Small-Signal Model and H parameters

A CE transistor circuit can be modeled by a two-port network as shown below:

The two-port network is described by the relationship between two pairs of variables ( $v_1=b_{be},\;i_1=i_b,\;v_2=v_{ce},\;i_2=i_c$) associated with the input and output ports. Any two of the four variables can be assumed to be the independent variables, while the other treated as the dependent variables, the functions of the independent variables. In general there are $C_4^2=4!/[2!(4-2)!]=6$ ways to choose any two out of the four variables. For example, three of these six choices are:

$$\left\{ \begin{array}{l} v_1=f_1(i_1,i_2) \\ v_2=f_2(i_1,i_2) \en... ...eft\{ \begin{array}{l} v_1=f_5(i_1,v_2) \\ i_2=f_6(i_1,v_2) \end{array} \right.$$ (63)

We use the third hybrid model to describe the CE transistor circuit with $v_1=v_{be}$, $i_1=i_b$, $v_2=v_{ce}$, and $i_2=i_{ce}$:

$$\left\{ \begin{array}{l} v_{be}=v_{be}(i_b,v_{ce}) \\ i_c=i_c(i_b,v_{ce}) \end{array} \right.$$ (64)

Taking the total derivative of the above, we get:

$$dv_{be}=\frac{\partial v_{be}}{\partial i_b} d i_b +\frac{\partia... ...b} d i_b +\frac{\partial i_c}{\partial v_{ce}} d v_{ce} =h_f d i_b+h_o d v_{ce}$$ (65)

where $h_i, h_f, h_r, h_o$ are the hybrid model parameters:

• $h_i=\partial v_{be}/\partial i_b=r_{be}$: input AC impedance with $v_{ce}=0$ (output AC short-circuit). This is the AC resistance between base and emitter, the reciprocal of the slope of the current-voltage curve of the input characteristics.

• $h_r=\partial v_{be}/\partial v_{ce}$: reverse transfer voltage ratio representing how $v_{ce}$ affects $v_{be}$ with $i_b=0$ (input AC open-circuit). In general $h_i$ is small and can be ignored.

• $h_f=\partial i_c/\partial i_b=\beta$: forward transfer current ratio or current amplification factor with $v_{ce}=0$ (output AC short-circuit). Typically, $h_f=\beta$ is in the range of 100 to 200.

• $h_o=\partial i_c/\partial v_{ce}=1/r_{ce}$: output admittance, with $i_b=0$ (input AC open-circuit). This is the slope of the current-voltage curve in the output characteristics. In general $h_o$ is small and can be ignored.

If the variations of the AC components of all these variables $i_b$, $v_{be}$, $i_c$ and $v_{ce}$ are small ( $\Delta x\rightarrow dx$) around the DC operating point $Q$ and far away from either the cutoff or the saturation region, the non-linear quantities that describe the input and output characteristics can be linearized as the following small signal model:

$$v_{be}=\frac{\partial v_{be}}{\partial i_b} i_b+\frac{\partial v_... ...+\frac{\partial i_c}{\partial v_{ce}} v_{ce}=h_f i_b+h_o v_{ce} \approx h_f i_b$$ (66)

In general, $h_r$ and $h_o$ are small and could be assumed zero to further simplify the model (right of the figure above) containing only two components, a resistor $h_i=r_{be}$ and a current source $h_f I_B=\beta I_B$.

The base and emitter forms a PN-junction with a resistance $r_{be}$

$$h_i=r_{be}\approx \eta\,\frac{V_T}{I_B}$$ (67)

as discussed in the section of diodes, which is not a constant, but a function of current $I_B$ through the PN-junction between base and emitter. Typically, at room temperature $(V_T=26\,mV)$, if $I_B$ is approximately in the range of $0.05\sim 0.1 mA$, then $r_{be}$ is a few hundred ohms.

Based on this small signal model, a transistor can be analyzed as a two-port circuit containing a resistor $r_{be}$ and a current source $\beta I_B$.

In summary, we see that there are two aspects of a transistor circuit:

• The DC operating point in terms of the DC currents $I_B$, $I_C$ and $I_E$ and voltages $V_{BE}$ and $V_{CE}$;
• The AC small-signal model by which all of nonlinear voltage-current relationships associated with the transistor are linearized based on the assumption that the signal is small and the dynamic range is totally within the linear region of both the input and output characteristic plots.

When analyzing the transistor circuit with an AC input signal (riding on top of a DC input), we need to consider both aspects. If the DC operating point is set up properly, i.e., in the middle of the linear region of output characteristic plot, and if the signal is small enough so that the dynamic range is inside the linear region, then the linear small-signal model applies and the circuit can be analyzed as a linear system.