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.
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You can find more background on this material at Electronics Tutorials
Definitions
Resistance
Symbol: R
Units: Ω (ohms)
A resistor is any part of a circuit that obstructs the flow of current.
Capacitance
Symbol: C
Units: `"F"` (farads)
A capacitor consists of 2 nonconnected plates:
Inductance
Symbol: L
Units: `"H"` (henrys)
An inductor is coil of wire in which current is induced.
Current: I (in amperes)
Voltage: V (in volts).
Ohm's Law: `V = IR`
Reactance
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:
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}
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:
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