Electrical engineers kept looking for ways to keep using Ohm’s law in circuits with
inductors and capacitors (avoiding the pain of solving differential equations) – and found
the conditions, under which a generalized Ohm’s law in the form
V=I
⋅
Z
applies.
The conditions are:
1.
All sources in the circuit should be sinusoidal, at the same frequency
ω
2.
All circuit elements should be linear
3.
The circuit should be in a sinusoidal steady state (all transients already vanished).
In general, the impedance
Z
is a complex number, which means that voltage
V
and
current
I
may have different phase angles.
•
For resistors,
Z
R
= R
(the impedance is the same as resistance)
•
For inductors,
Z
L
= j
⋅ω⋅
L
: the impedance is the product of
j=
√
(1)
,
the
frequency
ω
expressed in radians per second, and the inductance
L
in henry.
With these units, the impedance is in ohms
•
For capacitors
Z
C
= 1/(j
⋅ω⋅
C)
: the impedance is reciprocal of the product of
j
=
√
(1)
, the frequency
ω
in radians per second, and the capacitance in farads.
With these units, the impedance is in ohms.
Making sense of EE // 2nd ed
Phasors
© 2008 A. Ganago
Page 1 of 18
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In circuit equations with impedances, the voltage
V
and current
I
also become complex.
Of course, all voltages and currents in the lab are real, but
math with complex numbers
allows us to keep track of both the magnitude and the phase of each voltage and
current in the given circuit at the given (fixed) frequency.
The complex number that expresses both the magnitude and the phase of a voltage
or a current in the circuit at the given frequency is called a phasor
(note that the
handy weapon used by Star Trek heroes is called a phas
e
r with an
e
).
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 Spring '07
 Ganago
 Complex number, Electrical impedance, A. G anago, A. Ganago

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