Voltage and Current Dividers
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.
In the circuit above,
We can generalize this equation for N resistors in series with the equation:
where vk is the voltage across resistor k and v is the voltage across the whole string of resistors.
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.
In the circuit above
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:
where ik is the current flowing through resistor k and i is the current flowing through all the resistors.
Use voltage division to find vx in the circuit below:
Simplify the circuit and then use current division to find ix in the circuit below: