An Introduction to Bipolar Transistor H-Parameters
A bipolar transistor is fabricated from three regions of silicon doped with impurities and joining at two junctions. These regions are known as "n-type" or "p-type" semiconductor, depending on having an excess of either electrons or holes (negative or positive charge carriers).
The theoretical equations which predict the behavior of semiconductors are not practical for everyday use. These, borrowed from a text on semiconductor physics(1), represent only the minority-carrier-current densities.
If we consider the total AC hole-current density and designate it as jp*, we have:
Now by representing jp*(x,t) = Jp*e jwt and by letting x = -w0/2 and x = +w0/2 and by using other equations previously expressed in the text, we obtain emitter and collector minority-carrier-current densities, respectively, as follows:
Even if these equations were less complex, the constants embedded in them are not really known with any degree of accuracy. Therefore, simpler models have been developed to predict (at least approximately) how a bipolar transistor will perform as a small-signal amplifier. A model frequently used by semiconductor manufacturers is the hybrid model, and the values used with it are called hybrid or "h" parameters:
| Each of these h-parameters is defined as the ratio of a particular response of the transistor divided by a certain excitation which causes that response (don't try to apply the response and measure the excitation!): |  |
This particular implementation of
the hybrid model is for low frequencies only. Note that no reactive
components have been used. At high frequencies, several capacitors must
be added to the model to keep
it realistic.
In most modern texts the h11 term is renamed hie and represents the dynamic input resistance in the common emitter configuration(2). The h12 term is renamed hre, and represents a small input voltage developed as a result of reverse feedback from the output circuit - perhaps due to the bulk resistance of the emitter region. The term h21 is also called hfe and is the forward current gain of the transistor. The output conductance, h22, is renamed hoe. The resulting often-used model may be seen to the right.
For many amplifier circuits, the
values of hre and hoe are such that the
results are not seriously affected if they are ignored. The simplified
model becomes(3):
We might call this an "approximate hybrid equivalent circuit" for the common emitter configuration. It is quite useful in small-signal (AC model) analysis. Note that a close and useful approximation of hie may be obtained by dividing 30 mV/Ib (use the base bias current), and hfe is roughly the same as the factor b used in other models. (Purists will note that the theoretical numerator of the hie formula is 26 mV, but careful measurements on a variety of actual transistors yielded values between 30 and 38 mV.)
It should never be assumed that the results obtained via these models are anything other than approximations. This presents no great hardship, as the actual values of all the transistor parameters are already only approximately known. The predictions, however, can be surprisingly accurate; and the ease of using the approximate model makes it more likely that the predictions will actually be calculated instead of guessed at.
The question often arises whether to use hfe, the AC current gain; or hFE, the DC gain; in these analyses. The answer should be self-evident: use the DC gain for DC biasing predictions, and the AC gain for small-signal AC analysis. It is perhaps worth noting that the popular Motorola 2N3904 can have an hfe between 100 and 400 at 1 mA, and an hFE between 80 and 240 at the same current(4). Thus the variation of hfe or hFE values within a group of 2N3904's is greater than the difference between hfe and hFE. One cannot be very far wrong in simply assuming the current gain, AC or DC, is 100 unless told otherwise.
Once the approximate hybrid model for a transistor is drawn and labelled, prediction of a circuit's performance can proceed by adding the rest of the components comprising the amplifier and applying Ohm's, Kirchoff's, and Norton's Laws. Think carefully when adding the external components - often coupling and bypass capacitors may be replaced by short circuits if only the mid-band performance is of interest. Or, the coupling capacitors may first be ignored to find midband gain and input resistance, and then the appropriate resistor-capacitor combinations may be examined to determine frequency response. Sometimes it is beneficial to solve the simplest problem first, to gain an understanding of the circuit's basic performance; and then add the complications and exceptions afterwards. This treats the mid-band performance as an "ideal" case, and clearly demonstrates what departure from that ideal is caused by each reactance which must be considered.
Hybrid parameters are by no means the only set valid for approximating the performance of a transistor. Resistance (r-parameters), transmission (t-parameters), and admittance (y-parameters) have also been successfully used. Each naturally has its own advantages and disadvantages. Various reference works will help you decide which is best for the circuit you are examining.
1. Nanavati, Rajendra P., An Introduction to Semiconductor Physics, McGraw-Hill, 1963.
2. Paynter, Robert T., Introductory Electronic Devices and Circuits, Prentice-Hall, 1991.
3. Boylestad, Robert and Nashelsky, Lewis, Electronic Devices and Circuit Theory, Prentice-Hall, 1987.
4. (technical data book), Small-Signal Transistors, FETs and Diodes, Motorola Inc., 1989.
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