Lab Final Exam
• The question bank is posted on the ELMS
site under “Final Exam” in “Assignments”
Lab #6: the LRC Circuit and
Resonance
• Remember how AC circuits containing a cap,
an inductor, and a resistor in series behave
• Measure a resonance experimentally
• Upload a spreadsheet today. A full lab
report due by 2 PM same day next week
Crystal radio (AM radio)
Where is the
crystal? No
longer in there.
Modern crystal
radios use
diodes instead.
Today: inductor
and capacitor
together.
AM operates from 535 to 1605 kHz.
Series LRC Circuit
The phenomenon of
resonance is important in
physics
Impedance:
Resistor:
Capacitor:
Inductor:
R
-i
wC
iw L
(voltage in phase with current)
(voltage lags current by 90o)
(voltage leads current by 90o)
Current in the circuit
Start with a statement of Kirchoff’s law:
-i
V0e = IR + I
+ I (iw L)
wC
V0
- iwt
I=
e
1
( R + i (w L ))
wC
V0
I =
1 2
2
R + (w L )
wC
iwt
Current I is max when
denominator is min:
when ωL=1/ωC
w0 =
1
LC
Resonance
Resonance
t = L/R
Dw = 1 / t (width of resonance, VR =Vmax / 2)
w0
L
Q=
=
Dw
R 2C
phases
-i
statement of Kirchoff’s law:
V0e = IR + I
+ I (iw L)
wC
V0
I=
e - iwt = I e - i (wt -f )
1
( R + i (w L ))
wC
V0
I =
1 2
R 2 + (w L )
wC
1
Phase of current (and thus
wL voltage across R) with
wC
tan f =
respect to V0
R
iwt
Phase shift between voltage across resistor and
input is zero when at resonant frequency
phases
Note that since
VL leads by 90
degrees and VC
lags by 90
degrees, they
are always outof-phase by 180
degrees
Hints
• part A1.
200 mH: make this by putting two 100 mH inductors in
series. Because the mutual inductance is non-negligible, please be
sure to wire them together, measure the inductance, and then put
them into the circuit wired exactly as when you measured them.
• Part A2. assume the uncertainty on internal resistance of the
waveform generator is 2 Ohms. (50 ± 2) W
• C-1 at low frequency, the waveform can be ugly. Measure to the
average over the “features”. In other words, use cursors, not
“measure” on the oscilloscope.
• C-1 don’t assume that V0 is the same at all frequencies, monitor it
and record the values.
• C-1 note that the phase shift changes sign as a function of
frequency!
Some Derivatives
Q=
L
1 L
=
2
RC R C
¶Q 1 æ L ö
= ç 2 ÷
¶L 2 è R C ø
-1/2
¶Q 1 æ L ö
= ç 2 ÷
¶C 2 è R C ø
-1/2
1
1 1
= Q
2
RC 2 L
-L
1 1
=
Q
2 2
RC
2 C
¶Q -1 L
1
= 2
= -Q
¶R R C
R
d tan 2 q
1
=
dq
cos 2 q
Step-wave input
Charge on cap rings at resonant frequency while
decaying away
Like striking a bell
with a hammer
Tosc =
L
t =2
R
w
2
osc
=w 2
0
1
t2
2p
w osc
At large R
Critically damped: R is large
4 L no1oscillation
2
enoughRso
that
»
2
C
p
+1
occurs
Hints
• Capture a wave form of the ringing with
wavestar
• for part C, only vary R and only give a
qualitative answer