voltage across the entire string will be divided up among the resistors. We can express the voltage across a single resistor as a ratio of voltages and resistances, without ever knowing the current.

Voltage division1.gif

In the circuit above,

\frac{v_1}{v}=\frac{R_1}{R_1+R_2}

or

v_1=\frac{R_1}{R_1+R_2}v

We can generalize this equation for N resistors in series with the equation:

v_k=\frac{R_k}{R_1+R_2+\cdot\cdot\cdot+R_N}v

where vk is the voltage across resistor k and v is the voltage across the whole string of resistors.

Current Division

Resistors in parallel divide up the current. When we have a current flowing through resistors in parallel, we can express the current flowing through a single resistor as ratio of currents and resistances, without ever knowing the voltage.

Current division1.gif

In the circuit above

\frac{i_1}{i}=\frac{R_2}{R_1+R_2}

or

i_1=\frac{R_2}{R_1+R_2}i

where i is the current flowing through all the resistors. Note that the numerator on the right is R2, not R1. Remember that a larger resistance will carry a smaller current.

We can generalize the equation for N resistors in parallel with the equation:

i_k=\frac{\frac{1}{R_k}}{\frac{1}{R_1}+\frac{1}{R_2}+\cdot\cdot\cdot+\frac{1}{R_N}}i

where ik is the current flowing through resistor k and i is the current flowing through all the resistors.

Practice Problems

Problem 1

Use voltage division to find vx in the circuit below:

Voltage division problem1.gif

click here for the solution

Problem 2

Simplify the circuit and then use current division to find ix in the circuit below:

Current division problem1.gif

click here for the solution

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