BIOEN 316
Spring 2016
Homework 8: Op-amp circuits
Problems 1-4: due the afternoon of Tuesday, May 24, location TBA.
Problem 5: due 11 p.m., May 24, via Catalyst.
As preparation for the derivations requested in this homework set, you might want to
review examples 12.1 (non-inverting amplifier), 12.2 (inverting amplifier), section 12.7.1
(differential or difference amplifier), and section 12.7.5 (first & second order analog filters)
in the textbook (2nd edition). The difference amplifier may be solved by superposition as
shown in the example, or directly by using both inputs simultaneously.
Each solution should start with a circuit schematic that defines currents and node voltages.
Problem 8.1. Summing amplifier
Derive the transfer function (gain) of the summing circuit in figure 12.29, showing that
each of the terms in equation 12.23 should actually be negative. This problem is similar to
problem 8 in the text, but note that the problem should suggest that you use Kirchhoff’s
current law, rather than the voltage law.
Problem 8.2. Integrator / differentiator
Start with the first-order low-pass filter shown in figure 12.33. Remove the feedback
resistor Rf, and replace capacitor Cf with an inductor L. Derive the transfer function to
show that this circuit produces the derivative of the input voltage.
Problem 8.3. What is it?
Start with the inverting amplifier shown in figure 12.9. Replace Rf with a diode that permits
current to flow from the input to output, but not from output to input. Recall that the
voltage-current relationship for a diode is i = Iseλv, where v = forward voltage and λ=q/kT,
where q is the charge of one electron, k is Boltzmann’s constant, and T is temperature.
a) Derive the transfer function for this peculiar circuit.
b) Draw a circuit that could be used to multiply two signals. My circuit involves four opamps. You do not need to derive the transfer function for this circuit, but do state the
function of each component.
Problem 8.4.
Figure 12.34 shows an active, second-order, low-pass filter circuit. It is simplified in figure
12.35 by setting the gain to 1, such that that the non-inverting amplifier portion of the
circuit becomes a voltage follower. A similar second-order filter circuit – this time a bandpass filter – is shown on the Wikipedia Sallen-Key circuits page:
http://en.wikipedia.org/wiki/Sallen%E2%80%93Key_topology#Application:_Bandpass_fil
ter . An advantage of this circuit over the cascaded first-order circuits is that the output of
the Sallen-Key circuit as a whole is also the output of the op-amp, whereas in the cascaded
BIOEN 316
Spring 2016
circuits a second voltage follower would be necessary in order to prevent loading of the
circuit if a finite resistance were to be connected at the output of the second RC or RL stage.
a) Look on the EE stores online catalog, Linear ICs section, for the NE5534 operational
amplifier, which you could buy to put into a low-pass filter circuit or as the second op-amp
in a pair of cascaded first-order filters. What is the price, and would it be worth the cost for
you to buy an extra op-amp in order not to have to calculate the R and C values for the
Sallen-Key band-pass filter?
b) Go to the Newark.com online catalog (they ship free to the UW) and search for the
MCP6021-E/ST operational amplifier. I chose this one because it has a high bandwidth (it
can handle frequencies up to 10 MHz) and in a TSSOP case (which is pretty small).
As you increase the number N of amplifiers that you buy, at what value of N does it become
more economical to buy 100 rather than N+1 of them?
c) Look at the package information for the TSSOP-8 case, which is found here:
http://www.ti.com/lit/ml/mpds568/mpds568.pdf
Wherever you see a pair of numbers appearing in a fraction – such as 3.1 over 2.9 – it
means that the dimensions may be as large as the top number and as small as the bottom
number; the nominal value is in the middle.
What are the overall dimensions of this entire integrated circuit package?
Problem 8.5. This may be submitted via the Catalyst drop box.
In the past few weeks, you have analyzed a second-order band-pass RLC circuit to apply
before sampling, applied an FIR filter to extract EEG beta waves, and seen MATLAB’s IIR
filter design tool. For this homework, use MATLAB to compare the behavior of these three
filter types, each with cutoff frequencies of 0.5 and 50 Hz. For the digital filters, assume a
sampling frequency appropriate for the presence of 60-Hz noise and its first harmonic.
a) Simulate the second pass RLC circuit with its standard gain function G(jω). Note that
you do not actually need the R, L and C values themselves, just ωHI and ωLOW.
b) Use MATLAB’s butter() function or filter design tool, fdatool, to create a secondorder Butterworth IIR filter. Note that when you create the filter coefficients using the
butter() function, you need to pass it two normalized frequencies in the format
[w1 w2] for the two cutoff frequencies.
c) Create an FIR filter. After doing the IFFT of the original (low) pass band, truncate the
sinc function to keep one positive side lobe on each side; this is equivalent to keeping
points out to x = ±3π in sinc(x). Then multiply by the cosine as necessary.
Use MATLAB to overlay the magnitude spectra of these three filters on a single dB vs.
log(ω) or log(Hz) figure. If you can’t get them onto one plot, then use two or three subplots
in the same figure.