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.
Capacitance:
Symbol: C
Units: F (farads)
A capacitor consists of 2 non-connected 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: 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
The voltage across a capacitor with reactance XC (voltage and current are RMS, or 'root mean square' values):
VC = IXC
The voltage across an inductor with reactance XL (once again, voltage and current are RMS values):
VL = IXL
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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