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.


You can find more background on this material at Electronics Tutorials



Symbol: R

Units: Ω (ohms)

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

Resistor symbol


Symbol: C

Units: `"F"` (farads)

A capacitor consists of 2 non-connected plates:

Capacitor symbol


Symbol: L

Units: `"H"` (henrys)

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

Inductor symbol

Current: I (in amperes)

Voltage: V (in volts).

Ohm's Law: `V = IR`


Reactance is the 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:


Resistor symbol

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


Capcitor symbol

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


Inductor symbol

Representing voltages using the complex plane

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:

Vc, VR and VL complex plane axes