# 12. Parallel AC Circuits

Recall Ohm's law for pure resistances:

V = IR

In the case of AC circuits, we represent the impedance (effective resistance) as a complex number, Z. The units are ohms (Ω).

In this case, Ohm's Law becomes:

V = IZ.

Recall also, if we have several resistors (R1, R2, R3, R4, …) connected in parallel, then the total resistance RT, is given by:

1/(R_T)=1/R_1+1/R_2+1/R_3+...

In the case of AC circuits, this becomes:

1/(Z_T)=1/Z_1+1/Z_2+1/Z_3+...

## Simple case:

If we have 2 impedances Z1 and Z2, connected in parallel, then the total resistance ZT, is given by

1/(Z_T)=1/Z_1+1/Z_2

We can write this as:

1/(Z_T)=(Z_2+Z_1)/(Z_1Z_2)

Finding the reciprocal of both sides gives us:

Z_T=(Z_1Z_2)/(Z_1+Z_2)

### Example 1

Find the combined impedance of the following circuit:

### Example 2

Given that Z1= 200 − 40j Ω and Z2= 60 + 130j Ω,

find

a) the total impedance

b) the phase angle

c) the total line current

### Example 3

A 100\ Ω resistor, a 0.0200\ "H" inductor and a 1.20\ "µF" capacitor are connected in parallel with a circuit made up of a 110\ Ω resistor in series with a 2.40\ "µF" capacitor. A supply of 150\ "V", 60\ "Hz" is connected to the circuit.

Calculate the total current taken from the supply and its phase angle.

Didn't find what you are looking for on this page? Try search:

### Online Algebra Solver

This algebra solver can solve a wide range of math problems. (Please be patient while it loads.)

Play a math game.

(Well, not really a math game, but each game was made using math...)

Sign up for the free IntMath Newsletter. Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents!

Given name: * required

Family name:

email: * required

See the Interactive Mathematics spam guarantee.