LEP
4.4.05
-15
Capacitor in the AC circuit with Cobra3
Related Topics
Capacitance, Kirchhoff’s laws, Maxwell’s equations, AC
impedance, Phase displacement
Principle
A capacitor is connected in a circuit with a variable-frequency
voltage source. The impedance and phase displacement are
determined as a function of frequency and of capacitance.
Parallel and series impedances are measured.
Equipment
Resistor in plug-in box 47 Ω
Resistor in plug-in box 100 Ω
Resistor in plug-in box 220 Ω
Capacitor (case 2) 1 µF/250 V
Capacitor (case 2) 2.2 µF/250 V
Capacitor (case 2) 4.7 µF/250 V
Connection box
Connecting cord, l = 500 mm, red
Connecting cord, l = 500 mm, blue
Cobra3 Basic Unit, USB
Measuring module function generator
PowerGraph Software
Cobra3 Universal writer software
Power supply, 12 VPC, Windows® 95 or higher
39104.62
39104.63
39104.64
39113.01
39113.02
39113.03
06030.23
07361.01
07361.04
12150.50
12111.00
14525.61
14504.61
12151.99
1
1
1
1
1
1
1
2
2
1
1
1
1
2
Tasks
1. Determine the impedance of a capacitor as a function of
frequency.
2. Determine the total impedance of capacitors connected in
series and in parallel.
3. Determine the phase displacement between current and
voltage over a RC network as a function of frequency.
Set-up and procedure
1. Impedance measurement
Connect the Function Generator Module to the Cobra3 unit
and set up the equipment according to Fig. 1. The “Analog In
2 / S2” should be connected in a way that it measures the
voltage drop over the capacitor. Connect the Cobra3 unit to
your USB port. Connect both Cobra3 and Function Generator
Module to their 12 V supplies. Start the “measure” program on
your computer. Select the “Gauge” “PowerGraph”.
Circuit for impedance measurement
Fig. 1: Experimental set-up
PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen
P2440515
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LEP
4.4.05
-15
Capacitor in the AC circuit with Cobra3
On the “Setup” chart of PowerGraph click the “Analog In 2 /
S2” symbol and select the module “Burst measurement” with
the following parameters to enable the “Analog In 2 / S2” to
perform ac measurements. The obtained values are ac amplitude values, i.e. the positive peak voltage. To obtain the effective voltage in case of sine waves the values have to be divided by 22 .
Add a “Virtual device” with two calculated channels like this:
Fig. 4: Virtual device settings (channel 1)
Fig. 2: “Analog In 2 / S2” settings for ac measurement
Click the “Function Generator” symbol and set the parameters
like this:
Fig. 5: Virtual device settings (channel 2)
Fig. 3: “Function Generator” settings
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Set the channels to be recorded like this (see Fig. 6) and configure a diagram to be seen during measurement like this (see
Fig. 7)
PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen
LEP
4.4.05
-15
Capacitor in the AC circuit with Cobra3
Plot the impedance against the inverse frequency – exchange
the set for the x-axis with “Measurement” > “Channel manager…”. The linear dependance of the impedance from the
inverse frequency can be seen. To put more curves into the
same diagram use “Measurement” > “Adopt channel…”
(Fig. 9). The impedance of the capacitor is independent of the
resistance value.
Fig. 6: PowerGraph settings
Fig. 9: Impedance dependence on the time scale
auto range
2. Total impedance of parallel and series connection
Also measure the impedance of capacitors in parallel and in
series connection.
auto range
Fig. 7: Display settings
Record curves for different values of resistance and capacitance. After clicking the “Continue” button the “Start measurement” button apperars. You may stop the measurement
when the current does not rise much any longer, but for easy
evaluation with “Adopt channel...” it is best to always record
the same number of values – the drop down menue under
“Stop condition” you find a feature for this. If you select only
current I and voltage U2 to be displayed, current and voltage
curves plotted against frequency may look like this:
Capacitors in parallel
Fig. 8: Current/voltage dependence on the frequency
Capacitors in series
PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen
P2440515
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LEP
4.4.05
-15
Capacitor in the AC circuit with Cobra3
The rule for adding capacitances in parallel is Ctot = C1 + C2.
Capacitances in series sum up like
Use the “Survey” function for phase shift evaluation. With a
sample rate of 200 kHz one channel corresponds to 5 µs.
C1C2
1
1
1
or Ctot Ctot
C1
C2
C1 C2
3. Phase shift measurement
Connect the “Analog In 2 / S2” terminals so as to measure the
voltage drop over both capacitance and resistance.
Fig. 11: Sample measurement for phase shift
Here a curve obtained with 2,2 µF and 100 Ohm:
Start the “Gauge” “Cobra3 Universal Writer” and select the
“Fast Measurement” chart so that your Cobra3 can be used
similar to an oscilloscope. Set the parameters like this:
Fig. 12: Phase shift
Fig. 10: Fast Measurement and Funcion Generator settings
Record curves with different frequencies for each combination
of resistance and capacitance and take down the phase shift
to be plotted in a separate curve. For low frequencies it may
be better to put the voltage higher to get lower current noise
as the current is quite low for low frequencies. You may check
the phase shift with a resistor (without capacitance) to be zero.
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Fig. 13: Tangent of the phase shift vs. frequency
PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen
LEP
4.4.05
-15
Capacitor in the AC circuit with Cobra3
Theory and evaluation
The voltage UC on a capacitance C with charge
Q 1t 2 2
U0
1
R2 a
b .
I0
B
v·C
t
I1t 2 dt
is
UC 1t2 0
Q 1t2
C
and the impedance of the capacitance alone (R = 0) is
1
ˆ .
R
C
v·C
.
The voltage on the resistance R is with current I 1t2 UR 1t 2 R · I 1t2 R
U 1t 2 UC 1t 2 UR 1t2 Q 1t2
C
The impedance is then
dQ
dt
dQ
.
dt
In a plot of impedance vs. inverse frequency is the slope m
hence with v = 2 p · f
m dQ
R
U0 cos 1v · t 2 .
dt
for a resistor and a capacitance in series connected to an ac
voltage source. Differentiating this equation yields
dI
I
R
v · U0 sin 1v · t 2 .
C
dt
This differential equation has the solution
1
1
and C 2p·C
2p·m
From Fig. 9 the slope values read:
imprinted value / µF
µF slope / Ohm/ms
measured value / µF
4.7
36 ±2
4.42 ±0.2
2.2
72 ±4
2.21 ±0.1
1
142 ±7
1.12 ±0.05
I 1t 2 I0 cos 1v · t w2
with
tan 1w 2 m 36 Ohm>ms 1 C As
1 ms
4.42 · 10 6
4.42 mF
V
V
36 · 2 · p
A
m 72 Ohm>ms 1 C As
1 ms
2.21 · 10 6
2.21 mF
V
V
72 · 2 · p
A
m 142 Ohm>ms 1 C 1 ms
As
1.12 · 106
1.12 mF
V
V
142 · 2 · p
A
1
7 0
v · CR
i.e. the current is ahead of the voltage and
I0 U0
2
1
R a
b
B
v·C
2
.
PHYWE series of publications • Laboratory Experiments • Physics • PHYWE SYSTEME GMBH • 37070 Göttingen, Germany
P2440515
5
LEP
4.4.05
-15
6
Capacitor in the AC circuit with Cobra3
PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen
P2440515