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RC Circuits
Equipment DataStudio with 750 interface, RLC circuit board, 2 voltage sensors (no alligator clips), 2x35 in. leads, 12 in. lead
Reading Review operation of DataStudio oscilloscope. Review operation of the DataStudio signal generator.
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Introduction
The 3 basic linear circuits elements are the resistor, the capacitor, and the inductor. The
voltage across a resistor is proportional to the current through the resistor. The voltage
across a capacitor is proportional to the time integral of the current into the capacitor. The
voltage across an inductor is proportional to the time derivative of the current through the
inductor. This lab is concerned with the characteristics of capacitors and circuits consisting
of a resistor and a capacitor in series (RC circuits). The primary focus will be on the
response of an RC circuit to a step voltage.In this case will be using an AC square wave
as a step voltage. (A step voltage is a constant voltage which instantaneously changes to a
different constant voltage.) Also, you will be varying the frequency of a square waveform.
The frequency will vary the time of the period. A period is the overall time it takes to
complete one cycle. In the lab write up the period will be labeled as capital T.
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Capacitors
A capacitor is a 2 terminal circuit element that stores energy in its electric field. In its
simplest form a capacitor consists of 2 parallel plates separated by a small gap which is
usually filled with a non-conducting dielectric material. In practice many capacitors are
made by sandwiching a dielectric sheet such as mylar between 2 pieces of metal foil and
rolling this assmebly up into a cylinder.
If a positive current I enters one plate of a capacitor and exits the other plate a positive
charge Q builds up on the first plate and a negative charge -Q builds up on the second plate.
A voltage VC develops across the capacitor. The charge Q and voltage VC are linearly related
by the equation
Q = CVC ,
(1)
where C is a constant called the capacitance. Eq. 1 can also be written as
VC =
1 Z
Idt.
C
(2)
The unit of capacitance in S.I. units is the farad (F). One farad is a huge capacitance and
more usual units are the micro farad (µF) and the pico farad (pF). The constant C depends
on the geometry of the capacitor and the dielectric properties of the insulator between the
plates.
The most important specification for a capacitor of a given value is its maximum voltage
rating.
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3
RC Circuits
A series RC circuit with a voltage source V(t) connected across it is shown in Fig. 1. The
voltage across the resistor and capacitor are designated by VR and VC , and the current around
the loop by I . The signs are chosen in the conventional way. I is positive if it is in the
direction of the arrow. Kirchoff’s law, which says that the voltage changes around the loop
are zero, may be written
VR + VC = V.
(3)
Note: I =change in charge(Q) overtime.
Letting VR = IR = ( dtd Q)R and VC = C1 Q
Eq.(3) may be written as
d
1
R Q + Q = V.
(4)
dt
C
Of particular interest in this lab is the solution of Eq.(4) when V is a constant voltage.
The solution is
t
Q = (Q0 − CV )e− RC + CV,
(V = constant)
(5)
where Q0 is the charge on the capacitor at time t=0. The quantity RC is the time constant
of the circuit. Time constant references the time it takes for the current to flow in out of the
capacitor and resistor. Energy is stored in capacitor and the flow of charge and discharge
will be controlled by the resistor. The time dependence of the circuit is exponential when
V=constant. Beginning at time t = 0 the parameters of this circuit exponentially approach
constant values. VC approaches V and I approaches 0. In a time 3RC those parameters are
within 95% of the final values. As a function of time the voltage across the capacitor is C1 Q,
the current in the circuit is I = dQ
, and the voltage across the resistor is VR = IR.
dt
3.1
A time varying voltage to an RC circuit.
We can use this result to understand what happens when a constant voltage is applied to
a RC series circuit, or if a constant voltage is instantaneously changed from one constant
value to another constant value. A key point is that when the voltage is switched, the
charge on and the voltage across the capacitor do not change instantaneously. VC and Q
are continous but not differentiable at the switching point. The voltage across the resistor
and the current in the circuit do change instantaneously. VR and I are neither continuous
or differentiable at the switching point. Consider the situation shown in Fig.(2). For times
t < 0 the applied voltage is 0, and we assume that the applied voltage has been zero long
enough so that VC = Q = VR = I = 0. At t=0 a constant voltage V is applied to the circuit
for a time T /2 RC. For the time interval 0 < t < T /2 the voltage across the capacitor
rises exponentially from 0 toward V and the voltage across the resistor decays exponentially
for V toward zero.
As shown in Fig.(2) at time T/2 the applied voltage is returned to 0. As T RC at
time T/2 the voltage across the capacitor is essentially V. For times t > T /2, VC decays
exponentially to 0. Let be a very small increment of time. At time t = (T /2 − ) VR is
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almost 0. At time t = (T /2 + ) VR has jumped to −V . Can you explain why? For t > T /2,
VR goes exponentially to 0.
The above response of an RC circuit to a constant voltage which is first zero, then V,
and then zero again can be observed by applying a square wave to the circuit where half
of the square wave has zero voltage and the other half voltage V. If T is the period of the
square wave, we continue to assume that T RC. Make a single plot that shows the square
wave, VC , and VR , but read the next paragraph first and check that your plot agrees with
the statements made.
Regardless of the relative magnitudes of T and RC, if a periodic voltage is applied to a
series RC circuit, VC , VR , and I must obey the following conditions.
1. The time average value of VC must equal the time average value of the applied voltage.
You can prove this by integrating Eq.(3) over one period.
2. The time average value of the I (and V R ) must be zero. If they were not, what would
happen to the voltage across the capacitor?
Exercise. If T RC, graph the response of an RC circuit to an applied symmetric square
wave where the voltage oscillates between +V and -V? Your graph should show the applied
square wave, VC , and VR . Hint. At the instant the square wave “switches” the magnitude
of the voltage across the resistor is either 2V or −2V .
If a high frequency square wave, such that T RC, is applied to an RC circuit, the
voltages changes across the capacitor are minimal. The capacitor does not have time to
charge much before the voltage is reversed. Most of the voltage changes occur across the
resistor. As mentioned previously, the average voltage of the capacitor will be the time
average voltage of the square wave, and the average VR will be zero. Fig. 3 shows the
voltages for a square wave that oscillates between the constant voltage V and 0 (ground),
and Fig. 4 shows the voltages for a square wave that oscillates between +V and -V. In both
these figures, the exponential dependences of VC and VR are approximated as straight lines.
Neither VC or VR approach steady state values.
3.2
Charging and discharging of an RC circuit
The capacitor does not charge instantaneously. The charge is proportional dQ/dt. In time
when the capacitor does become charged the current will go to zero. This occurs at a certain
time constant which is related to the capacitance of the capacitor and the resitance value.
Discharging
t
Charge Q0 e− RC
t
Current I0 e− RC
t
Voltage V0 e− RC
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Charging
t
Q0 (1 − e− RC )
t
I0 (1 − e− RC )
t
V0 (1 − e− RC )
Experiments
In the following sections you will be observing the charging and discharging of the capacitor.
This is related to the time constant of the RC circuit. You will be applying voltage square
waves that have various periods. Depending on the time constant of the circuit and period
of the square being applied you will see various response curves across the capacitor and
resistor.
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The period of the square wave is T. Both the situation T RC and T RC will be
examined. For the former case, due to limitations of the equipment, the peak values of VR
will be less than the theoretical values. One equipment limitation that you can easily see on
the scope is the finite rise time of the square wave.
4.1
T RC, Positive Only Square wave
If necessary, remove the Power Amplifier II connector from input channel C of the 750
interface. Hook up the circuit shown in Fig.(5) taking care with the polarities. On the RC
circuit board connect one lead across the induction coil to short it out, choose R = 10 Ω and
C = 100 µF . See Fig.(6) for an overview of the setup. Keep in mind the after you’re done
wiring the circuit you need to plug in the analog voltage sensors. Plug the voltage sensor
that is across the capacitor to channel A and the voltage sensor that is across the resistor
to channel B. Open DataStudio and click on the output terminals of the 750 interface. A
signal generator window will pop up. Set the sample rate to 2000Hz. This is done in the
measurements and sample rate tab of the signal generator. In the signal generator window,
choose the positive only square wave, a voltage of 1.5 V, and a frequency of 50Hz. Open the
scope display by dragging the scope icon to the voltage ouput terminals icon of the set-up
window. Set up the voltage sensors on analog channels A and B. Make the input to channel 2
of the scope voltage sensor A and the input to channel 3 of the scope voltage sensor B. This
is done by draging the Scope icon from the Displays window to the Output Voltage icon in
the data window. Next, drag Scope 1 icon from the Displays window to the Voltage ChA
icon in the Data window. Last, again drag the Scope 1 icon form the displays winodw to
the Voltage ChB icon in the Data window. The input to channel 1 of the scope should be
the voltage ouput of the signal generator.
For the parameters you are using, calculate T and RC. Is the assumption that T RC
valid?
Some remarks.
• When using the oscilloscope after you click Stop, the last traces are stored. The stored
trace is usually better for examination than the “live” trace. The “live” trace may have
jitter.
• You can you the smart cursor on the stored trace. When measuring scope trace parameters, judicious use of the sweep speed, vertical gain, and vertical trace position
help.
• You can use many of the scope functions on the stored trace.
• You can print out the stored trace.
• Label your printed out traces immediately by V, VC , and VR as the printer is not a
color printer.
• If you use same vertical sensivity for all 3 scope traces you can easily compare the trace
values.
• The voltage of 1.5 V for the signal genertor is satisfactory for all the experiments.
Higher values may saturate VR .
• For reliable triggering with a positive only square wave, make the trigger voltage a little
positive.
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Click Start and examine VC , VR and V. Are the results what you expect? Use the smart
cursor to measure the time constant of the exponentials and compare your measurements to
RC (1/e = 0.37). Note that VC is fairly similar in form and magnitude to the input square
wave, and that VR looks a lot like the derivative of the square wave. Referring to the left
hand side of Eq.(4), dQ
will be of the order of Q
. With the assumption that T RC this
dt
T
Q
term can be neglected compared to the term C . We now have, to some approximation, that
VR
= VC . The result of differentiating this expression is dV
= CI = RC
. You have
V = Q
C
dt
constructed an electronic differentiator!
4.2
T RC, Symmetric Square Wave
Use the same parameters as in section 4.1 except use a square wave that is symmetric with
respect to ground Oscillates between (+V and −V ). Compare your results to what you
expect.
4.3
T RC, Positive Only Square Wave
Apply a 100 Hz positive only square to an RC circuit with R=100 Ω and C=330 µF . What
is the time constant and how does it compare to T? Compare your results to Fig. 3. Note
that VR is fairly similar in form to the input square wave and that VC is similar in form to
its integral. Using Eq.(4) can you show that this is plausible? In Fig. 3 is approximating
the exponentials by straight lines reasonable?
4.4
T RC, Symmetric Square Wave
Use the same parameters as in Section 4.3 except use a square wave that is symmetric with
respect to ground. Compare to Fig. 4. Is approximating the exponentials by straight lines
reasonable?
4.5
T ≈ RC, Symmetric Square Wave
Use the same RC components as in Section 4.3. Apply a symmetric square wave with a
frequency of 30 Hz. Are the results what you expect? If you have time try 10 Hz and 5 Hz.
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Finishing up and additional questions
1. What would occur on the time to discharge the capacitor if you increased the size of
the capacitor? Explain.
2. What real world applications can an RC circuit be used in?
Please leave your bench in a state of maximum order. Thank you.
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Figure 6