4
C
AUSE OF
P
HASE
S
HIFTS
When the input voltage and currents to a
circuit composed of resistors, inductors and
capacitors are sinusoidal (sines or cosines),
then all the voltages and currents in the circuit
are sinusoids of the same frequency. The only
changes that occur to the signals are scaling
and delays. Delays are measured as "phase
differences"- the original and resultant
waveforms are compared, and the difference is
found as a fraction of a full period (360).
Effect of Resistors
The relationship between the current flowing
through a resistor and the voltage applied
across it is a simple scaling. Resistors change
the magnitude of signals, but do not cause any
change in phase.
Effect of Capacitors
Because the current through a capacitor is
proportional to the derivative of the voltage
across it, a sine voltage results in a cosine
current. Since a cosine is the same as a sine
moved to the left by a quarter of a full period
(look at the diagram), the effect of the
capacitor is to introduce a phase shift of 90,
with the current leading in front of the voltage
cos
sin(
)
A
A
90
.
Effect of Inductors
Because the current through an inductor is
proportional to the integral of the voltage
across it, a cosine voltage results in a sine
current. Since a sine is a cosine moved to the
right by a quarter of a full period, the effect of
the inductor is to introduce a phase shift of
90, with the current lagging behind the
voltage,
sin
cos(
)
A
A
90
Since all the signals in a linear circuit with sinusoidal inputs will also be sinusoids of the same
frequency, the only properties of the sinusoid that need to be modelled are its magnitude and phase
angle (measured relative to some input chosen by the analyser). Complex numbers can carry this
information, since in polar form a complex number is represented as a magnitude and an angle.
Polar form
r
a
b
b
a
2
2
1
tan
Rectangular form
r
jr
sin
cos
a jb
The complex impedances of capacitors and inductors (
1 j C
and
j L
) contain information
encoding the phase shifts that those components cause (multiplying or dividing by "j" is equivalent
to a phase shift of 90, since
j
1 90
).
The magnitude of a complex number represents the amplitude of the corresponding sinusoid.
The angle of a complex number in polar notation is the phase shift of the corresponding sinusoid.
The frequency of the sinusoid is not represented, since it must be the same for all signals in the
circuit.