Download 3.reactance_and_impedance

Lecture 03:
AC RESPONSE
( REACTANCE N
IMPEDANCE )
OBJECTIVES
 Explain the relationship between AC voltage and AC
current in a resistor, capacitor and inductor.
 Explain why a capacitor causes a phase shift between
current and voltage (ICE).
 Define capacitive reactance. Explain the relationship
between capacitive reactance and frequency.
 Explain why an inductor causes a phase shift between
the voltage and current (ELI).
 Define inductive reactance. Explain the relationship
between inductive reactance and frequency.
 Explain the effects of extremely high and low
frequencies on capacitors and inductors.
AC RESISTOR
AC V AND I IN A RESISTOR
 Ohm’s Law still applies even though the
voltage source is AC.
 The current is equal to the AC voltage
across the resistor divided by the resistor
value.
 Note: There is no phase shift between
V and I in a resistor.
v R (t )
iR (t ) 
R
AC V AND I IN A RESISTOR
vR(t)
PHASE
ANGLE
FOR R,
=0
iR (t ) 
v R (t )
R
AC CAPACITOR
CURRENT THROUGH A CAPACITOR
 dvc 
 ic  C 

 dt 
 The faster the voltage changes, the
larger the current.
PHASE RELATIONSHIP
 The phase relationship between “V”
and “I” is established by looking at the
flow of current through the capacitor
vs. the voltage across the capacitor.
Graph vC(t) and iC(t)
vc(t)
90°
ic (t)
Note: Phase
relationship
of I and V in
a capacitor
 dvc 
 ic  C 

 dt 
PHASE RELATIONSHIP
 In
the Capacitor (C), Voltage LAGS
charging current by 90o or Charging
Current (I) LEADS Voltage (E) by 90o

I. C. E.
IC
90
VC
CAPACITIVE REACTANCE
 In resistor, the Ohm’s Law is V=IR,
where R is the opposition to current.
 We will define Capacitive Reactance,
XC, as the opposition to current in a
capacitor.
V  I XC
CAPACITIVE REACTANCE
 XC will have units of Ohms.
 Note inverse proportionality to f
and C.
1
1
XC 

2 fC  C
Magnitude of
XC
Ex.
Ex: f = 500 Hz, C = 50 µF, XC = ?
C1
VS
PHASE ANGLE FOR XC
 Capacitive reactance also has a phase
angle associated with it.
 Phasors and ICE are used to find the
angle
V
XC 
I
PHASE ANGLE FOR XC
 If V is our reference wave:
I.C.E
V0
_
XC 
 Z 90
I90
AC INDUCTOR
 The phase angle for Capacitive
Reactance (XC) will always = -90°
 XC may be expressed in POLAR or
RECTANGULAR form.
X C  90
_
or
 jX C
 ALWAYS take into account the phase
angle between current and voltage when
calculating XC
VOLTAGE ACROSS AN INDUCTOR
vind
di 

 L 
 dt 
 Current must be changing in order to create
the magnetic field and induce a changing
voltage.
 The Phase relationship between VL and IL (thus
the reactance) is established by looking at the
current through vs the voltage across the
inductor.
Graph vL(t) and iL(t)
vL(t
)
Note the phase
relationship
90°
iL(t)
 In the Inductor (L), Induced Voltage
LEADS current by 90o or Current (I)
LAGS Induced Voltage (E) by 90o.

VC
E. L. I.
IC
INDUCTIVE REACTANCE
 We will define Inductive Reactance,
XL, as the opposition to current in an
inductor.
V  I XL
INDUCTIVE REACTANCE
 XL will have units of Ohms (W).
 Note direct proportionality to f and L.
X L  2 fL   L
Magnitude of
XL
Ex1.
f = 500 Hz, L = 500 mH, XL = ?
L
VS
PHASE ANGLE FOR XL
 If V is our reference wave:
V0
XL 
 Z  90
I  90
E.L.I
 The phase angle for Inductive Reactance
(XL) will always = +90°
 XL may be expressed in POLAR or
RECTANGULAR form.
X L 90
or
jX L
 ALWAYS take into account the phase
angle between current and voltage when
calculating XL
COMPARISON OF XL & XC
 XL is directly proportional to frequency and
inductance.
X L  2 fL   L
 XC is inversely proportional to frequency and
capacitance.
1
1
XC 

2 fC  C
SUMMARY OF V-I RELATIONSHIPS
ELEMENT
TIME DOMAIN FREQ DOMAIN
R
V  Ri
V  RI
L
di
V L
dt
V  j L I
C
dv
iC
dt
I
I
V
j
jC
C
Extreme Frequency effects on
Capacitors and Inductors
 Using the reactances of an inductor
and a capacitor you can show the
effects of low and high frequencies
on them.
1
1
XC 

2 fC  C
X L  2 fL   L
Frequency effects
 At low freqs (f=0):
an inductor acts like a short circuit.
a capacitor acts like an open circuit.
 At high freqs (f=∞):
an inductor acts like an open circuit.
a capacitor acts like a short circuit.
Ex2.
 Represent the below circuit in freq
domain;
REVIEW QUIZ
-
-
What is the keyword use to remember the
relationships between AC voltage and AC current
in a capacitor and inductor
.
What is the equation for capacitive reactance?
Inductive reactance?
-
T/F A capacitor at high frequencies acts like a
short circuit.
-
T/F An inductor at low frequencies acts like an
open circuit.
IMPEDANCE
IMPEDANCE
 The V-I relations for three passive elements;
V  RI,
I
V  jLI, V   j
C
 The ratio of the phasor voltage to the phasor
current:
V
 R,
I
V
 jL,
I
V
1
j
I
C
 From that, we obtain Ohm’s law in phasor
form for any type of element as:
V
Z
I
or V  IZ
 Where Z is a frequency dependent quantity
known as IMPEDANCE, measured in ohms.
IMPEDANCE
 Impedance is a complex quantity:
Z  R  jX
R = Real part of Z = Resistance
X = Imaginary part of Z = Reactance
 Impedance in polar form:
Z  R  jX  Z θ
where;
Z  R  X ,  tan
2
2
1
X
R
R  Z cos θ, X  Z sin θ
IMPEDANCES SUMMARY
Impedance
Phasor form:
ZR
R0
ZL
X L 90
ZC
X C   90
Rectangular
form
R+j0
o
0+jXL
o
o
0-jXC
ADMITTANCE
ADMITTANCE
 The reciprocal of impedance.
 Symbol is Y
 Measured in siemens (S)
1 I
Y 
Z V
ADMITTANCE
 Admittance is a complex quantity:
Y  G  jB
G = Real part of Y = Conductance
B = Imaginary part of Y = Susceptance
Z AND Y OF PASSIVE ELEMENTS
ELEMENT
IMPEDANCE
R
ZR
L
  j L
C
1
Zj
C
ADMITTANCE
1
Y
R
1
Y
jL
Y  j C
TOTAL IMPEDANCE FOR AC
CIRCUITS
 To compute total circuit impedance in
AC circuits, use the same techniques as
in DC. The only difference is that
instead of using resistors, you now have
to use complex impedance, Z.
TOTAL IMPEDANCE FOR
PARALLEL CIRCUIT
1
Z total
1
1
1
1

 
  

Z x Z1 Z 2
Zx
1
Z total

 1
1 
1
1 
     
   
Zx 
 Zx 
 Z1 Z 2
1
 As a conclusion, in parallel circuit,
the impedance can be easily
computed from the admittance:
Z total 
1
Ytotal
 Ytotal 
1
Ytotal  Y1  Y2  ...  Yx 
Ex3: SERIES CIRCUIT
R=20Ω


Vs  10 sin 105 t  60 V
L = 0.2 mH
C = 0.25μF