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5
In this situation of positive, imaginary impedance, the impedance is purely inductive , as
an ideal inductor would be. The impedance of an ideal inductor with inductance L is a
linear function of frequency, given by
Z
L

j

L.
If the phase of the voltage is 90 degrees (

/2 radians) behind the phase of the current,
then the impedance is a negative imaginary number:
 
| |
| |
| |
| |
2 /
I
V
j e
I
V
Z
j
   

(6)
In this situation of negative, imaginary impedance, the impedance is purely capacitive , as
an ideal capacitor would be. The impedance of an ideal capacitor with capacitance C is
the inverse of a linear function of frequency, given by
Z
C

1 /
j

C
 
j
/.

C.
Actual circuit components are not purely resistive, inductive, or capacitive. Actual ca-
pacitors and inductors have impedances with resistive parts, and their impedances may
not be linear functions of frequency or independent of the voltage. The general expres-
sion for impedance, therefore, considers that it has a real, resistive part R and an
imaginary,
reactive part: X:
Z

R

j
X, (7)
Where X
 
L for an inductor and X
 
1/

C for a capacitor. Since the quantity X is
traceable to the ratio of a voltage to a current, it is expressed in ohms. Often, it is desira-
ble to express the impedance in ohms as a scalar (real) quantity; in that case, its
magnitude
2 2
| | X R Z
 
is used.
Units
The unit of resistance is the ohm, with the symbol

(omega). A 1-

resistor produces a
one-volt voltage across it when the current is one ampere.
The unit of inductance is the Henry, with the symbol H. For a one-amp AC current, a
1-H inductor would produce an AC voltage across it whose magnitude is numerically
equal to 2

times the frequency in Hertz.
The unit of capacitance is the Farad, with the symbol F. For a one-amp AC current, a
1-F capacitor would produce an AC voltage across it whose magnitude is numerically
equal to the inverse of 2

times the frequency in Hertz.
Series and Parallel Equivalent Circuits
The impedances of Actual resistors, inductors and capacitors are combinations of re-
sistance, inductance, and capacitance. The simplest models for actual inductors and
capacitors are the series and parallel equivalent circuits shown in Figure 1-1.
For example, the complex impedance of an inductor is
L j R jX R Z

   
(series equivalent circuit) (8a)
 
 
2
/ 1
/
p p
p p p p
p p
p p
L R
L R jR R
L j R
L j R










(parallel equivalent) (8b)