Chapter Contents

• Complex Numbers
• 1. Basic Definitions
• 2. Basic Operations
• 3. Graphical Representation
• 4. Polar Form
• Convert polar to rectangular using calculator
• 5. Exponential Form
• 6. Products and Quotients
• 7. Powers and Roots
• 8. AC Circuit Definitions
• 9. Impedance and Phase Angle
• 10. Reactance and Angular Velocity
• 11. AC Circuit Exercises
• 12. Parallel AC Circuits
• Vocabulary
• Fractals
• Complex Numbers Problem Solver

• Comments, Questions?

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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 R_T, is given by:

In the case of AC circuits, this becomes:

Simple case:

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

We can write this as:

Finding the reciprocal of both sides gives us:

Example 1

Find the combined impedance of the following circuit:

Answer

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

Answer

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.

Answer