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Dr Andrew French. September 2020.
TEACHER NOTES
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Equipment Signal generator + mains cable kettle lead + BNC to
4mm converter
Dual input oscilloscope + mains cable kettle lead + 2 x BNC to
4mm converter
Capacitor block (non-electrolytic) range 0.220mF to 15mF.
Fixed resistors (only 100ohm will be used)
Multimeter (set in ACA mode i.e. gives RMS amplitude of (AC)
current)
4 x blue wires (main RC circuit loop)
2x green wires (PD across AC input, connected to oscilloscope
CH1)
2x green wires (PD across capacitor, connected to oscilloscope
CH2)
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Kit list
Signal generator + mains cable kettle lead + BNC to 4mm
converter
Dual input oscilloscope + mains cable kettle lead + 2 x BNC to
4mm converter
Capacitor block (non-electrolytic) range 0.220mF to 15mF.
Fixed resistors (only 100ohm will be used)
Multimeter (set in ACA mode i.e. gives RMS amplitude of (AC)
current)
4 x blue wires (main RC circuit loop)
2x green wires (PD across AC input, connected to oscilloscope
CH1) 2x green wires (PD across capacitor, connected to oscilloscope
CH2)
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Suggested experiments 1. Fix Vin at about 3.0V (the middle value
of the signal generator amplitude).
2. Fix R at about 100W (measure it accurately with a
multi-meter).
3. Fix capacitances at 1.0mF, 2.2mF and 4.7mF. Do the middle
value only if time permits.
4. Record and plot gain |Vc/Vin| vs frequency f for ten to
twenty measurements over the
frequency range 0 to 3,000Hz.
5. Also record and plot the RMS current vs frequency.
6. Fixing the frequency at 200Hz, find the RMS current for the
full range of capacitances. You should find that current is
proportional to capacitance.
7. For a 4.7mF capacitor, keep the frequency at 3,000Hz and
switch to a square wave, and then a triangle wave output from the
signal generator. Observe that the RC circuit integrates the input,
if the output is taken across the capacitor. i.e. an output of
triangle waves or parabolae, respectively.
8. See if you can observe a differentiation effect for low
frequencies (i.e. about 100Hz) when the output is taken across the
resistor instead of the capacitor.
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Channel 1 (CH1): Potential difference across output of signal
generator (upper trace). Channel 2 (CH2): Potential difference
across capacitor (lower trace).
Dual channel oscilloscope
-
Input voltage from signal generator (CH1)
Potential difference across the capacitor (CH2)
Timebase = 0.5ms
Voltage divisions = 2.0V
2 1.4 2.0V
1.4V
c
c
V
V
2 3.4 2.0V
3.4V
in
in
V
V
3
1
2.95 0.5 10 s
678Hz
f
f
Phase difference
o o0.30360 36.62.95
0.3
2.95
3.4
1.4
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Gain reduces as frequency increases. 1
2c
fRC
Characteristic frequency
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Phase shift tends to 90o as frequency increases
1
2c
fRC
Characteristic frequency
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Current is proportional to frequency (and capacitance) when
frequencies are much less than the characteristic frequency.
1
2c
fRC
When frequencies are much greater than the characteristic
frequency, current tends to a constant:
2
in
RMS
VI
R
2
in
RMS
VI RC
R
Characteristic frequency:
-
2
in
RMS
VI
R
2
in
RMS
VI RC
R
Current is proportional to capacitance when frequencies are much
less than the characteristic frequency.
When frequencies are much greater than the characteristic
frequency, current tends to a constant:
-
C C
in C R
V Z
V Z Z
1
R
C
Z R
Zi C
2 f
2 2 2
1 1 1
1 1 1
C
in
V i C i RC
V i C R i RC R C
1tan2 2 22 2 2
11
1
i RCC
in
VR C e
V R C
i t i tC C in in
V V e V V e
2 2 2
1
1
C
in
V
V R C
1tan RC
1
RC
Mathematical model of voltage gain vs frequency
Complex impedence Potential divider
Capacitor
Resistor ‘driving’ AC input inV RV
I
CV
i t
in inV V e
Argand (phasor) diagram
Assume steady state
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2 2 2
1
1
C
in
V
V R C
2 f
1tan RC
1
2c
fRC
Characteristic frequency
Note linear phase response
1
RC
RC
if phase in radians
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Finding the RMS current
1
1
tan
2 2 2
tan
2 2 2
2 2 2
2 2 2
1
1
1
2 1
i t RCin
C
C C
C C
i t RCin
in
in
RMS
VV e
R C
V IZ
V I i C I i CV
i C VI e
R C
V RCI
R R C
V RCI
R R C
1
2c
fRC
Characteristic frequency
1
22
in
RMS
VI f
RCR
1
22
in
RMS
VI RC f
RCR
Low frequency limit: Current is proportional to frequency and
capacitance.
High frequency limit: Current tends to a constant, as one would
expect for a DC circuit.
2 f
-
2 2 22 1
in
RMS
V RCI
R R C
2 f
1
2c
fRC
Characteristic frequency
1
22
in
RMS
VI f
RCR
1
22
in
RMS
VI RC f
RCR
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RC circuit as an integrator
1
2
tan
2 2 2
1
1
1
i t RCin
C
i tin in i t in
C
i t
in in
ini t i t
in in
RCC in
VV e
R C
RC
V V VV e e
RC i RC i RC
V V e
VV dt V e dt e c
i
V V dt
High frequency limit
1
2f
RC
So a square wave input will produce a ‘sawtooth’ (triangular
wave) capacitor voltage.
2
2 1
i
i
e i
e i
-
1RCC in
V V dt High frequency limit
1
2f
RC
Square (ish!)-wave input
Saw-tooth output Saw-tooth input
Parabola output
RC circuit as an integrator
15μ
107
3.4V
3,125Hz
199Hz
2
in
C F
R
V
f
RC
W
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RC circuit as an differentiator
1 1
1
R
in
R in
i t
in in
in
in
in
R
V R i RC
V i C R i RC
RC
V i RCV
V V e
dVi V
dt
dVV RC
dt
Low frequency limit
1
2f
RC
So a ‘sawtooth’ (triangular wave) input will result in a square
wave resistor potential difference.
Note: For an unknown reason I couldn’t get this to work, even
when using the 0.22mF capacitor and frequency turned down to 100Hz.
In this case fc = 6,786Hz. AF 26/09/2020
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Excel sheet of measurements taken using RC circuit setup
As frequency and capacitance changes, the amplitude of the
signal generator also changes slightly. In this example, the signal
generator amplitude has to be changed for every measurement to
ensure a consistent value of 3.4V. The latter was measured using
the oscilloscope.
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Experimental data underlaid with model curves
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Model (line) vs measurement (+). R = 107W as measured.
Model appears to be an overestimate. Internal resistance not
accounted for?
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Does the internal resistance of the signal generator change as
capacitance increases? R increased to 120W as a guess.
