One of the first steps in analyzing and designing transistor circuits is to repre-sent the transistor by a satisfactory equivalent circuit. Many equivalent circuits are in use. Each is used to represent a different type of transistor at different operating conditions. In general, equivalent circuits may be divided into two types:
those which regard the transistor as a black box upon which measurements are made, and those which regard the transistor as being made up of physically real-izable active and passive components. The following discussion illustrates several of the more common equivalent circuits used in transistor circuit design.
The equivalent circuits in this chapter represent the a-c or incremental equivalent circuits, as opposed to the d-c or static representation. All the small-signal parameter symbols with which we are dealing will be given in lower case to distinguish them from static parameter symbols, given in upper case.
The Two-terminal Network. Consider a two-terminal black box. At one frequency, the behavior of a linear device may be specified in terms of two measure-ments: one an open-circuit voltage, the other a short-circuit current. Equivalent circuits, good at the frequency of measurement, may then be drawn for the device (as shown in Fig. 6.1).
voc-Open-circuit voltage.
i.c-Short-circuit current.
Fig. 6.1. Two-terminal network.
86
Y = -isc Voc
Equivalent Circuits and Parameter Interrelationships 87
loop
~
"Black box"Fig. 6.2. Four-terminal network.
The Four-terminal Network. A set of measurements may be made on a linear four-terminal device similar to those made on the two-terminal device. The three-terminal transistor is considered a special case of a four-three-terminal device, with two terminals common to both input and output. Consider the black box of Fig. 6.2.
It is possible, by making appropriate measurements of the various voltages and currents, to arrive at a useful equivalent circuit for any linear device, active or passive. The voltages and currents are shown to establish the measurement convention.
Figure 6.3 is shown to illustrate that a three-terminal network is just a special case of the four-terminal network of Fig. 6.2.
Two statements may be made concerning Figs. 6.2 and 6.3:
1. There are only two independent voltages and two independent currents.
2. If any two quantities are fixed by external means, then the other two are fixed by the black-box parameters. Note that generally we are not free to specify any two quantities arbitrarily; the quantities picked must be com-patible with the black-box parameters. Of the six possible circuit repre-sentations relating the voltages and currents, three have proved particularly helpful in describing junction transistors: (a) open-circuit impedance measurements, (b) short-circuit admittance measurements, and (c) a com-bination of the two, hybrid parameter measurement.
Open-circuit Impedance Parameters. Two equations may be used to define the black box as given in Fig. 6.2:
-
--+
"Black box"
=Vi
[ 1
-Fig. 6.3. Three-terminal network: a special case.
+
V2=
-(1) (2)
Figure 6.4
i1 and i2 are independent variables.
(3) electrical properties of Fig. 6.4 may be calculated by using the impedance param-eters. Current gain of the black box shown in Fig. 6.4 is given as an example.
Short-circuit AdmiHance Parameters. The black box of Fig. 6.2 may also be represented by the following two equations:
it
=
Y11 V1+
yi2V2Equivalent Circuits and Parameter Interrelationships 89 Like the z parameters, all the electrical characteristics of a black box are known if the y parameters are known.
Since z parameters are open-circuit parameters, a network can best be charac-terized by z parameters whose input and output may easily be open-circuited (e.g., low z circuit). A similar statement can be made about the y parameters. Since y parameters are short-circuit parameters, a network can best be characterized by y
parameters whose input and output may easily be short-circuited.
Hybrid Parameters. A third set of very useful parameters combines part of the open-circuit measurements and part of the short-circuit measurements to form a hybrid parameter system.
The hybr~d or h parameters may be defined by the following equations:
VI = huit
+
h I2V2 The admittance parameter set finds its chief usefulness at very high frequencies, where open-circuit measurement may lead to difficulties. It is almost impossible to avoid stray capacitance between transistor elements, and with open-circuited terminals these may cause regenerative effects which are often very unstable andunpredictable. If only short-circuit parameters are measured, then the stray (or parasitic) capacitances merely act as shunt reactances to ground, and do not enter into the active transistor measurements.
Of the three sets of parameters listed, the h parameters are the most often used in general transistor audio work and low-frequency video.
Comparison of z,.y, and h Parameters. Representation by hybrid parameters is the most useful scheme for two reasons. First, the h parameters are easy to measure. It must be remembered that h parameters are measured at some bias point; therefore, the terminals of the device cannot merely be shorted for a short-circuit measurement or opened for an open-short-circuit measurement. The short and open circuits must take place with regard to the biasing network attached to the transistor. The z and y parameters require either all open-circuit or all short-circuit measurements. Since the input impedance is rather low and the output impedance is rather high for a transistor (common base and common emitter), one type of measurement for both input and output is difficult. The Zll and Z21 meas-urements require the output to be open-circuited, which is difficult to do, particu-larly at high frequencies; it is equally difficult to short the input circuit for the Y12
and Y22 measurements. With the hybrid parameters, however, it is necessary only to short-circuit the output or open-circuit the input; this is easily accomplished at both low and high frequencies.
The second advantage of h parameters is that the input impedance, output admittance, and current gain of the device as used in a circuit approximate hll' h22' and h21' respectively, if certain assumptions are made. As an example: current gain
=
h2t!(1+
h22RL)' If RL is small compared to Ijh22' then the current gain of the device in the circuit equals h21.Representation of Equivalent Circuits. In four-terminal network theory, the small-signal parameters (z, y, and h) are characterized by numerical subscripts; in transistor circuit work, these parameters are usually designated by letter subscripts.
The first subscript designates whether the parameter is input or output, forward or reverse; the second subscript indicates the transistor configuration. The follow-ing shows the relationship between the numerical and literal designations.
11 = i ... Input parameter Common-collector configuration.. . .... c
Using letter subscripts, hoe would designate the output h parameter for the com-mon-emitter configuration.
Representation of Equivalent Circuits. The h-parameter equations having been written and defined, it is necessary to show how an equivalent circuit is derived from the equations.
Common-base Equivalent Circuit. The hybrid equivalent circuit for the h parameters of the common-base configuration (Fig. 6.5a) is shown in Fig. 6.Sb.
Equivalent Circuits and Parameter Interrelationships 91
e c
bo---+---~---+---~b
(a) Common·base configuration (b) Hybrid equivalent circuit
Fig. 6.5. Common-base configuration and hybrid equivalent circuit: (a) common-base con-figuration; (b) hybrid equivalent cjrcuit.
The reverse voltage transfer ratio, hrb , appears as a voltage generator in the input circuit; the forward current transfer ratio, hfb' appears as a current generator in the output circuit. By calling VI and V2, and il and i2 of Eqs. (15) and (16), Veb and
Veb, and ie and ie, respectively, and adding the base subscript b to the h parameters, the equations (for input and output) relating the h parameters for the common-base configuration become
(21) (22) Common-emitter Equivalent Circuit and Equations. The hybrid equivat.ent circuit for the h parameters of the common-emitter configuration (Fig. 6.6a) is shown in Fig.6.6b.
The reverse voltage transfer ratio, hre, appears as a voltage generator in the input circuit; the forward current transfer ratio, h{e, appears as a current generator in the output circuit. By calling VI and V2, and il and i2 of Eq. (15) and (16), Vbe and
Vee, and ib and ie, respectively, and adding the emitter subscript e to the h param-eters, the common-emitter equations (for input and output) relating the h parameters for the common-emitter configuration become'
Vbe
=
hieib+
hrevee (23)ie
=
h{eib+
hoevee (24)Common-collector Equivalent Circuit and Equations. The hybrid equivalent
b ~---~---Oc
eo---~---~---4----__oe (a) Common· emitter configuration (b) Hybrid equivalent circuit
Fig. 6.6. Common-emitter configuration and hybrid equivalent circuit: (a) common-emiHer configuration; (b) hybrid equivalent circuit.
b r---~--~-oe
co---~---~---~~----oc (a) Common-collector configuration (b) Hybrid equivalent circuit
Fig. 6.7. Common-collector configuration and hybrid equivalent circuit: (0) common-collector configuration; (b) hybrid equivalent circuit.
circuit for the h parameters of the common-collector configuration (Fig. 6.7a) is shown in Fig. 6.7b.
The reverse voltage transfer ratio, hre, appears as a voltage generator in the input circuit; the forward current transfer ratio, hfe , appears as a current generator in the output circuit. By substituting Vbe for Vi, Vee for V2 and h for h in Eq. (15) and by substituting h for h, ie for i2, and Vee for V2 in Eq. (16), and adding the collector subscript c to the h parameter of Eqs. (15) and (16), the common-collector equa-tions (for input and output) relating the h parameters for the common-collector configuration become
Vbe