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)


Voltage: V (in volts).


Ohm's Law: V = IR


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° .


For voltage: V = IX


The voltage across a resistor with resistance R:

VR = IR

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The voltage across a capacitor with reactance XC:

VC = IXC

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The voltage across an inductor with reactance XL:

VL = IXL

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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 capcitor we have a negative imaginary quantity. Our axes are as follows:

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