# 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\ mu"F" capacitor are connected in parallel with a circuit made up of a 110\ Ω resistor in series with a 2.40\ mu"F" 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.

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