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Chapter Contents

• Complex Numbers
• 1. Basic Definitions
• 2. Basic Operations
• 3. Graphical Representation
• 4. Polar Form
• 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

• Comments, Questions?

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8. An Application of Complex Numbers: AC Circuits

by M. Bourne

Before we see how complex numbers can help us to analyse and design AC circuits, we first need to define some terms.

Definitions

Resistance:

Symbol: R

Units: W (ohms)

A resistor is any part of a circuit that obstructs the flow of current.

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Capacitance:

Symbol: C

Units: F (farads)

A capacitor consists of 2 non-connected plates:

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Inductance:

Symbol: L

Units: H (henrys)

An inductor is coil of wire in which current is induced.

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Current: I (in amperes)

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Voltage: V (in volts).

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Ohm's Law: V = IR

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Reactance: Effective resistance of any part of the circuit. This could be from an inductor or a capacitor. See more in the next section Impedance and Phase Angle.

Symbol: X

Voltage in alternating current circuits

• The voltage across a resistance is in phase with the current .

• The voltage across a capacitor lags the current by 90° .

• The voltage across an inductance leads the current by 90° .

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For voltage: V = IX

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The voltage across a resistor with resistance R:

V_R = IR

The voltage across a capacitor with reactance X_C (voltage and current are RMS, or 'root mean square' values):

V_C = IX_C

The voltage across an inductor with reactance X_L (once again, voltage and current are RMS values):

V_L = IX_L

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Using the complex plane, we can represent voltages across resistors, capacitors and inductors.

The voltage across the resistor is regarded as a real quantity, while the voltage across an inductor is regarded as a positive imaginary quantity, and across a capacitor we have a negative imaginary quantity. Our axes are as follows: