Transistor - small signal model and small signal parameters

Analog Circuits /Electronics Circuits syllabus is packed with models. 

( I can see twinkling in your eyes, but… plz don’t disappoint. Just scan through below lines.)

The model 

To study the behavior of a transistor for different input conditions, we make use of different transistor models. These models are designed by taking various parameters of transistor into consideration. Hence having the same behavior as that of the corresponding transistor. To test the behavior of a transistor for a new input condition, instead of conducting a practical experiment, these inputs are applied on the corresponding transistor model and outputs are measured. Thus 'the model' helps to predict the nature of a transistor(or circuit) for various input or bias conditions without actually wiring it.
Here we are thinking about 'Small signal model of a transistor' only. Of course there is 'Large signal model' also.

Small signal model of a transistor

Small signal model is related to 'Small signal analysis'. Don't confused, it is here, of course in a chewable way...small signal analysis.
To conduct a small signal analysis, a small signal model of a transistor is used. The parameters used along with this small signal model are known as small signal parameters.

One of the widely using small signal model of a transistor is Hybrid - pi model.  

There are two versions for this  model as shown below.

fig 1

 Fig 1 shows a voltage controlled current source version.

fig 2

Fig 2 is current controlled current source version.

Lets familiar with the parameters used in small signal model.

1. Transconductance(g_m)

  As we know conductance is the ratio of current to voltage. For BJT, transconductance is find by taking  ratio of output current to input voltage.

 Transconductance  g_m   = i_c / v_be 

One more equation for g_m in terms of dc bias current Ic and V_T where V_T is the voltage equivalent of temperature.               g_m   =  Ic / V_T

2. Input resistance (r_π )

Small signal input resistance between base and emitter is called as input resistance and denoted by r_π.     
         r_π  = v_be /i_b    It can also be expressed in terms of  g_m as
         r_π  =  β / g_m   where  β is the current gain. 
We can find out more equations for r_π by substituting for g_m. 
         r_π  =  βV_T / Ic   =   V_T / I_B  

3.Output resistance (r_o)

It is the output resistance r_o = V_CE/ Ic     

T- model of  transistor

Another small signal model of a transistor is T -model. Both pi and T models can be  used alternatively. But choose the one which will make the circuit less complicate.

fig 3

 T- model with output resistance is below.

fig 4

These are the different small signal models of npn transistor.