ASSIGNMENT #4

1. (30%) One of the enduring themes in building amplifiers was to reconcile the transistor turn-on non-linearity with achieving a reasonable amplifier gain with good linearity. Gilbert gain cells represent the "compensating inverse non-linearity" approach, emitter (or source) degeneration resistors represent a kind of camouflage effect. Op amps programmed with resistors are the reward of achieving a powerful new building block and then putting it back into a simple (retrograde, one might almost say) configuration.

(a) (10%) Discuss the three approaches above in terms of the "price paid" and what was gained in return.

(b) (20%) An entirely distinct and unique approach is the use of the "differential quartet." (I'll hand out in class). Simulate this circuit for V+=15V, using 2N3904 transistors (in simulation devices are always matched. Enjoy it while you can.).

Show an input/output plot to document the linearity of this differential amplifier. Use a common-mode input voltage level of +2V and a differential input large enough to exercise the full available linear range. Compare this to the linearity you would have had without Q3 and Q4, where Q5 and Q6 bias Q1 and Q2 directly, and a resistor of suitable value connects the emitters of Q1, Q2 to achieve the same small-signal gain of 36. Use the same range of differential input voltage as before. Make printouts from Multisim to document this performance comparison.

Extra credit (10%): Can you offer any explanation why this circuit is so linear? Is there any price paid for the improved linearity besides the two extra transistors? I.e. is there any unusual behavior introduced by Q3 and Q4 that could be detrimental?

2.(a) (10%) Consider the recent EENG 229Lb lab, particularly the common-source output amplifier stage, Fig.3, using the 2N5459 JFET. With reference to the 2N5459 datasheet, and the minimum and maximum transconductance (more formally called Forward Transfer Conductance on the data sheet) given there, what is the minimum and maximum voltage gain you expect from this stage?

(b) (15%) In the 324 schematic and gain calculation I handed out, the current sources are left in symbolic form. In the spirit of S&S Fig. 6.18, p.513, where a single resistor is used as "master current source", assume the LM324 operates on +12V, use a single resistor (not greater than 110k-ohm) as "master current source" and generate all other needed currents with current mirrors, regular or Widlar-type (S&S, p.517).
Extra credit (10%: generate the needed currents using an active "master current source" that is not as directly dependent on supply voltage as a resistor. This is the approach used in modern op amps. Discuss your circuit choice.)

3. (15%) Sedra & Smith 2.74 (Recall your work in EENG 229 with Bode plots of open-loop and closed-loop gain. What is called f_T here in S&S, is also called Gain-Bandwidth product and also corresponds closely to open-loop unity gain frequency.)

4. (5%) Sedra & Smith 2.93

5. (5%) Sedra & Smith 5.57 (also, what would the voltage gain become at I_D=500 uA for R_G=1 Meg-ohm?)

6. (5%) Sedra & Smith 5.108

7. (10%) Sedra & Smith 6.91

8. (5%) Sedra & Smith 6.96

Please submit your project (Lab#3) report along with the final questions (below). I will schedule time in the afternoon of Monday, May 12th (or earlier, by appointment) for you to give me the required brief demo of your project.

Longer-term Questions (in lieu of a Take-Home Final) - these are due Monday, May 12th.

You have choices. Do [(1) OR (2)] AND [(3) OR (4)]

(1) (30%) In our discussions of circuit "shapes" throughout the semester I have often referred to "design principles." Pick four from among the following five design principles, give a circuit example that you think exemplifies that principle, draw the circuit and discuss it thoughtfully.

• Symmetry (give two examples for this one)
• Equilibrium
• Eliminating "excess baggage"
• Superposition
• Analogies or Boundary Testing (pick one or the other)

(2) (30%) Derive and explain the output waveform of the circuit below, for the given triangular wave input. (Assume the input frequency is low enough to avoid any high-frequency limitations of transistor operation. Take the emitter-collector saturation voltage to be a fixed 0.3V. Assume the diodes are diode-connected transistors of the same type, i.e. the diode drops match the base-emitter drops. What might this circuit be used for?

(3) (70%) Our model for the BJT was based on the Schottky equation and its exponential relationship between junction curent and junction forward voltage. This led to some nice opportunities, such as the transconductance multiplication we used in the power meter and Gilbert gain cells.

The MOS transistor's governing relationship between drain current and gate voltage is quadratic. This means you should be able to build a "squaring" circuit, subject to suitable normalization. Apart from the MOS devices (for which you should state your assumed values of V_t and K), you would use op amps for the various voltage manipulations, offsetting, scaling, current-to-voltage conversion, making things "ideal", etc.

What you are really after is building a multiplier, based on this ability to square, because you realize that

This is called the "quarter-square" multiplier. If your squaring circuit permitted negative as well as positive inputs, such a multiplier would permit direct multiplication of both negative and positive variables. We will keep it simpler here and assume only positive values for V₁ and V₂ . Further assume that the inputs V₁ and V₂ and the output V_o are all to be normalized to 10V, i.e. V_o = 0.1V₁V₂

So there you have the premise. Starting with a squaring circuit (well thought-out, since everything else depends on it), design the overall "quarter-square" multiplier circuit, using op amps to manipulate the various voltages. Keep in mind that an op amp's output swing is constrained by the +/-15V supplies to (assume) around +/-12V max.

This problem will be graded with strong emphasis on clarity of thought. Please proceed very methodically in terms of circuit modules, so that assignment of partial credit is easier.

Extra Credit: (10%) Describe how you would elaborate the squaring circuit so that it could handle negative inputs also.

(4) (70%) Think for a moment: voltage and current are distinct electrical "degrees of freedom," independent electrical parameters. Therefore one ought to be able to send two separate signals on a single wire (with respect to a ground wire), one signal represented by voltage, the other by current. Ideal op amps are good at asserting voltages with very low source resistance, converting current to voltage and voltage to current, all things that are useful in this connection.

Because that's what I want you to do with op amps—design the circuits that go into the two boxes below, so that two signals (from dc to some modest frequency) can travel over the same pair of wires, of negligible resistance, (and think of one of them as ground) in opposite directions, with unity gain. I.e. V_i1=V_o1 and V_i2=V_o2, with -10V<V_i1<10V, -10V<V_i2<10V. The inputs should be high impedance and the outputs able to drive the kind of loads that op amps can drive, i.e. a few k-ohms. Use dual 15V power supplies. This problem will be graded with strong emphasis on clarity of thought and simplicity of implementation.

Lest you think this very strange and implausible, let me remind you of the telephone, where a single pair of wires carry two signals in opposite directions. Now you do hear yourself in your own headset, but this is done deliberately, to reassure you that the phone is working properly. Electrically it could be arranged instead that you do not hear your own voice in the earpiece at all, that only the other person does. But psychologically this would be unsettling, so telephone engineers have designed in what they call the "side-tone", so you can also hear yourself, but more softly than the other person's voice. (For modem communication over a phone line, elimination of the side-tone would be an advantage.)

The telephone circuit traditionally accomplishes this two-way signalling arrangement with a circuit based on transformers, a possibility at audio frequencies. But I want you to use the advantages of modern op amps and make the circuit work down to DC.

Hints:
- Work in well-explained steps toward the final solution (also a good way to get partial credit).
- Start with one signal V_i1 going across to V_o1. Easy. Just a wire.
- Now imagine that while V_i1 is asserted on the wire, on the V_i2 side a current is injected into the wire.
- It is sensed at the V_i1 side where it is turned into V_o2.

Configured according to those hints, the two sides of the circuit are not symmetrical. But since the intended function is symmetrical, shouldn't there be a circuit possible that is the same on both sides? It is possible. Does it violate the notion of the independence of the electrical parameters? No, because each signal could be represented by a symmetrical but still orthogonal combination of voltage and current.

Extra Credit: (20%) Evolve your op amp design to this symmetrical form. (That's clearly the kind of circuit you want for telephones, otherwise a "type A" phone could only talk to a "type B" phone.)
There is even a further degree of symmetry one could go after, not having one wire grounded, but having them electrically balanced. I mention that only to show a further aspect of symmetry, this important design principle, in this problem, not because I want you to "unfold" your design and make it symmetical in this additional way.