Lesson 19
Impedance
Learning Objectives

For purely resistive, inductive and capacitive elements
define the voltage and current phase differences.

Define inductive reactance.

Understand the variation of inductive reactance as a
function of frequency.

Define capacitive reactance.

Understand the variation of capacitive reactance as a
function of frequency.

Define impedance.

Graph impedances of purely resistive, inductive and
capacitive elements as a function of phase.
Review
R, L and C circuits with Sinusoidal
Excitation

R, L, C have very different voltage-current
relationships
vR  iR R
dvC
iC  C
dt
diL
vL  L
dt

(Ohm's law)
(capacitor current relationship)
(inductor voltage relationship)
Sinusoidal (ac) sources are a special case
The Impedance Concept

Impedance (Z) is the opposition that a circuit
element presents to current in the phasor
domain. It is defined
V
V
Z = = Ðq = ZÐq
I I

Ohm’s law for ac circuits
V  IZ
Impedance

Impedance is a complex quantity that can be
made up of resistance (real part) and
reactance (imaginary part).

Unit of impedance is ohms ().
Z
X
q
R
Resistance and Sinusoidal AC

For a purely resistive circuit, current and
voltage are in phase.
Resistors

For resistors, voltage and current are in phase.
VR VR q VR
ZR 

 0  R0  R
I
I q
I
Z R  R0
Example Problem 1
Two resistors R1=10 kΩ and R2=12.5 kΩ are in series.
If i(t) = 14.7 sin (ωt + 39˚) mA
a)
b)
c)
d)
Compute VR1 and VR2
Compute VT=VR1 + VR2
Calculate ZT
Compare VT to the results of VT=IZT
Inductance and Sinusoidal AC

Voltage-Current relationship for an inductor
diL
d
 L  I m sin t 
dt
dt
  LI m cos t   LI m sin t  90
vL  L
vL  LI m sin t  90 
ZL 

iL
I m si n  t
 LI m


90
2
Im
0
2
  L90
( )
It should be noted that for a purely inductive circuit
voltage leads current by 90º.

Inductive Impedance

Impedance can be written as a complex
number (in rectangular or polar form):
Z L   L90  j L ( )

Since an ideal inductor has no real resistive
component, this means the reactance of an
inductor is the pure imaginary part:
X L  L
Inductance and Sinusoidal AC

Voltage leads current by 90˚
Inductance

For inductors, voltage leads current by 90º.
VL VL 90 VL
ZL 

 90   L90  j L
I
I 0
I
Z L  jX L  X L90
X L   L  2 fL
Impedance and AC Circuits

Solution technique
1.
Transform time domain currents and voltages into phasors
Calculate impedances for circuit elements
Perform all calculations using complex math
Transform resulting phasors back to time domain (if reqd)
2.
3.
4.
Example Problem 2
For the inductive circuit:
vL = 40 sin (ωt + 30˚) V
f = 26.53 kHz
L = 2 mH
Determine VL and IL
Graph vL and iL
Example Problem 2 solution
vL = 40 sin (ωt + 30˚) V
iL = 120 sin (ωt - 60˚) mA
iL
Notice
90°phase
difference!
vL
Example Problem 3
For the inductive circuit:
vL = 40 sin (ωt + Ө) V
iL = 250 sin (ωt + 40˚) μA
f = 500 kHz
What is L and Ө?
Capacitance and Sinusoidal AC

Current-voltage relationship for an capacitor
dvC
d
iC  C
 C Vm sin t 
dt
dt
 CVm cos t  CVm sin t  90
Zc 
vc
Vm sin t

ic CVm sin t  90


Vm
0
1
2


  90 ( )
Vm
C
90 C
2

It should be noted that, for a purely capacitive circuit
current leads voltage by 90º.
Capacitive Impedance
Impedance can be written as a complex number
(in rectangular or polar form):
 1 
 1 
Zc  
   90   
 j ( )
 C 
 C 

Since a capacitor has no real resistive
component, this means the reactance of a
capacitor is the pure imaginary part:
 1 
Xc  

 C 
Capacitance and Sinusoidal AC
Capacitance

For capacitors, voltage lags current by 90º.
VC VC 0 VC
1
 1 
ZC 

   90 
  90   
j
I
I 90
I
C
 C 
ZC   jX C  X C   90
 1   1 
XC  



C
2

fC

 

Example Problem 4
For the capacitive circuit:
vC = 3.6 sin (ωt-50°) V
f = 12 kHz
C=1.29 uF
Determine VC and IC
Example Problem 5
For the capacitive circuit:
vC = 362 sin (ωt - 33˚) V
iC = 94 sin (ωt + 57˚) mA
C = 2.2 μF
Determine the frequency
ELI the ICE man
E leads I
When voltage is applied to an
inductor, it resists the change
of current. The current builds
up more slowly, lagging in time
and phase.
I leads E
Since the voltage on a
capacitor is directly
proportional to the charge on it,
the current must lead the
voltage in time and phase to
conduct charge to the capacitor
plate and raise the voltage