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I = (1/Z)Vext Z must have
LRC circuit
amplitude & phase
L
I
C
Vext
R
Imag
|I| = |1/Z| |V|
f
V
Real
Usually write: Vext
Z is IMPEDANCE
= IZ(w )
(generalized resistance)
But Z is ratio of applied voltage to
resulting current, or "in/out"
1
I=
Vext
Z(w )
1/Z (also Y) is ADMITTANCE
Is a ratio of "out/in"
1/Z is a frequency-dependent, complex quantity
that describes the system's response to a driving
voltage. It is a "response function".
Is there a phase shift?
R circuit
I
V=IR
Vext
R
Vext = V0 e
iwt
Vext -V = 0
Look for
Vext = IR
iw t
iw t
I = I0e
if iw t
= I0 e e
V0
I 0 = ;f = 0
R
independent of w
V0 e
I=
R
Purely resistive circuit:
• Current in phase with driving
voltage at all frequencies
• Magnitude indep. of frequency
R circuit
Vext = V0 e
I
V=IR
Vext
R
iwt
V0 i 0 iw t
I= e e
R
drive
response
Phasor diagram
Vext
=R
I
Impedance, Z
I
1
=
Vext R
Admittance, 1/Z
C circuit
Vext = V0 e
I
V=q/C
Vext
I
C
Look for
iwt
= I0e
if iwt
= I0 e e
iwt
Vext -V = 0
q
= Vext
C
I
= Vext = iwV0 eiw t
C
i p2 iw t
I = w CV0 e e
p
I 0 = wCV0 ; f =
2
dependent on w
Purely capacitive circuit:
• Current leads driving voltage (ICE)
•Magnitude depends on frequency
C circuit
Vext = V0 e
I
iwt
input
V=q/C
Vext
C
Phasor diagram
ICE
i p2 iw t
I = w CV0 e e
output
Vext
1 -i p2
=
e
I
wC
Impedance, Z
I
i p2
= wCe
Vext
Admittance, 1/Z
(sometimes Y)
L circuit
VL =LdI/dt
I
+
Vext
-
L
V
Look for
I = I 0 eiw t = I 0 eif eiw t
V0
p
I0 =
;f = wL
2
dependent on w
Vext = V0 e
iwt
Vext -VL = 0
LI = Vext
iw LI = V0 e
iw t
V0 iw t V0 -i p2 iw t
I=
e =
e e
iw L
wL
Purely inductive circuit:
• Current lags driving voltage (ELI)
• Magnitude depends on frequency
L circuit
Vext = V0 e
I
+
Vext
-
L
V
Phasor diagram
ELI
iwt
V0 -i p2 iw t
I=
e e
wL
input
output
Vext
i p2
= wLe
I
Impedance, Z
I
1 -i p2
=
e
Vext wL
Admittance, 1/Z
LRC circuit
L
I
dI
q
VL = L ;VR = IR;VC =
dt
C
C
R
L (inductance), C (capacitance),
cause oscillation, R (resistance)
causes damping
2
q˙˙ + 2bq˙ + w 0 q = 0
dI
q
-L - IR - = 0
dt
C
q
Lq + Rq + = 0
C
R
1
q+ q+
q=0
L
LC
I = (1/Z)V
LRC circuit
L
I
C
Vext
R
Imag
|I| = |1/Z| |V|
f
V
Real
q ( t ) = Re éë q0 e ùû
iw t
iw t
Vext = Re éëV0 e ùû
i (fq + p /2 ) iw t ù
é
I(t) = q ( t ) = Re w q0 e
e
ë
û
I0 =
wV0 L
(
é w2 -w
0
ë
)
2 2
1/2
+ 4b w ù
û
2
-2bw
f I = + arctan 2
2
2
w0 - w
2
p
12
I0 =
wV0 L
(
é w -w
ë
2
0
)
2 2
1/2
+ 4b w ù
û
2
2
What is the best variable to plot for the LRC lab?
ADMITTANCE
Amplitude
of I/Vapp
Frequency, w ->
p
-2bw
f I = + arctan 2
2
2
w0 - w
1
-wL
tan f = w C
R
Phase of I
(rel to Vapp)
Frequency, w ->
Q factor of an
underdamped
oscillator
Damping time or "1/e" time is t = 1/b > 1/w0
(>> 1/w0 if b is very small)
How many T0 periods elapse in the damping time?
This number (times π) is the Quality factor or Q of the
system.
t
w0
Q=p =
T0 2 b
large if b is small compared to w0
Max Amplitude
Current
Amplitude
|I0|
1
of max amp
2
Dw
w0
Q=
Dw
Driving Frequency------>
Find frequencies where POWER
drops to half maximum (current drops
to 0.707 of max). These define w.
Find resonant frequency, w0
Homework problem to show the two definitions are the same.
QUETSION: FM radio stations have broadcast frequencies of
approximately 100 MHz. Most radios use a series LRC circuit similar
to the one you used in the lab as part of the receiver electronics.
Estimate the spacing of the broadcast frequencies of FM stations if
typical receivers have a Q of 500 or better. Explain your reasoning,
and include a graph.
station 1
w
station 2
w
w0
f0
Qº
=
Dw Df
QUETSION: FM radio stations have broadcast frequencies of
approximately 100 MHz. Most radios use a series LRC circuit similar
to the one you used in the lab as part of the receiver electronics.
Estimate the spacing of the broadcast frequencies of FM stations if
typical receivers have a Q of 500 or better. Explain your reasoning,
f0
and include a graph.
Q=
station 1
w
Df
= 500
station 2
f0 100 MHz
Df = =
= 0.2 MHz
Q
500
Therefore, stations 99.3
and 99.5 FM are allowed,
but 99.3 and 99.4 FM are
not!
They have cross-talk!
You should be able to:
• Calculate & plot the magnitude and phase of 1/Z
• Convert between the mag/phase and Re/Im forms
• Draw phasor diagrams of Vext, I, 1/Z (or Z)
• Express 1/Z (or Z) in terms of R, L, C or w0, b
You should be able to discuss:
• The amplitude of the response and resonance
• The phase of the response
• The nature of the behavior at all frequencies
• The transfer of the series LCR circuit analysis to
analogous oscillatory systems