Transcript Lecture - UMD Physics

Lab 5
The AC Circuit, Impedance , High-Pass and
Low-Pass Filters
This experiment requires only a spreadsheet upload
Crystal radio (AM radio)
Where is the
crystal? No
longer in there.
Modern crystal
radios use
diodes instead.
Today: AC
circuits with a
capacitor or an
inductor
AM operates from 535 to 1605 kHz.
Lab #5: RC and RL AC Circuits
• remember how AC circuits containing
capacitors and resistors, as well as inductors
and resistors, behave.
First, remember how to describe a sine (or cosine)
w = 2p f
1
f =
T
Capacitors and AC sources
Voltage across cap is the same
as the voltage from supply.
When the voltage is changing
quickly, the charge also has to
change quickly -> big current
1
∆𝑉 = 𝑉0 sin 𝜔𝑡 = 𝑄
𝐶
𝑄 = 𝐶𝑉0 sin 𝜔𝑡
𝑑𝑄
𝐼=
= 𝐶𝑉0 𝜔 cos 𝜔𝑡 = 𝐶ω𝑉0 sin 𝜔𝑡 + 𝜋/2
𝑑𝑡
𝑉0
𝐼=
sin 𝜔𝑡 + 𝜋/2
1/ ω𝐶
𝐼0 =
𝑉0
1/ ω𝐶
Size of the current depends on the frequency
Get biggest currents at high frequencies
AC RC Circuits
As before, be
careful with the
grounds!
• Presence of the capacitor affects the size of the current in the circuit
in a frequency-dependent way.
• “phases” of signals across voltage source, resistor, and capacitor
differ
• math is most easily done by modeling the voltage source as V = V0eiwt
instead of V = V0 cos(wt ) and an imaginary reactive impedance for
the capacitor (to shift its affect on the current by 90 degrees) and then
taking the real part at the end.
The Math
What is the current?
𝑉0 𝑒 𝑖𝜔𝑡 = 𝐼(𝑡) ∙ 𝑍
1
𝑖
=𝐼 𝑡 ∙ 𝑅+
= 𝐼(𝑡) ∙ 𝑅 −
𝑖𝜔𝐶
𝜔𝐶
Any complex number can be written as a magnitude and an angle in the
complex plane.
Z = R2 +
tan f =
𝑉0
𝑒 𝑖𝜔𝑡
1
=𝐼 𝑡 ∙𝑅 1+
𝜔𝑅𝐶
𝐼(𝑡) =
𝑉0 /𝑅
1+
1
𝜔𝑅𝐶
𝑒𝑖
2
2
𝑒 𝑖𝜑
𝜔𝑡−𝜑
1
1
=
R
1
+
w 2C 2
w 2 R 2C 2
-1
w RC
Easy to read off mag of current.
Current (and thus voltage across the
resistor) is shifted in phase from the
voltage source by φ
What is the current at very large ω?
Voltage across R and across C
𝑽 𝒕 = 𝑽𝟎 𝐜𝐨𝐬 𝝎𝒕
𝑽𝑹 𝒕 = 𝑽𝟎 cos 𝝋 cos 𝝎𝒕 − 𝝋
VR(t) leads V(t) by 
𝑽𝑪 𝒕 = −𝑽𝟎 sin 𝝋 sin 𝝎𝒕 − 𝝋
VC(t) lags V(t) by p/2 - 
Inductive Impedance
L
Z = [R2 + (wL)2]1/2
XL = iwL
t = L/R
tan = wL/R = wt
As with the RC circuit, the current can be written as
𝐼(𝑡) =
𝑉0 /𝑅
1+
1
𝜔𝜏
𝑒
2
𝑖 𝜔𝑡−𝜑
𝐼(𝑡) =
𝑉0
cos 𝜑 𝑒 𝑖
𝑅
𝜔𝑡−𝜑
lab
• For a fixed frequency, measure the phase shift f
two ways and “compare” (using a c2 test)
• Measure the phase shift versus frequency two
ways and use to extract RC. Compare to RC
calculated directly.
hints
• Include systematic errors for R and C measurements, but not for t and V
measurements with scope.
• MAKE SURE DUTY CYCLE IS ALL THE WAY COUNTER CLOCKWISE!
• phase shift can not be greater than p
• remember “compare” is a mathematical operation involving a c2 test
• make sure the voltage oscillates around zero (using the offset knob). Make
sure there is no dc offset.
• remember, VR leads VIN by f
• If your wave form looks funny, your amplitude is too big for the
instrumentation amplifier. Make it smaller
Error Propagation
cos f =
VR
V0
2
2
ö
æ ¶f
ö æ ¶f
sf = ç
s VR ÷ + ç
s V0 ÷
è ¶VR
ø è ¶V0
ø
¶f
1
¶f -VR
from the first: sin f
=
and sin f
= 2
¶VR V0
¶V0 V0
2
æ1
ö æ VR
ö
1
sf =
ç s VR ÷ + ç 2 s V0 ÷
sin f è V0
ø è V0
ø
æ 1
ö
2
æ1
2
ö
s f = cot(f ) ç s V ÷ + ç s V ÷
è VR
ø è V0
ø
R
0
2
Error Propagation
You measure dt. Calculate y = cotan(2 p f dt).
What is error in y?
y = cot(j )
¶y
1
=- 2
¶j
sin (j )