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LECTURE 26
Deadline for this week’s homework
assignment is extended until
Friday, March 30, after lecture
Pick up
• lecture notes
RLC circuits with generators
Lecture 25
NO generator
Decaying
oscillations
Lecture 26
With generator
Sustained
oscillations
Problem 1 A resistor and an inductor are
connected in series with an AC generator whose
emf is given by E(t) = Eo cos(wt). Find
(a) the effective impedance Z(w) of the circuit
R
Z (w )
Definition of Z: E0  I 0 Z (w )
(b) the phase difference between the emf E(t) and
the current I(t).
Problem 2 A resistor and an inductor are now
connected in parallel with the generator
E(t) = Eo cos(wt). Calculate for this configuration
(a) the effective impedance Z(w) of the circuit
Z (w )
Definition of Z: E0  I 0 Z (w )
(b) the phase difference between the emf E(t) and
the current I(t).
Complex impedance formalism
GOAL: Reduce AC problems to DC problems
Typical problem:
• generator signal given: E(t) = Eo cos(wt),
• need to find the current I(t) = Io(w)cos(wt-d)
Note:
eiwt  cos(wt )  i sin( wt )
Choose complex form for E
E (t )  E0 e
iwt
Re[ E (t )]  E0 Re[ eiwt ]
Re[ E (t )]  E0 cos(wt )
Look for a current of the form
I (t )  I 0 (w )eiwt
where Io(w) is complex:
I 0 (w ) | I 0 (w ) | e -id
Complex impedance formalism
Voltage
across R
VR (t )  I 0 (w )eiwt R  VR (w )eiwt
VR (w )  I 0 (w ) R
ZR  R
Just like in the DC case!
Voltage
across L

)
d
I 0 (w )eiwt  iwLI 0 (w )eiwt
dt
VL (w )  I 0 (w ) Z L (w ) Z L (w )  iwL
VL (t )  L
Looks like a “resistor” with
Complex impedance ZL
Voltage
across C
1
1
iwt
iwt
I
(
w
)
e

I
(
w
)
e
0
0
C
iwC
Vc (w )  I 0 (w ) Z c (w )
1
VC (t ) 
Z C (w ) 
Looks like a “resistor”
with complex impedance ZC
iw C
ALL RULES FOR DC CIRCUITS APPLY
WITHIN THE COMPLEX IMPEDANCE
FORMALISM
Problem 2’ Use the complex impedance
formalism to find:
(a) the effective impedance Z(w) of the circuit
Z (w )
(b) the phase difference between the emf E(t) and
the current I(t).
Problem 3
(a) Calculate the total effective impedance for the
following circuit:
(b) Calculate the phase shift between E(t) and the
current I(t) flowing through the generator.
Problem 4 Find the frequency at which the
current is the largest in the series RLC circuit.
Problem 5 Show that in the parallel configuration,
the current flowing through the generator has a
minimum.