Jonathan Roderick
Hakan Durmus
Scott Kilpatrick Burgess
Experiment #6
BJT Dynamic Circuits
Introduction:
The point of biasing an analog circuit correctly is so the active devices within the circuit operate in a desirable fashion (linearly) on signals that enter the circuit. These signals are perturbations about the bias point (or quiescent point, a.k.a. Q-point); for instance, you might bias your input port at 2V, and then add a 50 mV peak-to-peak sine wave to this bias voltage. Ideally, amplifiers are linear devices, meaning the output signal is some multiple of the input signal, independent of the input amplitude. Transistors are normally non-linear devices (recall their I-V characteristics), so the output amplitude does depend upon the input amplitude. However, by suitably restricting the input swing and correctly biasing the circuit (Q point), the resultant output will show very little curvature, meaning that the non-linear circuit acts approximately linear for small-signal deviations about the bias point.
In this lab, dynamic circuits using BJTs will be introduced. Once a BJT is biased in such a way that it operates in a linear region, then the “small-signal” BJT model may be used for analysis and design of circuits that contain the transistor. This model forms the basis for understanding the dynamic performance of several commonly encountered circuits.
Theory:
Small-Signal Model for the Bipolar Transistor
The small-signal model for an NPN bipolar transistor is shown in figure 6.1. For the purposes of this lab, the models and theory presented will focus on the NPN Bipolar Junction Transistor. The following models also apply for the PNP transistor with the slight modification of: reversing the direction of all controlled current sources and branch currents, and a reversal in polarity of all port and branch voltages.
Fig. 6.1 Small-signal model for the bipolar transistor.
Note: The small signal model is just a tool that is used to help circuit designers analyze circuits utilizing BJTs. This tool is only valid if the transistor is operating in its linear range. Therefore it should be understood that when using the small signal model, that significant effort has been made to ensure that the signal being processed in the amplifier is not too large, thus validating the “small signal” model accuracy. A large enough signal may cause the transistor to leave its linear operation if its signal change has a magnitude large enough to offset the set Q (biasing) point.
The key element in the small signal model is the controlled current source, which can be shown as depending on the internal base current i (or the internal base-emitter voltage ve). The quantity gm is defined as
gm= (6.1)
signifying how responsive the collector current is to changes in the driving voltage ve. The small signal model accounts the various internalresistances associated with each terminal. Resistor re is the small resistance associated with the highly doped emitter. Resistor rb is a distributed, non-linear resistance, and thus hard to characterize with a single value, but it corresponds to the resistance between the base contact and that region of the base material lying underneath the emitter. Resistor rc likewise is hard to characterize with a single value, but represents the net resistance between the collector contact and the bottom portion of the base material. Resistor r, known as the emitter-base junction diffusion resistance, is not a physical resistance (it is a mathematical model conceived from a Taylor series expansion of the base-emitter current, IBE, about the Q-point) like the preceding three, but rather a dynamic quantity defined as
r = (6.2)
It represents how resistant the input base current is to changes in the internal base-emitter voltage (i.e., that voltage not including the voltage drop across rb, represented as “ve” in Fig.6.1). The controlled source indicates how much the collector current changes for a change in base current (or equivalently, base-emitter voltage). Like r, resistor ro is a dynamic resistance and it is know as the forward Early resistance. It represents the influence of changes in collector-emitter voltage on collector current, and thus is calculated by
ro = (6.3)
For high Early voltages, AE, this resistance is negligible, and thus the collector voltage has a negligible impact on the current flowing out of the collector contact. The internalresistance does have profound effects on overall circuit performance. Large base, collector, and emitter resistances reduce circuit gain, diminish gain-bandwidth product, and increase electrical noise. However, rb, re, and rc are inversely proportional to the emitter-base junction injection area and a price is paid for increasing the area to lower resistances. Increasing the area of the device results in larger parasitic capacitances, so increasing the size of the transistor to reduce internalresistance reduces the circuit response speed. Power consumption is also a trade-off. All internalresistance , particularly rb, decrease monotonically with increasing the base and collector bias currents, IB and IC respectively. In the world of wireless and mobile electronics, we want the batteries in our cell phone to last longer, so large power consumption in wireless electronics is avoided. Welcome to the wonderful world of circuit design, where hindering constraints are inversely proportional to each other. Your job as a circuit designer is to find a nice median that allows you to meet all the specifications for your design.
The small signal model also accounts for internal parasitic capacitances found with in the BJT. C represents the depletion capacitance of the base-collector junction. C is composed of two parts: 1) a diffusion capacitance given by
C = (6.4)
and 2) a depletion capacitance, which is usually negligible compared to the diffusion capacitance when the base-emitter junction is forward-biased.
To develop numerical values for the symbols in the small-signals model, the defining derivatives must be evaluated symbolically, then evaluated about the Q-point. With the bias quantities specified, numerical values may be assigned to each small-signal parameter.
Canonic Cells of Linear BJT Technology.
The BJT Transistor has four basic topologies that are building blocks for more complicated circuit architecture. The BJT transistor may be connected in a diode, common emitter, common collector, or common base configuration. A quick and simple way to determine the difference between the common base, collector, or emitter is: First, determine what terminals where the input and output are connected. Then, the particular canonic cell receives its name from the terminal that is leftover. For example, if you are looking at the ac BJT configuration in figure 6.2, you will notice that the input is at the base, while the output is located at the collector. Hence, the leftover terminal is the emitter and this canonic cell is deemed a “common emitter” amplifier.
Figure 6.2 An AC schematic diagram of a Common
Emitter Amplifier.
Diode-Connected Transistor.
The simplest canonic cell for the BJT is the diode-connected transistor. The collector is tied to the base of the transistor, so it exhibits I-V behavior of a conventional PN junction diode. Figure 6.3 depicts a transistor connected this way and its small-signal equivalent circuit. This model assumes the transistor is biased in the linear region and leaves out the Q-point currents.
The diode-connected transistor reduces the number of terminals of a typical BJT to two (the base and collector are now the same terminal). Thistwo terminal device may be modeled as a two terminal resistor seen in figure 6.3. Using the low-frequency small-signal model of BJT (neglecting all capacitance), the equivalent resistance of the diode-connected can be found to equal Rd.
(6.5)
If ro>rc+rb+r, then equation 6.5 reduces to
(6.6)
Figure 6.3(a) Diode-Connected BJT and (b) its small-signal low frequency equivalent model.
Common Emitter Amplifier.
The common emitter amplifier was shown in figure 6.2. Replacing the schematic symbol of a BJT in figure 6.2 with the small signal model, one can calculate the gain, input impedance and the output impedance. Figure 6.4 shows a common emitter amplifier utilizing the small signal model. Assuming that rc is negligible and ignoring the early effect (ro=) the gain, input resistance (Rin) and output resistance (Rout)may be calculated.
Figure 6.4A low frequency common emitter canonic cell using the small signal model.
(6.7)
where rx is the resistance seen by emitter.
(6.8)
and
(6.9)
assuming is large, then the gain reduces to
(6.10)
The common emitter canonic cell is used to achieve an inverting gain that is independentofthe transistor. Rindepends on what the value of rx is, but since it is multiplied by it is assumed not to be too small.Rout is very large.With a Rin that can be made fairly large and a Rout is very large, the common emitter is not a very ideal voltage amplifier. Additional circuit features are usually added to enhance performance.
Common Collector Amplifier.
A common collector canonic cell is shown in figure 6.5. Notice, the input nor the output of the canonic cell is connected to the collector of the transistor.
Figure 6.5A common collector BJT canonic cell.
Using the Small signal model it can be shown that the gain, input resistance and the output resistance are the following.
(6.11)
by choosing a small Reethe gain will reduce to
(6.12)
with
(6.13)
and
(6.14)
Since the gain can be designed nearly equal one, rin can be made fairly large, while rout is small (due to it being inverselyproportional to ) the common collector canonic cell can be designed to be a decent voltage buffer.
Common Base amplifier.
The common base canonic cell is shown in figure 6.6. The input is a current source at the emitter, while the output is taken at the collector. Hence, this is a common base configuration of a BJT.
Figure 6.6 An AC common base BJT canonic cell.
Using the small-signal model it can be shown that the current gain (Ai), rin, and rout are the following.
(6.15)
(6.16)
(6.17)
The common base has a current gain of about one, a large output resistance and a small input resistance, therefor it is used as a current buffer.
Common-Emitter Amplifier Example
In the previous lab, the common-emitter amplifier was biased, but no mention was made of why it is called an amplifier. To answer this, we analyze the circuit in the previous lab, with a few modifications. First, we need to feed our input signal into the base, without the DC bias of the signal source and the common-emitter amplifier interfering with each other. This is accomplished by adding an AC coupling capacitor (Cin)to the input port, large enough so that it will act like an AC short at the frequencies at which we operate, thus eliminating any transfer of DC offsets. Second, the amplifier needs to drive a resistive load which we don’t want to upset our bias point, so we append another AC coupling capacitor (Cout) to the output port. Third, we replace the transistor symbol used in the previous lab with the small-signal model, leading to the following circuit:
Fig. 6.7 An AC common-emitter amplifier for AC analysis (from experiment #5 figure 5.4)
Assuming that we are at low-enough frequencies that the parasitic capacitances of the BJT don’t affect our results, but a high enough frequency where the coupling capacitors are acting like shorts, and further neglecting the Early resistance and internal emitter resistance, analysis of our model leads to:
Av= (6.18)
With large beta, this reduces to
Av = (6.19)
a rather simple expression independent of transistor parameters. As long as , the transfer function has a magnitude greater than 1, explaining why the common-emitter is called an amplifier. For the values derived in the previous lab, this requires that Rload be at least 375 for gain.
Wilson Current Source
Another type of current source is the Wilson current source, shown below:
Fig. 2Wilson current source.
This current source has two advantages, namely, it gives very good matching between output and reference currents, and it provides very high output resistance. Analysis of both the matching and output resistance are left as pre-lab exercises. The advantage of high output resistance is that the output current is insensitive to whatever load is attached to it. Thought of another way, if you consider the current source as a parallel combination of a Norton current and Norton resistance, and your load as a pure resistance, then the resultant current divider is such that nearly all of the Norton current flows into the load resistance, with very little lost to the internal resistance of the current source.
Pre-lab Exercises
1)Given the definitions of the small-signal parameters for the bipolar transistor, express r, gm, and ro in terms of bias parameters (e.g., collector current, Vce, etc.). For a 1mA current and Vce of 3.2V as well as an Early voltage of 200 and a Beta of 100, what are the values of r, gm and ro?
2)Derive the expressions given for the Gain, input resistance and output resistance for the common collector circuit in figure 6.5.
3)Derive the expressions given for the Gain, input resistance and output resistance for the common base circuit in figure 6.6.
4)Derive the expressions for the gain, input resistance, and output resistance for the low-frequency common-emitter amplifier in Fig. (schematic representation in experiment #5 figure 5.4).Using a 5v supply, correctly choose values for the resistors,pictured in figure 5.4 of the last lab,so the transistor is properly biased in linear operation, while also achieving a gain of 50. Use SPICE to verify your design and use the Spice models given to you in your homework. Use the .op command to verify the bias currents and voltages as well as the transistor gm for your circuit, then perform a .ac analysis to observe the frequency response. Place a large (e.g., 0.1 uF) capacitor across resistor Re. What happens to the gain? Do a hand analysis and SPICE simulation to confirm your response.
Lab Exercises
1)Build the common-emitter amplifier of Fig.. Determine an appropriate input signal level so that the output signal is large enough to measure, but small enough so that the transistor acts linearly. Determine the low-frequency gain at 10 kHz. Does it match reasonably well with your hand results? Change the input signal level by +/- 6-dB. Does the circuit behave linearly? Place a 0.1F capacitor across Re and re-measure the AC gain (you may need to adjust your input signal amplitude once more to ensure linear behavior). Does this match your prediction as well? At really low frequencies, the gain rolls off. Why is this? At high frequencies, the gain diminishes as well. What causes this decrease?
2)Build the Wilson current source discussed in this lab, using the reference resistor value you calculated in the pre-lab. Determine the current through this resistor (it may differ from 1 mA depending on the Vbe values of the transistors), and then measure the output current (use a DVM with several digits accuracy). What is the percentage error? How does this compare to the current sources of the previous lab? Based on the expression you determined for the error, determine a value for the of the transistors.
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