Transcript Impedance and Ohm’s Law

Objective of Lecture
 Describe the mathematical relationships between ac
voltage and ac current for a resistor, capacitor, and
inductor.
 Discuss the phase relationship between the ac voltage and
current.
 Chapter 9.4 Fundamentals of Electric Circuits
 Explain how Ohm’s Law can be adapted for inductors and
capacitors when an ac signal is applied to these
components.
 Derive the mathematical formulas for the impedance and
admittance of a resistor, inductor, and capacitor.
 Chapter 9.5 Fundamentals of Electric Circuits
Resistors
Ohm’s Law
if i(t) = Im cos(wt + q)
then v(t) = Ri(t) = R Im cos(wt + q)
V = RIm q = RI where q = f
The voltage and current through a resistor are in phase
as there is no change in the phase angle between them.
Capacitors
i(t) = C dv(t)/dt where v(t) = Vm cos(wt)
i(t) = -Cw Vm sin(wt)
i(t) = wCVm sin(wt + 180o)
i(t) = wCVm cos(wt + 180o - 90o)
i(t) = wCVm cos(wt + 90o)
Capacitors
V = Vm0o
I = wCVm cos(wt + 90o)
Vm cos(wt + 90o) = V ej90 = V90o = jV
I = jwCV = wCV90o
or
V = (1/jwC) I = - (j/wC) I = (1/wC) I-90o
Capacitors
 90o phase difference between
the voltage and current
through a capacitor.
 Current needs to flow first to
place charge on the electrodes of
a capacitor, which then induce a
voltage across the capacitor
 Current leads the voltage (or
the voltage lags the current)
in a capacitor.
Inductors
v(t) = L d i(t)/dt where i(t) = Im cos(wt)
v(t) = - Lw Im sin(wt) = wLIm cos(wt + 90o)
V = wLIm90o
I = Im cos(wt)
Im cos(wt + 90o) = I ej90 = I 90o = jI
V = jwLI = wLI90o
or
I = (1/jwL) V = - (j/wL) V = (1/wL) V
-90o
Inductors
 90o phase difference
between the voltage and
current through an
inductor.
 The voltage leads the
current (or the current
lags the voltage).
Impedance
If we try to force all components to following Ohm’s
Law, V = Z I, where Z is the impedance of the
component.
Resistor:
Capacitor:
Inductor:
ZR = R
ZC =  j wC 
ZL = jwL
R0o
1 wC   90o
wL90o
Admittance
If we rewrite Ohm’s Law:
I = Y V (Y = 1/Z), where Y is admittance of the
component
o
1
/
R
=
G
G

0
Resistor:
YR =
wC90o
Capacitor:
YC = jwC
Inductor:
YL =  j wL 
1 wL   90o
Impedances
Value at w =
Admittance
s
0 rad/s ∞ rad/s
Value at w =
0 rad/s ∞ rad/s
ZR = R = 1/G
R
R
YR = 1/R = G
G
G
ZL = jwL
0W
∞W
YL =-j/(wL)
∞ W1
0 W1
ZC = -j/(wC)
∞W
0W
YC = jwC
0 W1
∞ W1
Inductors act like short circuits under d.c. conditions and
like open circuits at very high frequencies.
Capacitors act like open circuits under d.c. conditions and
like short circuits at very high frequencies.
Impedance
Generic component
that represents a
resistor, inductor, or
capacitor.
Z = Z f
Z = R + jX
Z = R +X
2
2
f = tan  X R 
1
R = Z cosf 
X = Z sin f 
Admittance
1
Y =1 Z =
R + jX
R
G= 2
2
R +X
X
B= 2
2
R +X
Y = Y 
Y = G + jB
Y = G2 + B2
 = tan1 B G 
G = Y cos 
B = Y sin  
Summary
 Ohm’s Law can be used to determine the ac voltages
and currents in a circuit when impedance or
admittance are used.
 A resistor’s voltage and current are in phase.
 Voltage leads current through an inductor by 90o.
 Current leads voltage through a capacitor by 90o.
Component
Impedance
Resistor
ZR
Capacitor
ZC
Inductor
ZL
R
Admittance
R 0o
 j wC 1 wC   90o
jwL
wL  + 90
G
jwC
G 0o
wC  + 90
 j wL 1 wL   90o