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:

Circuit diagram

Example 2

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

Circuit diagram

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.