Notes on Lab #1 - dc Circuits
Physics 219 – Fall, 2002
Notes on Lab #2 – Capacitors
2.1 Introduction 1
2.2 Capacitors in Parallel 4
2.3 Capacitors in Series 5
2.4 Reactance of a Capacitor 6
2.5 Complex Impedances 8
2.6 RC Circuits 11
Response of a RC Circuit to a "Step Input" (Student Manual section 2-1) 12
2.7 Differentiator (Student Manual section 2.2) 14
2.8 Integrator (Student Manual Section 2-3) 17
2.9 Low Pass Filter (Student Manual Section 2.4) 18
Analysis Using Complex Impedances 20
Decibels 22
2.10 High Pass Filter (Student Manual 2.5) 23
Analysis Using Complex Impedances 24
2.11 Blocking Capacitor (Student Manual 2.8) 25
2.1 Introduction
We now add a new circuit element to our repertoire, the capacitor.
[pic]
[pic]
For convenience we'll drop the "Δ" and write
[pic]
which can be rewritten
[pic].
(Recall from Physics 108 that generally
[pic]
where A is the area of the plates, [pic] is the dielectric constant of the material between the plates and d is the separation between the plates. Because the capacitance is proportional to the plate area, the physical size of a capacitor tends to increase as the capacitance becomes larger.)
In our hydrostatic analogy for electrical circuits, capacitors can be thought of as bathtubs. "q" is the amount of water in the tub, and "V" is the depth of the water in the tub.
[pic]
The "capacity" of the tub is proportional to its area. Thus, a tub with large "capacity" will acquire a small depth of water for each gallon of water that is added to it.
If we change the value of the voltage on the capacitor charges will flow onto (or off of) the capacitor’s plates according to the defining relation
[pic]
That is, for a capacitor:
Changes in voltage lead to currents.
(In the water analogy, we increase the pressure at the bottom of the tub, say by connect to some tall water tower, then water will flow into the tub.) But if the voltage doesn’t change then no currents flow.
[pic]
[pic]
The amount of current that flows in the capacitor will be bigger when C is bigger and when the voltage is changing rapidly. As we shall see below, in many ways a capacitor behaves like a “frequency-dependent resistor” whose effective resistance is, roughly,
[pic]
where [pic] is the frequency at which the voltage is varying.
Note that this is quite different to the behavior of a resistor, where a constant voltage across the resistor will cause a current that is directly proportional to the size of the voltage.
[pic]
Below we will undertake a more formal (that is, more mathematical) analysis of the behavior of circuits with capacitors. This more formal analysis will produce a good quantitative description of how these circuits perform. But this very simple view of thinking of capacitors as “frequency-dependent resistors”, along with a Thevenin view of the world which reduces just about everything to voltage dividers, will be amazingly useful in leading to a good qualitative understanding of how these circuits behave.
The SI unit for capacitance is the farad:
1 farad = 1 F = 1 coulomb / 1 volt
One farad is a huge capacitance. (You’ll rarely use a 1 F caacitor.) Typical capacitances encountered in the lab are in the microfarad range:
1 µF = 10-6 F
or even smaller. The picofarad is another useful unit:
1 pF = 10-12 F
Due to an international conspiracy of unknown origin, capacitor manufacturers appear to go to great lengths to obscure the value of the capacitance of a particular capacitor. (All kidding aside, it’s probably in part because it makes it harder for people to reverse engineer a circuit.) The situation is made worse by the fact that capacitance meters are relatively scarce (we don't, for example, have one in the lab) and are more awkward to use than ohmmeters.
2.2 Capacitors in Parallel
[pic]
Suppose we place an amount of charge +q on the top terminal and an amount -q on the bottom terminal of the above arrangement of parallel capacitors. These charge will distribute themselves over the top and bottom plates respectively:
[pic]
By equivalent capacitance we mean
[pic]
Since charge is conserved we have
[pic]
Therefore
[pic]
=> parallel capacitors "add"
=> largest capacitor “dominates”
2.3 Capacitors in Series
Consider now the following arrangement of capacitor in series:
[pic]
Suppose we again place an amount of charge +q on the one terminal and an amount -q on the other terminal. We can calculate the equivalent capacitance from
[pic]
[pic]
Thus
[pic]
=> smallest capacitor dominates series capacitance
2.4 Reactance of a Capacitor
Capacitors become most interesting when there is some time dependence in the circuit. Consider the circuit shown below:
[pic]
From the definition of capacitance we have that
[pic]
If we differentiate both sides of this equation with respect to time, keeping in mind that C is a constant, we have
[pic]
Since
[pic]
this means that
[pic]
or
[pic]
Suppose that
[pic]
This implies that
[pic]
where [pic] is the amplitude of the time varying current:
[pic]
Note that there is a phase shift of 90 degrees between V(t) and I(t) and that the ratio of the voltage amplitude to the current amplitude is given by
[pic]
Recall that for resistors driven by a sinusoidal voltage we found (from Ohm’s Law) that V(t) and I(t) were "in phase" and that the ratio of the voltage amplitude to the current amplitude was
[pic]
Therefore, for capacitors the quantity [pic] is somewhat analogous to a resistance in that it gives the ratio of the voltage amplitude to the current amplitude. The quantity [pic] is called the capacitive reactance and is denoted by the symbol [pic].
In many ways a capacitor behaves like a resistor whose frequency dependent resistance is given by [pic].
Thus at high frequencies capacitors start to look like a "short circuit" while at low frequencies they start to look like an "open circuit".
2.5 Complex Impedances
A more sophisticated way of dealing with these matters is to introduce complex numbers. When capacitors (and also inductors) are added to the mix, Ohm's Law can be generalized as
[pic]
where Z is called the complex impedance. For resistors the complex impedance is real and is given by
[pic]
while capacitors have an imaginary complex impedance given by
[pic]
where [pic].
The rules for finding the equivalent complex impedance for combinations of impedances look just like the rules for resistors:
Series combination:
[pic]
[pic]
Parallel combination:
[pic]
[pic]
Examples:
1)
[pic]
[pic]
From the generalized form of Ohm's Law [pic] it can be shown (see Horowitz and Hill pp. 32-33) that:
The magnitude of [pic] gives the ratio of the amplitude of V(t) to the amplitude of I(t).
The angle that [pic] makes with the real axis gives the "phase difference" between V(t) and I(t).
2)
[pic]
[pic]
3)
[pic]
[pic]
Note that in this case [pic] as expected.
2.6 RC Circuits
[pic]
Ohm's Law implies
[pic]
or
[pic] (1)
For the capacitor
[pic] (2)
Differentiating this last equation with respect to time gives
[pic]
or
[pic] (3)
Using 2.4.3 in 2.4.1 gives
[pic] (4)
Eqn. 4 is the differential equation that governs the behavior of the RC circuit. If [pic] is known then this equation can be solved to find [pic].
Response of a RC Circuit to a "Step Input" (Student Manual section 2-1)
Consider the specific case of a "step function" input. At t = 0, with the capacitor uncharged, the input voltage changes abruptly from 0 to A.
[pic]
Example: Find [pic] for times or t > 0, given the "step function" input shown above. That is, find [pic] when the input voltage given by
[pic]
For times t > 0, equation 2.4.4 above implies
[pic]
We seek a function [pic] that is a solution to the differential equation 2.4.4. It is easy to verify that any function of the form
[pic]
where B is any constant, is a solution to this equation.
We must choose the constant B so that the initial conditions are satisfied. In this case we require that
[pic]
since the capacitor is assumed to be uncharged at t = 0. This condition will be satisfied if B = -A so that the solution becomes:
[pic]
A graph of this solution looks like
[pic]
(Recall that e-1 = 0.37)
The quantity RC is called the time constant. It corresponds to the amount of time that it takes for the output voltage to get ~2/3 of the way to its final value.
Conclusion: The RC combination introduces a time lag ~RC. Because of the presence of the capacitor, the output voltage cannot instantaneously follow changes in the input voltage.
2.7 Differentiator (Student Manual section 2.2)
[pic]
Recall that if
[pic]
then
[pic]
Also, for this input voltage we have
[pic]
Now let us add a small resistor R in series with the capacitor. By “small” we mean that [pic] so that the capacitor is still predominately responsible for the total complex impedance. Thus adding a resistance R that is “small” in this sense should not appreciably alter the behavior of the circuit.
Let us take the voltage across the resistor to be the output of this circuit:
[pic]
Ohm's Law implies
[pic]
or, since the current is only slightly changed by the addition of R:
[pic]
Therefore
[pic]
[pic]
Conclusion: If [pic] then
[pic]
Thus in this case the above circuit acts as a differentiator.
The Fourier Theorem states that any periodic function can be constructed as a linear combination of sin and cos functions. This allows us to extend the above discussion to cover a wide variety of input functions. The circuit will act as a differentiator provided all the "frequency components" present in the input voltage satisfy the condition[pic].
For example, suppose Vi(t) is a square wave.
[pic]
The output voltage in this case will be approximately equal to the derivative of the function, but the peaks won't be as sharp as expected:
[pic]
The peaks have a width ~RC. The circuit is unable to differentiate the high frequency components of the square wave. Alternatively, we can view this result as a consequence of the time lag introduced by the RC combination in responding to the input voltage.
2.8 Integrator (Student Manual Section 2-3)
Now suppose that [pic]so that the resistor dominates the total impedance of the RC combination. We can write this condition as [pic]. (Note: This means that this condition is violated if the input voltage has a dc component.)
[pic]
Again we apply a square wave input:
[pic]
where A is the amplitude of the square wave and T is the period so that
[pic]
If [pic]then [pic] so that the capacitor doesn't charge up very much during the charging portion of the cycle. This implies that
[pic]
for all time t.
Ohm;s Law implies that
[pic]
The voltage across the capacitor is given by
[pic]
[pic]
[pic]
Hence the name integrator. (It is important to keep in mind that in deriving this result we assumed that [pic].)
In words: The output voltage [pic] is proportional to the charge on the capacitor which in turn is equal to the integrated current. But since the current is approximately equal to [pic] we have that[pic] is approximately proportional to the integral of [pic].
2.9 Low Pass Filter (Student Manual Section 2.4)
Circuits containing capacitors will obviously behave differently as the frequency changes. For example, consider
[pic]
We can think of this circuit as being a kind of voltage divider, if we think of the capacitor as being a kind of "frequency-dependent resistor". Let us reconsider the case of a voltage divider circuit consisting of two resistors:
[pic]
We found in that case that
If [pic] then [pic]
If [pic] then [pic]
Similarly, for the RC circuit shown above:
If [pic] then [pic]
If [pic] then [pic]
Therefore, for low enough frequencies[pic] while for high enough frequencies[pic]. Therefore the circuit is a low pass filter. Low frequencies "make it through to the output" unattenuated while high frequencies are attenuated.
[pic]
The cutoff frequency [pic]roughly represents the boundary between frequencies that pass and those that don't pass through the filter.
Analysis Using Complex Impedances
A more exact analysis of the low pass filter can be made using complex impedances:
[pic]
[pic]
For the low pass filter
[pic]
and
[pic]
so that
[pic]
[pic]
[pic]
[pic]
[pic]
where A is a complex number. The magnitude of the complex number A gives the ratio of the amplitudes of the output voltage and input voltage. The magnitude of A is found to be:
[pic]
Note that when [pic] the magnitude of A is equal to[pic]. Thus at the cutoff frequency [pic] the signal is attenuated by a factor of [pic].
The "phase angle" of the complex number A gives the phase difference between [pic] and [pic]. It is easy to see that at frequencies well below the cutoff frequency, the phase angle of A approaches zero (that is, A becomes almost entirely real) while at frequencies well above the cutoff frequency, the phase angle of A approaches 90 degrees (that is, A becomes almost entirely imaginary.)
Decibels
The ratio of two voltages is usually given in decibels, which are defined by
[pic]
where A2 and A1 are the amplitudes of the two voltage signals. Thus, in the above low pass filter circuit we can see that at the cutoff frequency 0, where the ratio of the signal is being attenuated by a factor of [pic] the attenuation expressed in dB is given by
attenuation in decibels = 20 log10 [pic] = -3.01 dB
You'll often hear people say things like the beyond the cutoff the response of a low pass filter drops at a rate of 6 dB per octave. Can you figure out what this means?
2.10 High Pass Filter (Student Manual 2.5)
[pic]
For the RC circuit shown above, if we think of it as a voltage divider in which the capacitor acts like a “frequency dependent resistor” we find that:
If [pic] then [pic]
If [pic] then [pic]
Therefore, for high enough frequencies[pic] while for low enough frequencies[pic]. Therefore the circuit is a high pass filter. High frequencies "make it through to the output" unattenuated while low frequencies are attenuated.
[pic]
Again, the cutoff frequency [pic]roughly represents the boundary between frequencies that pass and those that don't pass through the filter.
Analysis Using Complex Impedances
A more exact analysis of the high pass filter can be made using complex impedances:
Using an analysis similar to that used for the low pass filter:
[pic]
[pic]
[pic]
[pic]
[pic]
Once again, the magnitude of the complex number A gives the ratio of the amplitudes of the output voltage and input voltage. (The "phase angle" of A gives the phase difference between [pic] and [pic]) The magnitude of A is found to be:
[pic]
Note that when [pic] the magnitude of A is equal to [pic]. Thus at the cutoff frequency [pic] the signal is attenuated by a factor of [pic].
2.11 Blocking Capacitor (Student Manual 2.8)
Suppose you connect two different power supplies set to two different voltages to a common point, as is shown in the example below where a 15 volt supply and a 5 volt supply are both connected to point A:
[pic]
A "tug-of-war" results, with each supply trying to force the common point to different voltages. Who wins?
The way to answer this sort of question is to think in terms of the Thevenin equivalent circuits for each of the power supplies. If [pic] is the Thevenin equivalent resistance (or output impedance) of the 15 volt supply and [pic] is the Thevenin equivalent resistance of the 5 volt supply, then the circuit really looks like this:
[pic]
which is equivalent to
[pic]
The voltage at point A can be readily calculated:
[pic]
From this expression in is easy to see that if [pic] then [pic] is close to 15 V while if [pic] then [pic] is close to 5 V.
Thus the power supply with the smaller output impedance tends to win out.
Suppose you have a sine wave generator with a 50 Ω output impedance and you wish to add a "dc offset" of 5 V to the sine wave. How can you do this? One bad idea is shown in the circuit below:
[pic]
Since the voltage divider shown has an output impedance that is much larger than 50 Ω , the sine wave generator "wins the tug-of-war" and the offset is much less than 5 V. (Can you calculate what it is?)
A much better idea is to insert a "blocking capacitor":
[pic]
Then, for dc voltages, the sine wave generator is effectively not connected to point A, while for ac voltages it is. The result is that the ac voltage supplied by the function generator will "ride on top of" the 5 V dc voltage supplied by the voltage divider.
[pic]
Another example of the use of a blocking capacitor can be found on the input channels of your oscilloscope. The "ac coupling mode", which you can select with a toggle switch, works by inserting a large blocking capacitor (~10 µF) into the signal path. This "blocks" any dc component the signal might have, so that only the ac component of the signal will be seen by the scope. Another way of thinking about this is view the RC circuit formed by the blocking capacitor and the 1 MΩ input impedance of the scope as a high pass filter with a very small cutoff frequency:
[pic]
In the "dc coupling mode" there is no blocking capacitor in the signal path so that both the ac and dc components of the signal are seen on the scope. The dc coupling mode is the default setting for your scope; ac coupling should only be used in those special circumstances where to are trying to view a small ac signal that's riding on top of a large dc offset.
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