Laboratory Exercise 2 – Capacitors and Resistors (RC circuits)
We learned about capacitors in physics ‘lo those many years ago (maybe not so long ago for
some of you). The basic idea of these devices is charge (or energy) storage. The simplest
conceptual version of a capacitor (a.k.a., cap) is two metal plates connected to the rest of the
circuit and separated by empty space. Real caps have different physical arrangements, but share
many of the properties of the idealized case: the cap looks like an open circuit to a time-invariant
applied voltage and the outcome of forcing current to flow in the circuit is that an excess of
electrons will build up on one of the plates and an excess of holes on the other. Because electrons
and holes do not like to be close to one another, the amount of charge that can be accumulated
depends on how big the plates are, or the capacitance, and how hard the circuit pushes (the
voltage). If the circuit pushes too hard, the electrons and holes jump the gap with disastrous
results for the cap (and often the rest of your circuit) so caps also have a maximum voltage
rating.
Units: Capacitance is measured in farads, or rather some small fraction of a farad like pF, nF, or
F since F is a very large unit. The two commonly used units are pF and F, a fact that can be
useful in figuring out the somewhat bizarre labeling scheme used on caps. {see mini-module on
Caps} The significance of this choice of units is that resistance in ohms () times capacitance in
farads (F) is equal to time in seconds. The importance of this relationship (RC = time) will
become more apparent as you work through this lab.
Capacitors are frequently used in electronics: to produce a desired time or frequency response, to
store charge for enhanced current delivery, or to help hold voltages constant. In this Laboratory
Exercise, we will investigate some of the more common circuit blocks that help us attain these
objectives.
An analogy to the physical principle of capacitance is a tub filling with water. The bigger the tub,
the longer it takes to fill and the more it will store. Capacitance is analogous to the diameter of
the tub, the flow of water is current, and the height of the water in the tub is analogous to voltage
across the cap. If we stop putting water in and open the drain we expect it to take some time for
the tub to empty, during which it would still be providing flow (current). This describes the
storage function of caps. Unfortunately, this analogy doesn’t provide insight into the frequency
dependent impedance behavior of the capacitor, which is what we use capacitors for most often.
So it’s not a very useful analogy.
Concept Question 1 – What is the RC time product for a 1F cap and a 100 K resistor?
If your scope probe has a capacitance of 3 pF, what resistance will produce an RC time constant
of 1 nsec? (This has some real-world implications for measuring fast-changing signals.)
Design and Usage Considerations
Maximum Voltage: As mentioned above, the maximum voltage for a cap is a practical issue: if
voltage in excess of the rated value is used, catastrophic discharge can result. Discharges (like
lightning) occur when the voltage applied is sufficient to cause gain during the transport of ions
and electrons present in the space (or insulator) between the electrodes. Electrons and ions,
typically formed at a sharp defect within the capacitor structure, where the electric field is
especially strong, accelerate through the field and bombard neutral species producing more ions
and electrons. Like overloading a power supply, a discharge failure will cause the cap to function
improperly due to an internal short or lack of storage of charge (at best) or to be destroyed
(usually). The small electrolytic caps can explode like little firecrackers, smelling awful and
generally spewing pieces and smoke all over the place (don’t try this at home.)
Take home lesson: Always use capacitors that are specified for voltages well in excess of what
you would expect to encounter in your circuit, and otherwise you can ignore this specification
(a.k.a., “spec”).
Capacitance: The capacitance spec is far more important to circuit design, since it will determine
both the frequency (or time) behavior of the circuit and the potential for charge storage in current
buffering applications.
Polarization: Many capacitors are polarized with preferred plus and minus voltage connections:
connect the plus side of the cap to the higher voltage node in the circuit. The polarization is
usually indicated by a stencil on the cap or by a longer lead or tab.
Since we will be producing and measuring time varying voltages, you will need to use the
oscilloscope (a.k.a., Oscope or just scope) and the function generator. If this is the first time that
you have used these two tools, it might be prudent to run through the mini-modules (scope,
function generator) on their use.
The definition of the capacitance (a physical property) of a capacitor (a device) is based on the
charge (in coulombs) Q stored at a given voltage
Q = CV
(1)
where C and V are capacitance and voltage. This is fine, but we don’t actually have any easy
way of measuring charge, so this relationship turns out to be less useful than the one we get by
taking the time derivative of eqn. 1.
dQ/dt = I = dCV/dt = C dV/dt, hence
I = C dV/dt
(2)
where I is the time dependent current in amperes (a.k.a., amps or A). This equation says that the
current that flows through a cap depends on the capacitance and how fast the voltage is
changing. Static voltages (a.k.a., DC) shouldn’t cause current to flow through a cap at all (after a
short charging period) consistent with our picture of a couple of plates separated by vacuum. The
thing that intuition doesn’t always give us is that rapidly varying (AC) voltages and their
associated currents can “pass through” the capacitor fairly easily (little impedance). We appear to
have a violation of Ohm’s law, where resistance to current flow depends on the frequency f of
the source. Of course, Ohms law only refers to resistors, so we are ok.
Concept Question 2 – If a steady DC current of 1 amp is to be maintained across a 0.1 F
capacitor, what rate of change of the voltage (dV/dt) must be applied to the cap?
How long would it take to go from 0 V to 15 V at that rate?
If a 0.1 F capacitor is charged to 100 volts and then made to discharge to ground (100 V to 0 V)
in 1.0 sec, what is the average current that flowed during the discharge? (Keep this in mind
when you reach into a TV set or other electronic appliance filled with large high voltage storage
capacitors! The “bite” of getting shocked is largely due to current.)
Now to the matter of periodically varying voltage sources – how fast does the voltage have to
change (average of the absolute value of dV/dt) in a sinusoidal voltage waveform that goes from
-1 V to +1 V with a frequency of 100 Hz?
Clearly, the voltage in the question above does not change at a constant rate. What is the
functional form of the time rate of change for V = sin (f t)?
Reactance
The frequency-dependent impedance (a general term for resistance to current flow, represented
by the symbol Z) of capacitors and inductors is called reactance and is given the symbol Xc for
capacitors and XL for inductors. The best way to characterize impedance and reactance uses
complex numbers. This leads to elegant representations that are very powerful, but most
chemists cringe when they see that little i or j. I’ll wave my hands about the complex
relationships in lecture and you can read about them in the text, if you wish. Fortunately, there
are easier-to-use real scalar expressions for reactance that work for most of the calculations that
we need to do. The reactance (in ohms Ω) for capacitors is XC = (2  f C )-1 and that for inductors
is XL = 2  f L, where f is the frequency (in Hz = s-1) and L is the inductance in Henrys. (In the
complex form, XC is multiplied by –j and XL by j, making them imaginary numbers.) The
impedance (reactance) of capacitors decreases with increasing frequency, while that of
inductors increases with increasing frequency. The impedance (resistance) of resistors is
independent of frequency. To a first approximation, impedances in series add (Ztot = Z1 + Z2 +
…) while impedances in parallel add in inverse (Ztot-1 = Z1-1 + Z2-1 + …) wherever the
impedances are coming from. The complex number treatment modifies these relationships to
give better answers, but the scalar versions are useful as a “first guess”.
Concept Question 3 – At what frequency does a 100 pF capacitor possess a reactance of 100
k?
Estimate the impedance of a circuit made up of a 100  resistor in series with a 1 μF cap at DC
(0 Hz) and at 10 kHz?
Estimate the impedance of a circuit made up of a 100  resistor in parallel with a 1 μF cap at DC
(0 Hz) and at 10 kHz?
Which element has the smaller impedance at each of the frequencies ( 0 and 10 kHz) in the
question immediately above? Which way would you go if you were current?
Circuit Exercise 1 - For an illustration of the somewhat foreign concept that capacitors have
impedances that depends on how fast voltage is changing, we will construct one of the two
commonly-used simple RC circuits, the high pass filter (a.k.a., differentiator or AC coupled
circuit). Set up the circuit shown below. The connection between the function generator and
ground is made internally within the trainer. We sometimes say the function generator is groundreferenced or that it “knows where ground is”. Proper schematics will show this connection, but
you don’t have to make it. There will be other examples of this as we go along. Set the function
generator FG to produce a 100 kHz sine wave by monitoring the input (between FG and the cap)
with one channel of the scope and the output (between the resistor and cap, as shown) with the
other channel of the scope. Trigger the scope off the input, since it will change in amplitude less
than the output. The size (amplitude) of the signal out is going to change a lot in this experiment,
and it might be easier to see with AC coupling on the scope. Experiment with changing the
frequency of the input before you start taking measurements. Make sure you know how to
measure the frequency and the voltage. Note that we are measuring the voltage drop across the
resistor (to ground). What we have constructed is a special type of voltage divider, where one of
the elements has impedance (and hence voltage drop) that depends on the frequency of the input.
Func. Gen.
100 pF
Scope
100 K
Predict what the output voltage will be at the low frequency limit (DC or f = 0 Hz).
Plot the measured voltage out (peak to peak amplitude) divided by the voltage in vs. the
frequency on a log-log plot (i.e., log (Vout/Vin) vs. log (frequency) and paste it in below. This
graph is called a Bode plot and it provides a good overall description of the filtering capabilities
of this circuit.
Why is this circuit called a high pass filter (i.e., what part of the signal gets through)?
At what frequency does the filter begin to pass signal?
What characteristics would a low pass filter have?
What would you guess a low pass filter circuit would look like?
Circuit Exercise 2 –You probably guessed above that to get a filter that favors low frequencies;
you just switch the two impedance sources in the voltage divider above, the cap and the resistor.
Then you get the other of the two simplest RC circuits, the low pass filter (a.k.a., integrator). Set
up the circuit shown below, noting that the components are different here than before! Drive the
input with the lowest voltage square wave you can get from the function generator at about 10
kHz. Monitor the input and output with the scope as you did before.
Func. Gen.
10 K
.01 uF
Scope
Why is this circuit called the integrator?
Lower the frequency of the square wave to around 100 Hz. Have a close look (in time) at the
output waveform, focusing on the beginning and end of the squares. (Hint: use the trigger
controls.)
Remember when we said that a little current flows through a cap when you first switch on a DC
voltage source? It is hard to do that “single shot” experiment, but in effect, that’s what we’re
seeing here.
Describe the functional form of the output. Measure the characteristic time for this function.
Switch back to the sine wave, and experiment with changing the frequency of the input a bit
before you start taking measurements.
Again construct the Bode plot - the ratio of measured voltage out to voltage in vs. the frequency
in on a log-log plot (i.e., log (Vout/Vin) vs. log (frequency). On your plot mark the following: 1)
the pass band, where the output is relatively insensitive to frequency change, 2) the point where
the output has fallen to ½ the pass band value, this is called the –3dB point, and the
corresponding frequency is called f3dB, and 3) the region where the output falls off fairly linearly
(on the log plot) with frequency – the rejection region. These are all very important
characteristics of a given filter, and may be predicted in the circuit design using the equations
described above.
Don’t tear out the low pass filter circuit, unless you’re out of time.
A caution on simple RC filters: It should be noted at this point that neither of these filters
functions very well. They only work well (close to the ideal) when the voltage out is only a small
fraction of the voltage in. When we learn how to work with operational amplifiers in a few labs,
we’ll be able to make much more functional versions of the filter circuits that are only a little
more complex (and costly).
Storage
The falling edge of the low pass filter waveform (at low frequency) gives you a pretty good
mental picture of the storage function of capacitors. If you connect a cap between a voltage
source and ground, in parallel with the rest of your circuit, and then remove the voltage source
quickly from the circuit, the cap will have to discharge (get rid of the charge) by forcing current
through the rest of the circuit. Thus the capacitor tends to resist sudden changes in voltage. This
can be used to smooth out sudden wiggles in the power supply for a circuit, and you will often
see big electrolytic caps connected between voltages (both negative and positive) and ground on
circuit boards, usually very near some component that requires a constant voltage. This as a lowpass filter application, but an alternative view is that the cap is a “current battery”, able to
mitigate a sudden draw from a device, or a sudden failure from the source. (Often these two
effects are linked, and the cap can help prevent them from feeding on one another to damage the
circuit.)
An important wrinkle: Phase Shift
You’ve probably noticed in the experiments above that the phase (where the peaks and valleys
are in time) of the input and output of these circuits is not the same and changes with frequency.
This is an important property of RC circuits, but one that we don’t use all that often as chemists.
That’s why engineers like to characterize the reactance of a capacitor as an imaginary number.
Then, since the resistance of a resistor is defined as all real, the real part of the impedance of the
resulting RC circuit has both the phase and amplitude information encoded. Hopefully we will
get a chance to look at phasor diagrams in the lecture, which is a nice way of looking at this
relationship. The phase shift that we get from inductors is exactly the opposite for a given
reactance from the phase shift observed for capacitors.
Circuit Exercise 3 –We’ll use the same circuit (low pass filter) you just built to investigate the
phase shift as a function of frequency - hopefully it’s still on the breadboard.
Measure the phase shift as a function of frequency, starting at about 100 Hz, and then going
through 1, 10, and 100 kHz. Report the results below or make a graph.
What does it look like the limiting phase shift is going to be at high frequencies? What is it
(obviously) at low frequencies?
If you’ve seen the phasor diagram, does this make sense? (If you haven’t seen it yet and want to,
call the instructor over.)
A caveat: We have treated all of the components we’ve seen so far as pure. Resistors only have
resistance, capacitors only have capacitance, the oscilloscope is purely resistive, etc. The truth is
that the real physical components can have characteristics of more than one type. A resistor may
possess some capacitance, indeed a simple piece of coaxial cable like we use to hook up to the
scope actually has some capacitance, as does the scope, a phenomenon which you saw when you
calibrated your probe in the scope mini-module. Typically capacitors have some inductance.
These types of considerations should be kept in the back of your mind as you go through this
course and use electronics in real-life to work with chemically relevant signals. Most of the time,
if the circuit you are using has been properly designed; these considerations will impose small
effects. But sometimes they can be important in explaining unexpected phenomena and become
more evident when you start “pushing the envelope”, making very high voltage or very fast
circuits.
Real World Example
Provide a real world example where the concept of filtering could be used to improve a
chemically relevant electrical signal.