Capacitor in the AC circuit with Cobra3
TEP
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.
Related topics
Capacitance, Kirchhoff’s laws, Maxwell’s equations, AC impedance, Phase displacement
Equipment
1
1
1
1
1
1
1
2
2
1
1
1
1
2
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
Fig. 1: Experimental set-up
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Capacitor in the AC circuit with Cobra3
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
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”.
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 √ 2 .
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Capacitor in the AC circuit with Cobra3
TEP
Fig. 2: “Analog In 2 / S2” settings for ac measurement
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TEP
Capacitor in the AC circuit with Cobra3
Click the “Function Generator” symbol and set the parameters like this:
Fig. 3: “Function Generator” settings
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Capacitor in the AC circuit with Cobra3
TEP
Add a “Virtual device” with two calculated channels like this:
Fig. 4: Virtual device settings (channel 1)
Fig 5: Virtual device settings (channel 2)
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Capacitor in the AC circuit with Cobra3
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)
Fig 6: PowerGraph settings
Fig 7: Display settings
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Capacitor in the AC circuit with Cobra3
TEP
Record curves for different values of resistance and capacitance. After clicking the
“Continue” button the “Start measurement” button appears. 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
menu 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:
Fig. 8: Current/voltage dependence on the frequency
Plot the impedance against the inverse frequency – exchange the set for the x-axis with
“Measurement” > “Channel manager…”. The linear dependence of the impedance from the
inverse frequency can be seen. To put more curves into the same diagram use
“Measurement” > “Adopt channel…”
Fig 9: Impedance dependence on the time scale
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Capacitor in the AC circuit with Cobra3
2. Total impedance of parallel and series connection
Also measure the impedance of capacitors in parallel and in series connection
The rule for adding capacitances in parallel is Ctot = C1 + C2.
Capacitances in series sum up like
C C
1
1 1
=
+
or C tot = 1 2
C tot C1 C 2
C 1+C 2
3. Phase shift measurement
Connect the “Analog In 2 / S2” terminals so as to measure the voltage drop over both
capacitance and resistance.
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Capacitor in the AC circuit with Cobra3
TEP
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 10: Fast Measurement and function 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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Capacitor in the AC circuit with Cobra3
Use the “Survey” function for phase shift evaluation. With a sample rate of 200 kHz one
channel corresponds to 5 μs.
Fig. 11: Sample measurement for phase shift
Here a curve obtained with 2,2 μF and 100 Ohm:
Fig. 12: Phase shift
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Capacitor in the AC circuit with Cobra3
TEP
Fig. 13: Tangent of the phase shift vs. frequency
Theory and evaluation
The voltage UC on a capacitance C with charge
2
Q(t) = ∫0 I ( t)dt
is
U C (t) =
Q(t)
.
C
The voltage on the resistance R is with current I ( t) =
U R (t) = R⋅I (t) = R
U (t) = U c (t)+U R (t) = Q
dQ
dt
dQ
dt
(t)
dQ
+R
= U 0 cos(ω t ).
C
dt
for a resistor and a capacitance in series connected to an ac voltage source. Differentiating
this equation yields
I
dI
+R
= −ω⋅U 0 sin(ω t)
C
dt
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Capacitor in the AC circuit with Cobra3
TEP
This differential equation has the solution
I ( t) = I 0 cos( ω t+ φ )
with
tan (φ ) =
1
>0
ω⋅CR
i.e. the current is ahead of the voltage and
I0 =
U0
√
2
( )
R 2+
1
ω⋅C
The impedance is then
√
U0
1
= R 2+
I0
ω⋅C
2
( )
and the impedance of the capacitance alone (R = 0) is
1
= R^ c .
ω⋅C
In a plot of impedance vs. inverse frequency is the slope m hence with ω = 2 π⋅f
m=
1
1
and C =
2 π⋅C
2 π⋅m
1 ms
−6 As
= 4.42⋅10
= 4.42 μ F
v
V
36 ⋅2⋅π
A
1 ms
−6 As
m = 72 Ω /ms⇒C =
= 2.21⋅10
= 2.21 μ F
v
V
72 ⋅2⋅π
A
1 ms
−6 As
m = 142 Ω /ms⇒C =
= 1.12⋅10
= 1.12 μ F
v
V
142 ⋅2⋅π
A
m = 36 Ω/ms⇒ C =
12
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