Engineering Spring Homework Assignment 4: BJT Biasing and Small Signal Properties

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1 Engineering Spring 2011 Homework Assignment 4: BJT Biasing and Small Signal Properties 1.) The circuit below is a common collector amplifier using constant current biasing. (Constant current biasing is common in integrated circuits and is the main reason the Early effect is ultimately important.) I want you to find its output impedance algebraically and then to apply that result after setting up the biasing. The resistor R o is the effective source impedance of the I EE current source. VCC= 15 volts R2 R3 CIN h fe = 150 R1 I EE R o 1.1) Using the h-model of a transistor, find an algebraic expression for the small-signal output impedance including in that expression r o of, R o of the current source, the source resistance of the signal (R3), and R BB. (The h-model is the first one I used in class - it has the collector current controlled by the base current.) You may assume that the reactance of CIN is very small at all frequencies of interest. [Hints: substitute the small signal model for and apply a test signal to the output rather than the input. The shown on the left would be zero. You can do that when the model is linear. The ratio of the test signal voltage to the resulting test current is the output impedance. You can simplify the algebra by forming the Thevenin equivalent of R1, R2, R3, and before applying the test signal with set to zero. Call that equivalent resistance R EQ.] 1.2) Suppose the compliance of the current source I EE is 0.5 vout VCC. This sets the lower limit on. It is rarely desirable to forward bias a collector-base junction and that sets the upper limit. Find the optimum Q-point for the circuit, ignoring r o, R o for the moment. [The optimum formula for the common emitter circuit that I developed in class is not quite appropriate here. Use the idea of maximizing the possible symmetrical output swing directly.]

2 1.3) Suppose that I EE is 0.1ma, R1 = 100 K, R3 = 10 K, V BE = 0.7, and h FE = h fe = 150. Find R2 for optimum operation. [Hint: use KCL at the base node.] 1.4) Suppose V A = 200 volts and Ro ro. Find the output impedance. 1.5) Suppose the lower 3DB point is to be 20 KHz. What should CIN be? 2.) The growth, processing, distribution, and preparation of food is the largest single industry in the United States. Personally my favorite products of this industry are baked goods, sugar being my favorite food group. The circuit below is a small piece of a production mixing system for cookie dough. It (in my dreams) is a vibrator that shakes the sugar chute to keep the flow of sugar constant as part of the first continuous casting cookie factory. A flow sensor changes the input signal under closed-loop fuzzy-logic computer control. Your job is to select the biasing resistors R1 and R2 to ensure that the circuit will always be able to shake the chute enough. Determine a set of resistor values and show by direct calculation that the circuit will maintain between 80% and 125% of the optimum I CQ over transistor variations due to manufacture and temperature. (The plant is one of several to be built around the world, and each plant uses several of the shakers. No one wants to tweak each one!) The frequency of the input signal is always 60 Hz. Pick a value of C IN such that the loss in the input coupling circuit is less than 1 DB for nominal h fe = VCC = 40 volts R2 C 0.06 H C B 10 ohm - Darlington Pair R1 E 1.5 ohm The two transistors in the dashed box are called a Darlington pair and may be purchased as a single device. They act together like a single transistor with the effective terminals marked C, E, and B. The current gain, h FE is 750 min and 20,000 max. Because there are two base-emitter junctions in series between the base and emitter terminals of the compound device, the effective V BEQ is higher than for a single transistor. You may assume that the measured V BEQ of 1.35 volts at 45 C is reliable for all devices. However, re- 2

3 member that each of the internal transistors is temperature sensitive and each of the internal V BEQ s changes by -2.2 mv per degree C. The circuit must work over a range of temperature from about 10 C in winter on startup to 80 C in summer operation. Because of the internal compound structure, the saturation voltage of the Darlington pair is high -- about 1.1 volts. Also this circuit is an example of reactive load used over a restricted frequency range and the appropriate formula for the optimum quiescent current in circuits of this type is the same as the one I derived in class for resistive loads even though the load is highly reactive in this case. 3.) Here is a this-and-that problem, the sort of thing one puts together to point out ideas rather than practical applications. You might find it useful to be reminded of a result I cited in class but did not prove, namely that the output impedance of a common collector stage was ZE re ZS / 1 hfe where Z S was the input signal s source impedance. You may use V BE = 0.7 volts for all transistors. VCC= 12 v. C IN 180 3K R E1 0.1 uf b = 80 Q2 b = K 1K R E2 3.1) What is the stage type for each of and Q2 (CE, CB, CC)? 3.2) There are two undetermined resistors in the circuit. Choose one to make I C1 = 5 ma. 3.3) Select the other to make the circuit output impedance 18 ohms. 3.4) What is the midband gain of the circuit? 3

4 3.5) This circuit probably works best at frequencies around 1 MHz. Select C IN to give it a low frequency cutoff of 10 khz. 3.6) What is the function or purpose of C IN? 4.).) In class I included the Early effect in all models of the bipolar transistor because the effect is quite visible in measurements and is important to their use in integrated circuits. (The Early effect is the tilt of I C vs V CE curves such that if their flat sections are extrapolated to negative values of V CE, these lines all intersect the X-axis at approximately the same point, the Early voltage.) However, when I then calculated gains and output impedances for our first circuits, I omitted the r O component that the Early effect adds to the model. I made a hand-waving argument but here in an exercise for the reader is a better proof. The circuit below is the same kind of common emitter amplifier on which I gave the approximate analysis. The amplifier is shown with zero volts input signal and I want you to find its output impedance. The assumption in the base drive by V BB is that at signal frequencies the source impedance driving the base is negligible. The output impedance will be R C in parallel with a resistance that is mostly proportional to r O. Find an algebraic expression for that resistance in terms of R E, h fe, r O, and r b. To do so, redraw the circuit as a small signal model, remove R C, and compute the output impedance by looking back into the output terminals. (You look back by connecting a test source to the output and computing the ratio of voltage to current, which will be the dynamic output impedance.) VCC +15V R C V BB R E Suppose that in the original circuit, the collector current is 1 ma, h fe is 100, R E is 350 ohms, R C is 7.5K, and the Early voltage of is 75 volts. What are r O and r b? What is the numerical small signal gain of the original circuit? What is the transistor output impedance and what is the circuit s output impedance? 4

5 5.) Sedra and Smith problem from the 5th edition or 6.95 from the 6th edition reproduced below except do the problem twice, once with the hybrid-pi model and once with the Tee/common base model. Suppose you wanted to calculate input resistance. Which model is easier to use to find that by hand? 6.) Sedra and Smith problem from the 6th edition reproduced below. 5