Physics 202 Spring 2014
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You may find it useful to refer to Smith’s discussion of the low pass R-C filter in section 1.10.
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For the next few weeks you will need to be familiar with the operation and manipulation of the oscilloscope, the function generator, the small white ‘breadboards’ for wiring up circuits, and the various resistors, capacitors, and inductors stored in the cabinets in front of you. Today is a chance to practice up, building on the skills and knowledge you gained in 201 (and patching up any holes
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therein).
In the center of the room we have a fancy meter (“The Digibridge”) which can measure the capacitance, inductance and resistance of components -- feel free to use this to check component values as you need. Remember that the oscilloscope and the signal generator are both ‘grounded’ instruments and thus the black wires for each must always be connected to the same point
(electrically speaking) in the circuit (because they are already connected to each other by way of the power cord). If you are not sure what that means or why it would matter, today would be a good
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time to ask.
If you are not yet comfortable using the basic functions on the oscilloscope, today is the day to rectify that situation. Make sure you get your hands on the knobs. No just sitting there “recording
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data!”
1. Measuring the output resistance of the signal generator. The definition of an “ideal” voltage source is one that delivers the same voltage no matter what kind of load is hooked up to the output (for today we will be concerned with resistive loads). A real signal generator (or other voltage source) will always show some decline in
R
S voltage when hooked to a small load (i.e. one that draws a lot of current). Often devices are
V constructed so that this drop-off in voltage happens as if the real signal generator consisted of an ideal signal generator in series with a resistor. The resistance R
S
is called the
“output impedance.” (In practice a voltage source may behave this way over the range of output currents it is intended to drive and then simply shut off if the current draw is too large. As long as the circuit is within the intended range of use, this model is the simplest one that captures the basic fact that the voltage will drop as the current draw increases).
R
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Physics 202 Spring 2014
Determine the output impedance ( R
S
) of your signal generator as follows: measure the output voltage (using the scope) as you connect successively smaller resistors as the load. (The
“resistance substitution box” (the one with the switches on it) is handy for doing this). While you could measure the amplitude off of the scope display you can also have the scope give you a numerical value (hit the “Measure” button, select “pk-pk” (ie. peak-to-peak voltage) for the appropriate channel). Looking at the front of the signal generator it says “50 Ω ” next to the output so don’t expect to see much change when using resistors much larger than 50 Ω . Go ahead and take some measurements and see if you can observe the voltage change.
Plot your data and fit it assuming that, as shown in the diagram above, the output characteristics of the signal generator are the same as an ideal signal generator with a series resistor. [ Hint: that means you are supposed to write down a formula and plot your data in such a way that you can compare the data to the formula. This may require a moment or two of thought ]. Are your measurements consistent with this model? What is the numerical value of R
S
?
If you assume that the signal generator output is well described by the series resistance model then the quick way to find the numerical value of R
S
is measure the output voltage for R = ∞ and then lower R until the output voltage drops to ½ of its original value. Do this, and check that you get reasonable agreement with your fitted value for R
S
.
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2. Lowering the output impedance of the signal generator. There is a way to reduce the output impedance of the signal generator (at the expense of reducing the output voltage).
Hook a 10 Ω resistor across the output of the signal generator (see sketch). This V
0
R
S
10 Ω
To rest of circuit naturally drops the output voltage a lot
(but we don’t care, just expand your scale on the scope!). We want to think about the
10 Ω resistor as part of the source, the load is what gets hooked up to the terminals. Determine
(experimentally) the output resistance of the “new signal generator” (i.e. the entire circuit inside the dashed box). [H ow? Using what you just learned above ]
R
S
R’
S
V
0
10 Ω ’ V’
0
It’s not too hard to calculate what the output resistance of the new signal generator should be.
Imagine you were to measure (with an ideal infinite resistance voltmeter) the output voltage of
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Physics 202 Spring 2014 this circuit. Then imagine you were to measure (with an ideal zero resistance ammeter) the output current. From those two numbers (which you can calculate for the left hand circuit above) you can find the parameters that would be needed to construct the circuit on the right
(i.e V
0
′ and R ′
S
).
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3. Determining the decay time for an RC series circuit. Recall that the resistance and capacitance in an
RC circuit are related to the time-constant, τ , associated with the exponential decay of the voltage across the capacitor by τ = RC . Choose a resistance in the k Ω range, and capacitance in the 0.1 to 0.01 µ F range. Construct the circuit shown and drive it with a square wave at sufficiently
V
0
R
C
V
C
low frequency that you can observe the alternate periods of decay and charging. You might let CH 1 display the function generator signal, while CH 2 displays the voltage across the capacitor V
C
. Manipulate the scope knobs to determine the experimental value of the 1/e time, τ , making sure that it is reasonable agreement with the chosen component values .
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4. The frequency response of an RC series circuit: amplitude. The complex impedance of a capacitor
(Smith page 23) is Z
C
= 1 i ω C
. Remember that ω is an angular frequency (radians per second) whereas in the lab one always measures frequency, f , in Hz (cycles per second). When comparing “theory expressions” to measurements you need to remember the factor of 2 π between frequency and angular frequency.
If one considers the RC series circuit as a voltage divider the natural dividing line between low frequency and high frequency behavior is when
ω
C
= 1 τ
Z
R
= Z
C
i.e. ω
C
= 1
RC
which is to say
. This is often called the “corner frequency” since on a log-log plot of output voltage vs frequency this is where the response “turns a corner” and switches from low frequency to high frequency behavior. (Smith page 22, where ω
LO
= ω
C
).
Set the signal generator to produce a sinusoidal signal and experiment with driving your circuit with various frequencies (from below to above the corner frequency (in Hz) of f
C
= 1
2 π
1
τ
). Is this is a “low-pass” or “high-pass” filter? Measure the output amplitude as a function of frequency.
Plot your data in Kaleidagraph and look for any gaps in data or obvious errors. It is helpful to make a log-log plot (double click on the axis of your plot and choose log). Fit your data to the expected theoretical expression (calculated on the fly right now or check out pages 21 and 22 in
Smith).
Is your fitted value of f
C
in agreement with the value of τ that you measured above? with the
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value expected from the component values?
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Physics 202 Spring 2014
5. The frequency response of an RC series circuit: phase. Study the relative phase of the function generator voltage and the voltage across the capacitor as a function of frequency. Based on calculations, what do you expect for the relative phase to be when the drive frequency is:
a. much less than f
C
b. equal to f
C
and
c. much higher than f
C
.
For each case, draw yourself a phasor diagram like the ones shown in Figure 1.9.2 of the
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textbook. Are your observations consistent with these diagrams?
6. Using the oscilloscope in ‘X-Y’ mode.
Ready for a cute trick? Normally the scope displays voltage vs time (“YT” mode). However it has the option to instead display voltage on CH 2 vs voltage on
CH 1 (“XY” mode). This is an example of a “parametric plot,”a plot of y(t) vs x(t) with t being the parameter.
To put the scope in XY mode, hit the “Display” button, choose XY from the menu (instead of
YT). Set the two channels to be on the same scale (volts per division). Observe the behavior of the plot as you vary the signal generator frequency.
Working from the phasor diagram for the circuit, you should be able to qualitatively understand
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the shapes you see.
7. An LR circuit. An inductor has a complex impedance of Z
L frequency dependent quantity, unlike a capacitor the
= i ω L . Like a capacitor, this is a inductor’s impedance increases with frequency. Look at the circuit shown and predict whether it should act as a “low pass” or a “high pass” filter. Predict the corner frequency, f
C for L = 10 mH (milliHenry) and R = 1 k Ω . When the drive
, frequency is f
C
what should be the phase relation between
V
0
and V
R
? What if the drive frequency is well below well above f
C
?
f
C
?
V
0
L
R V
R
Construct the circuit and test your predictions (You do not need to take extensive measurements: you should quantitatively confirm the value of f
C qualitatively).
and check the phase behavior
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8. Trouble in paradise.
Replace the 1 k Ω resistor in your LR circuit with a 10 k Ω resistor and measure the corner frequency. Now try a 20 k Ω resistor. See anything odd?
In our theoretical calculations we assume ideal inductors, resistors and capacitors which only approximate real components. An actual coil of wire has some inductance, some capacitance and some resistance (even that is not the full story, as we will see next week). As you have just
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Physics 202 Spring 2014 observed, an “inductor,” all by itself, can exhibit the resonance behavior of an LC circuit (which we will examine in detail next week). In making electronic components manufacturers engineer components to behave as closely as possible to ideal inductors, capacitors and resistors but there are always tradeoffs and compromises depending on what other characteristics are desired. As a general rule inductors are often more problematic than capacitors and resistors and are less likely to be found in circuits.
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Today’s lab completion requirements: Make sure you have checked with me before you leave.
Show me your graph/fit for the RC filter (section 4) and give me a brief explanation of phase issues in this circuit. If you have any lingering questions about your observations, now is the time to
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discuss them.
Future labs will require some writing up and handing in. I will let you know about that in each week’s lab.
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