Download 2 Impedance and Transfer Functions

2 IMPEDANCE AND TRANSFER FUNCTIONS
2
Impedance and Transfer Functions
The concepts of impedance and transfer functions are are essential for a clear understanding
of analog electronic circuits. This exercise will explore these ideas in both dc and ac contexts,
and discuss simple applications in voltage dividers and filters.
This lab will require two days.
Reading: HH Sections 1.16–1.24 (pgs. 28–44)
2.1
Voltage Divider
The voltage divider, Fig. 1(a), is a simple but deceptively important circuit. It’s basic use
is to reduce a voltage, according to
Vout =
R2
Vin .
R1 + R2
Construct a voltage divider with the values shown. Measure Vout and check that it agrees
with the formula. Measure R1 , R2 and Vin so that you can make the comparison with some
precision.
The voltage divider introduces the important concepts of input and output impedance.
To understand the output impedance, short the divider output to ground through your
current meter, measuring the short-circuit current Iout (short). The output impedance Zout
is defined by
Vout
Zout =
Iout (short)
where Vout is the open-circuit output voltage you first measured. With this definition, Zout is
the Thevenin output resistance for the equivalent circuit of Fig 1(b). What Zout do you obtain
for your circuit? How does it compare to the expected value for a divider, Zout = R1 k R2 ?
Hook up a 7 kΩ resistor from the output to ground, and measure the resulting output
voltage. Compare to what the Thevenin model predicts based on Zout .
Vin = 15 V
Zout
Vout
R1 = 10k
Vout
R2 = 5k
(b)
(a)
Figure 1: (a) Voltage divider circuit. (b) Thevenin equivalent circuit.
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2.2 Cascading Circuits
2 IMPEDANCE AND TRANSFER FUNCTIONS
The input impedance Zin is the effective resistance seen from the input of the circuit (Vin )
to ground. It can typically be measured directly with an ohmmeter. For a voltage divider
with no load, Zin is evidently R1 + R2 . What it the Zin for your divider circuit with the 7 kΩ
load resistor? Compare a measurement and calculation.
2.2
Cascading Circuits
Suppose we want to divide a signal by 3, and then by 3 again. Design and assemble a pair
of voltage dividers to do this to an accuracy of at least 10%. To avoid drawing too much
current, don’t use resistors with R < 100 Ω. If you have trouble, recall that cascading circuits
is simple if you ensure that the input impedance of the second circuit is high compared to
the output impedance of the first circuit. Make sure you understand this point. Describe
the circuit and its performance in your report.
2.3
DMM Impedance
Construct Fig. 2 using a Fluke DMM on the voltmeter setting. Because the input impedance
of the DMM is finite, some current will flow through the 10 MΩ resistor, and the meter will
not read 10 V. Equivalently, the resistor forms a voltage divider with Zin for the DMM,
and the voltmeter reads the divided voltage. Use this measurement to determine Zin for the
meter. Compare to what you get for the ELVIS DMM.
Could you alternatively measure Zin for one DMM using the ohmmeter of the other?
What happens if you try?
If you made a voltage divider circuit with R1 = 1 MΩ, R2 = 2 MΩ, and Vin = 10 V, what
reading would you expect to see if you measured the output with a Fluke DMM?
Devise a method to measure the input impedance of a current meter, and determine its
value for both the Fluke and ELVIS meters. In what situation could the finite impedance of
the current meter cause a measurement error?
2.4
RC Filter
The RC circuit of Fig. 3 can also be considered as a voltage divider, with the resistance R2
replaced by the frequency-dependent impedance Z2 = 1/iωC. The output voltage can then
Figure 2: Measuring the DMM impedance. The DMM should be on the voltmeter setting.
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2.5 Bode Analyzer
2 IMPEDANCE AND TRANSFER FUNCTIONS
Figure 3: RC filter circuit.
be immediately expressed as
Vout =
Z2
1
Vin =
Vin .
R 1 + Z2
1 + iωRC
In general, a relationship like this defines a transfer function G(ω) via Vout = G(ω)Vin . Here
the magnitude
√
|G| = 1/ 1 + ω 2 R2 C 2
is nearly 1 for small ω, but decreases like ω −1 for large ω. This circuit can therefore be used as
a low pass filter. The cutoff frequency at which the filtering action begins is fc = (2πRC)−1 .
If G is expressed as |G|eiφ , the phase φ represents a phase shift applied to a sinusoidal
signal. For the RC circuit,
φ = − tan−1 (ωRC).
The phase shift can be measured by comparing the time delay ∆t between zero crossings of
the input and output wave forms, with φ = −ω∆t in radians, or φ = −360f ∆t in degrees
for frequency f . By convention, φ is negative when the output lags the input.
Set up a low pass filter using R = 10k and C = 10 nF. Use the oscilloscope to measure
the attenuation and phase shift of a sine wave at frequencies of 10, 100, 300, 1000, 3000,
10k, and 100k Hz. Plot the resulting values in Excel.
Because of the large range in frequency and amplitude, it is generally more convenient
to make plots on a log scale. Conventionally, the amplitude of the transfer function |G| is
measured in decibels (dB), given by g = 20 log10 (Vout /Vin ). Make another pair of plots for
your data with the gain in dB and phase shift in degrees vs. log(f ). This representation of
a transfer function is termed a Bode plot.
For comparison, use Excel to calculate the theoretical values for g and φ, and add them
to your plot.
2.5
Bode Analyzer
Bode plots are a powerful tool for analyzing circuit performance, but they can be tedious to
measure. The ELVIS Bode Analyzer instrument can simplify this process. Find the analyzer
instrument on the computer and open it. Set the Stimulus Channel to Scope Ch 0, and the
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2.6 Filtering Signals
2 IMPEDANCE AND TRANSFER FUNCTIONS
Response Channel to Scope Ch 1. Leave the function generator hooked up to your circuit,
but close (or at least turn off) the FGEN tool.
Make the following connections to use the analyzer tool:
1. Use a cable to connect the function generator output to the Scope 0 connector on the
side of the breadboard. This is how the instrument measures Vin .
2. Hook Vout from the circuit to the Scope 1 connector. This provides a measurement of
Vout .
Use the Bode tool to take a measurement from 10 Hz to 100 kHz. It should generate a
plot similar to the data you took by hand. Save the log file and import it into Excel, and add
the data to your previous graph. The graph will be clearest if you use lines for the ELVIS
data and points for the data you took by hand. Note any disagreements you observe.
Swap the resistor and capacitor, and use the analyzer to observe how the Bode plot
changes. What kind of filter is the circuit now? Plot this data in your report, and work out
a theoretical calculation for comparison.
2.6
Filtering Signals
Filters are mostly used for eliminating noise, so to see them in action we need to create
a noisy signal. This can be achieved by adding a high frequency signal from the function
generator to a 60 Hz signal derived from the wall power lines. To start, locate the 6.3 V
transformer and plug it in. Observe its output on the scope using a BNC cable and a banana
plug adapter (available in the supply cabinet). How does the amplitude of the signal compare
to what you expect? When working with the transformer, be careful not to short the two
outputs together, as this will blow the instrument’s fuse.
Because the transformer’s output is floating with respect to ground, it can be added to
the function generator signal using the simple circuit of Fig. 4. The 1k resistor sets the
output impedance high enough to limit the current flow to a few mA. Assemble the circuit
and observe the output on the scope. Depending on the time scale setting, you should see
either a 60 Hz signal with high-frequency “noise” or a 20 kHz signal with a fluctuating offset.
Suppose first that the low frequency signal is of interest. Pass the composite signal
through a low-pass filter with C = 10 nF and R = 10 kΩ and describe the output. Does
out
Figure 4: Composite signal generator.
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2.7 Cascaded Filter
2 IMPEDANCE AND TRANSFER FUNCTIONS
in
out
Figure 5: Cascaded filters.
the filter successfully isolate the low-frequency component? What is the amplitude of the
residual high frequency component, and does that amplitude agree with expectations?
Repeat these measurements with a high-pass filter using the same R and C.
2.7
Cascaded Filter
If you want to attenuate high frequency signals more effectively than the circuit of Fig. 3
achieves, you can cascade two filters, as in Fig. 5. Design such a filter with a cut-off frequency
of about 1.5 kHz. There are many ways to achieve this, but the design will be simple if
you ensure that the input impedance of the second filter is much larger than the output
impedance of the first filter. In that case, the attenuations of the two filters will simply
multiply. (Compare to the experiment with cascaded dividers from Section 2.2.)
Put your circuit together and measure its Bode plot with the analyzer. Include the data
in your report. Compare to what you saw for the filter of Fig. 3. Note that when the phase
φ drops below -180◦ , the Bode analyzer wraps the phase around to +180◦ . This makes the
plot hard to read, but you can fix it by manually subtracting 360◦ from the appropriate
points in Excel.
Run the composite signal of Fig. 4 through the double filter. Does it perform better than
the single filter did?
2.8
Scope Impedance
We noted earlier that a DMM has a finite input impedance that can affect measurement accuracy. The same is true for oscilloscopes, and the impact is often more significant. The situation here is more complicated because ac signals are involved, so the scope input impedance
is complex and frequency dependent.
You can measure Zin for the oscilloscope using the circuit of Fig. 6(a). Channel 1 monitors
the input voltage, while the difference between Ch 1 and Ch 2 reveals the amount of current
flowing through the resistor, and thus into the scope. The scope input is typically modeled
as a resistor in parallel with a capacitor, as in Fig. 6(b). At low drive frequencies, the resistor
will dominate the impedance, allowing R to be measured. At high frequencies, the capacitor
2-5
2.8 Scope Impedance
2 IMPEDANCE AND TRANSFER FUNCTIONS
Input
1M
Scope, Ch 2
Scope, Ch 1
(a)
(b)
Figure 6: (a) Circuit for measuring oscilloscope impedance. (b) Model for oscilloscope
impedance.
will dominate. Using this approach, what values for R and C do you observe? Note that
the type and length of cable used to connect to the scope can affect this measurement.
For even moderately high impedance sources, the scope can be a significant load on
the circuit at higher frequencies. This effect can be reduced by using a 10× scope probe,
described in Appendix A of the text. The probe contains a voltage divider that reduces the
signal level by a precise factor of 10. This is not itself particularly desirable, but the divider
also increases the input impedance by about the same factor. Replace the scope cable with
a 10× probe and measure the impedance again. Make sure that the probe is on the 10×
setting.
To get the best performance with a probe, it needs to be properly compensated. (Again,
see Appendix A.) Hook your probe up to the Probe Adj terminal on the scope and observe the
square wave it produces. Find the calibration adjustment screw on the probe, and adjust
it to minimize the overshoot and undershoot. This sets both the capacitive and resistive
parts of the voltage divider to 10×, so you get uniform attenuation at all frequencies. In
general, it is a good idea to use a properly compensated probe except when you are looking
at small signals for which the 10× attenuation is unacceptable. When using a probe, set the
oscilloscope input calibration appropriately so that the scale readings compensate for the
probe attenuation. Recall that this is done by holding down the Selector switch and rotating
the Variables knob.
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