# 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
(*R*_{1}, *R*_{2},
*R*_{3}, *R*_{4}, …) connected
in **parallel**, then the total resistance
*R*_{T}, 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 *Z*_{1} and
*Z*_{2}, connected in parallel, then the total
resistance *Z*_{T}, 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 *Z*_{1}= 200 − 40*j* Ω and
*Z*_{2}= 60 +
130*j* 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.

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