A mathematical model of the box

In order for our black box to be of any use as a model of an amplifier (or any

other type of system), we have to establish some mathematical relationship

between input and output. There are a number of ways of doing this. For

example, we could write:

where z11, z12, z21 and z22 are the circuit's parameters. A parameter is a

quantity upon which the characteristics of the circuit depend. In mathematical

jargon, the parameters are coefficients used in the pair of simultaneous

equations which relate the external voltages and currents. The subscripts

indicate if the parameter is a property of the input circuit (z11), output circuit

(z22) or if the parameter links input to output (z21) or output to input (z12).

The first equation of the pair is the input equation and shows how the input

voltage is related to the input and output currents.

The second equation is the output equation and shows how the output voltage

is related to the input and output currents.

Note that in the two equations, the left-hand term is voltage and will therefore have the units of volts.

Thus, the term z11I1 which occurs in the right-hand side of the first equation must also have units of volts.

If the transistor is operated at 'low frequency' these impedances can be

regarded as being purely resistive. In this context 'low frequency' is up to a few

hundred kilohertz. We shall assume such frequency limitation

applies so that all the z parameters are resistive.

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The equivalent circuit of the z parameter model

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We are now almost in a position to draw an equivalent circuit for the black box

containing the transistor. But before attempting this, let's note that the term

z12I2 is an input voltage source whose magnitude depends upon the output

current, and the term z21I1 is an output voltage source which is dependent upon

the input current. The parameter z12 represents a reverse trans-resistance

which connects output to input, whilst the term z21 represents a forward transresistance connecting input to output. (It is, incidentally, the property of z21 in

linking input to output which has given the transistor its name, i.e. from 'transresistance'.)

In fact, z21I1 represents feedforward and z12I2 feedback. The two

voltage sources are represented by voltage generator symbols in the equivalent

circuit, as shown below.

Having made this point, let's now try and draw an equivalent circuit. This can

be done by applying Kirchhoff's voltage law to the equations. Consider the

input equation

The voltage V1 is equal to the sum of:

(i) a voltage derived from a current I1, flowing through a resistance z11

(ii) a voltage generated from the generator z12I2.

Thus, the input circuit can be represented as shown below

Note that the circuit forms a Thévenin equivalent, and that z11 represents the

input resistance to the black box with the output open circuit (so that I2 is

zero).

Now consider the output equation:

The output voltage can be seen to be the sum of:

(i) a voltage generated by the voltage generator z21I1

(ii) a voltage derived from a current I2 flowing through a resistance z22.

Thus, the output circuit can be represented as

Note that the circuit forms a Thévenin equivalent circuit and that

z22 forms the output resistance of the box, with the input open circuit. Also

note again the direction of I2. It is assumed to be flowing into the circuit.

The two circuits can now be combined to give a complete equivalent circuit:

This is the z equivalent circuit for our two port black box containing the

transistor.

OTHER POSSIBLE EQUIVALENT CIRCUITS

We shall consider two other pairs of equations, which are particularly appropriate to the representation of a transistor. These equations are:

the y parameter equations:

and the h parameter equations:

Determine the units of each of the parameters used in these two pairs of equations.

In the case of the y parameters, it can be seen that for I to equal yV, y must

represent admittance and have the units of siemens. At low frequencies the

y parameters can be considered to be purely conductive. We shall make this

assumption, so that 'y' could be replaced by 'g'.

In the case of the h parameters, the situation is a little more complex. We have

to consider each term on its merits. Take the first equation of the pair; as the

left-hand side is in volts, each term on the right-hand side must also represent

voltage and have the units of volts. Thus, in the term h11I1, h11 must represent

impedance and have the unit of ohms. At low frequencies h11 will be resistive.

In the term h12V2, V2 already represents volts and therefore h12 must be

dimensionless. In fact, it represents a reverse voltage gain from output to

input, i.e. it gives feedback.

In the second equation, each term must represent current and have the units of

amperes. Consider the first term h21I1. I1 is already current and therefore h21

is dimensionless. It, in fact, represents forward current gain. Finally, in the

term h22V2, h22 must represent admittance and have the units of siemens. At

low frequencies h22 can be regarded as purely conductive.

The 'h' stands for hybrid, reflecting the fact that the right-hand side of the h

equations are a mixture of voltages and currents, giving parameters that are a

mixture of impedance, admittance and dimensionless gains. Contrast this state

of affairs with the parameters of the z and y models, whose parameters have

units of ohms and siemens respectively.