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AC Analysis

Now that we know the DC operating point, and hence all DC currents and voltages, we can analyse the AC amplification.

The next step is to draw the AC equivalent circuit. To do this, first consider the amplifier circuit from the point of view of the small AC signals, as in Figure 111. Note that capacitors are replaced by short circuits, and DC sources are zeroed out.

For the moment we ignore the effects of the load resistance R_L, and also we assume v_out=v_c for AC signals. Note that $i_{in} \neq i_b$, and v_in=v_b.

  

Figure 111: AC equivalent circuit for the circuit of Figure 110 neglecting R_L.

The resistor R_E does not appear since it has been shorted out by the bypass capacitor C_E. The DC voltage source V_CC is also shorted to ground, so R_B1 and R_C connect directly to the AC ground.

Using the AC hybrid-$\pi$ model we get the circuit of Figure 112. It is now straightforward to determine the gain

$$A_{in,out} = \frac{v_{out}}{v_{in}}$$

and other properties of interest.

  

Figure: AC equivalent circuit with hybrid-$\pi$ model for the circuit of Figure 110 neglecting R_L.

The resistance r_e can be calculated from (91) as

$$r_e = 6.83 \ \Omega$$

and the resistance $r_\pi$ is therefore

$$r_\pi = (\beta + 1) r_e = 2.056 \ k\Omega$$

Now

$$i_b = \frac{v_{in}}{r_\pi}$$

(by Ohm's Law) and so

$$i_c = \frac{\beta v_{in}}{r_\pi} .$$

Also

$$v_{out} = v_{c} = - i_c R_C \ (= v_{ce} )$$ (103)

and so

$$v_{out} = - \frac{\beta R_C}{r_\pi} v_{in} .$$ (104)

This says that the amplifier has gain

$$A_{in,out} = - \frac{\beta R_C}{r_\pi} = - \frac{\beta R_C}{(\beta + 1)r_e} \approx - \frac{R_C}{r_e} .$$ (105)

The approximation is valid if $\beta >>1$. In this example we get

A_in,out = -321 .

The small signal AC input resistance   looking into the amplifier from the input is defined by

$$r_{in} = \frac{v_{in}}{i_{in}} = R_{B1} \parallel R_{B2} \parallel r_\pi$$

and this calculates to $r_{in} = 1.76 \ k\Omega$ The small signal AC output resistance   looking into the amplifier from the output is defined by

$$r_{out} = \frac{v_{c}}{i_{out}} = R_{C}$$

so r_out = 2.2 k$\Omega$.

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