Post on 18-Jan-2016
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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 RESISTORAC V AND I IN A RESISTOROhm’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.
( )( ) R
R
v ti t
R
vR(t)
( )( ) R
R
v ti t
R
AC V AND I IN A RESISTORAC V AND I IN A RESISTOR
PHASE PHASE ANGLE ANGLE FOR R, FOR R, =0=0
AC CAPACITOR
CURRENT THROUGH A CAPACITORCURRENT THROUGH A CAPACITOR
The faster the voltage changes, the larger the current.
dt
dvCi c
c
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.
PHASE RELATIONSHIP
Graph vC(t) and iC(t)
90°90°
vvcc(t)(t)
iicc (t)(t)
NoteNote: Phase : Phase relationship relationship of I and V in of I and V in a capacitora capacitor
dt
dvCi c
c
In the Capacitor (C), Voltage LAGS charging current by 90o or Charging Current (I) LEADS Voltage (E) by 90o
I. C. E. V C
IC
90
PHASE RELATIONSHIP
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.
CX IV
CAPACITIVE REACTANCE
XC will have units of Ohms.
Note inverse proportionality to f and C.
1 1
2CX
fC C
Magnitude of XC
Ex.Ex.
Ex: f = 500 Hz, C = 50 µF, XC = ?
V S
C 1
Capacitive reactance also has a phase angle associated with it.
Phasors and ICE are used to find the angle
PHASE ANGLE FOR XPHASE ANGLE FOR XCC
IV
XC
PHASE ANGLE FOR XPHASE ANGLE FOR XCC
If If V is our reference wave: is our reference wave:
90
900 _Z
IV
CX
I.C.E
AC INDUCTOR
The phase angle for Capacitive Reactance (XC) will always = -90°
XC may be expressed in POLAR or RECTANGULAR form.
ALWAYS take into account the phase angle between current and voltage when calculating XC
90_CX CjXor
VOLTAGE ACROSS AN INDUCTORVOLTAGE ACROSS AN INDUCTOR
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.
dtdi
Lvind
Graph vL(t) and iL(t)
Note the phase relationship
vvLL(t)(t)
iL(t)90°90°
In the Inductor (L), Induced Voltage LEADS current by 90o or Current (I) LAGS Induced Voltage (E) by 90o.
E. L. I. VC
IC90
INDUCTIVE REACTANCE
We will define Inductive Reactance, XL, as the opposition to current in an inductor.
LX IV
INDUCTIVE REACTANCE
XL will have units of Ohms ().
Note direct proportionality to f and L.
2LX fL L
Magnitude of XL
Ex1.
f = 500 Hz, L = 500 mH, XL = ?
V S
L
PHASE ANGLE FOR XPHASE ANGLE FOR XLL
If If V is our reference wave: is our reference wave:
90
90
0L Z
I
VX
E.L.I
The phase angle for Inductive Reactance (XL) will always = +90°
XL may be expressed in POLAR or RECTANGULAR form.
ALWAYS take into account the phase angle between current and voltage when calculating XL
90LXLjXor
COMPARISON OF XL & XC
XL is directly proportional to frequency and inductance.
XC is inversely proportional to frequency and capacitance.
2LX fL L
1 1
2CXfC C
SUMMARY OF V-I RELATIONSHIPS
ELEMENT TIME DOMAIN FREQ DOMAIN
RiV
dt
diLV
dt
dvCi
RIV
IjV L
Cj
Cj II
V
R
L
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.
2LX fL L
1 1
2CX
fC C
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
IMPEDANCEThe V-I relations for three passive elements;
The ratio of the phasor voltage to the phasor current:
CjjR
IV LI,V I,V
CjjR
1
I
V L,
I
V ,
I
V
From that, we obtain Ohm’s law in phasor form for any type of element as:
Where Z is a frequency dependent quantity known as IMPEDANCE, measured in ohms.
IZVor I
VZ
IMPEDANCE
Impedance is a complex quantity:
R = Real part of Z = Resistance
X = Imaginary part of Z = Reactance
jXRZ
Impedance in polar form:
where;
θZjXRZ
R
XXR 122 tan,Z
θsin ZX θ, cosZR
IMPEDANCES SUMMARY
Impedance Phasor form: Rectangular form
ZR R+j0
ZL 0+jXL
ZC 0-jXC
90oLX
90oCX
0oR
ADMITTANCE
ADMITTANCE
The reciprocal of impedance.Symbol is Y Measured in siemens (S)
V
I
Z
1Y
ADMITTANCE
Admittance is a complex quantity:
G = Real part of Y = Conductance
B = Imaginary part of Y = Susceptance
jBGY
Z AND Y OF PASSIVE ELEMENTS
ELEMENT IMPEDANCE ADMITTANCE
RZ
Lj
Cj1
Z
R
1Y
Lj1
Y
CjY
R
L
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 2
1 1
1 2
1 1 1 1 1
1 1 1 1
total x x
totalx x
Z Z Z Z Z
ZZ Z Z Z
As a conclusion, in parallel circuit, the impedance can be easily computed from the admittance:
xtotal
totaltotal
total
YYYY
YY
Z
...
1
21
1
Ex3: SERIES CIRCUIT
R=20Ω
L = 0.2 mH
C = 0.25μF
V6010sin10V 5s t