  24 February 2009
Author: Giorgos Lazaridis
BJT Transistor theory

The T and Î  (II) models

To analyze the AC transistor operation, we will use the T and Î  models. These models are used to replace the transistor in the circuit. The transistor is replaced by the emitter resistor and a current source.

The transistor T model

Suppose that we have a Common Emitter transistor amplifier from which we've make the AC equivalent circuit as follows: We can replace the transistor above with the T model. The circuit is then changed as follows: It is called "T model" because the transistor is replaced by a T-shaped circuitry. In our example you can locate this model if you search for this T-shaped circuitry rotated 90 degrees clockwise. The top side of the T has the collector current source, and the bottom side of the T has the internal AC emitter resistor. This resistor is marked with the symbol "r'e". The small "r" means that we are referring to an ac resistance, the "e" pointer means that we are referring to the emitter, and the prime symbol (') means that we are referring to an internal size of the transistor.

As you see, the base AC voltage is directly applied across the internal base-emitter resistance. Therefore we can extract the following equation:

ie = ub / r'e

The input impedance of the base is this:

Zin(base) = ub / ib

Finally, from the collector's side, the ac collector voltage is calculated with the following formula:

uc = ic x rc

The symbol rc is the total AC collector resistance. The collector's resistance in DC operation is different than the AC resistance. That is because, in AC operation, the coupling capacitor adds the load resistance RL in parallel with the DC collector resistance RC. Therefore, we use the symbol rc in short for the total resistance RL//RC.

The transistor Î  (II) model

Let's replace the transistor from the previous CE amplifier with the Î  model: It is obvious why this is called Î  model. The letter Î  comes from the Greek alphabet and is spelled like the letter "P". A double "I" letter can be used instead (II). From the T model, we know that:

Zin(base) = ub / ib (1)

And we also know that:

ub = ie x r'e (2)

(1)(2)=> Zin(base) = r'e x ie / ib

But from the theory we know that ie / ib is the current gain Î². Therefore:

Zin(base) = Î² x r'e

Both models can be used for the AC transistor analysis with the same results. If you happen to know the AC base voltage ub and the AC current ib, the T model can be then used to analyze the circuit, without needing to know the Î² value. On the other hand, if the current gain Î² is known, then you can use the Î  model for the analysis.

The Base-Emitter AC internal resistance of the transistor (r'e)

So far, we have seen how to do the DC analysis and the AC analysis separately, but we still do not know how the AC and the DC voltages are connected. The internal Base-Emitter AC resistance does exactly this: connects the emitter DC current with the base AC current. We use the symbol r'e which is different from the symbol re. The prime symbol shows that we refer to an internal size. The re is used for the AC external emitter resistance.

The Base-Emitter internal AC resistance of the transistor depends on the DC current of the emitter. The equation which connects these two is this:

r'e = 25mV / IE

You may wonder what these 25mV are. The story goes back in 1947, when William Shockley invented the first transistor. Shockley used the diode current to determine the resistance:

IE = IS (eVg/kT - 1)

IS is the reverse saturation current and V is the voltage across the diode. At 25 oC, the above equation can be rewritten like this:

IE = IS (e40V - 1)

After some calculations, the equation becomes like this:

r'e = 25mV / (IE + IS)

And since IE is many times greater than IS:

r'e = 25mV / IE

The above equation is valid for operation at room temperature (25 oC). For an accurate calculation at different temperatures, the following equation can be used:

r'e = (25mV x T+273) / (IE x 298)

T is the contact temperature in degrees Celsius. Let's see now what this equation means. Suppose that we have a common emitter amplifier like the one we saw in previous pages, and we want to use the Î  model to calculate the transistor input impedance. Suppose also that we did the DC analysis with the help of the DC equivalent, and found that the emitter current is 1.1mA. From this DC current, we can calculate the AC base-emitter resistance:

r'e = 25mV / 1.1mA = 22.7 Ohms

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