Question 8.8: Consider a class AB amplifier. Analyze the effect of power b......

Suppose that the input bias of the class AB amplifier is set so as to maximize the input voltage swing. In such conditions, the circulation angle is \alpha_M. We have then from (8.23), the input waveform, repeated here for convenience:

v_{G S}(\theta(t))=V_{G S, P} \cos \theta+V_{G S, D C} \hspace{30 pt} \text{(8.23)}

with V_{G S, D C} and V_{G S, P} given, respectively, as a function of the circulation angle by (8.26) and (8.27) evaluated for \alpha=\alpha_M :

\begin{gathered} V_{G S, D C}=-\frac{\left|V_T\right|}{1-\cos \left(\alpha / 2\right)} \hspace{30 pt} \text{(8.26)}\\ V_{G S, P}=\frac{\left|V_T\right|}{1-\cos \left(\alpha / 2\right)} . \hspace{30 pt} \text{(8.27)}\end{gathered}

\begin{gathered} V_{G S, D C}=-\frac{\left|V_T\right|}{1-\cos \left(\alpha_M / 2\right)} \hspace{30 pt}\\ V_{G S, P}=\frac{\left|V_T\right|}{1-\cos \left(\alpha_M / 2\right)} . \hspace{30 pt} \end{gathered}

We now back the amplifier off the maximum power conditions by decreasing the peak input voltage amplitude to V_{G S, P}^{\prime}<V_{G S, P}; since the input bias voltage is kept constant we have:

v_{G S}(\theta(t))=V_{G S, P}^{\prime} \cos \theta+V_{G S, D C},

while the new circulation angle α is given by the condition:

v_{G S}(\alpha / 2)=V_{G S, P}^{\prime} \cos (\alpha / 2)+V_{G S, D C}=-\left|V_T\right|,

or:

\cos (\alpha / 2)=-\frac{V_{G S, D C}+\left|V_T\right|}{V_{G S, P}^{\prime}}=\frac{\cos \left(\alpha_M / 2\right)}{1-\cos \left(\alpha_M / 2\right)} \frac{\left|V_T\right|}{V_{G S, P}^{\prime}} .

The input available power now reads:

P_{ \text{av} , i n}=\frac{\left(V_{G S, P}^{\prime}\right)^2}{4 R_g}=\frac{\left(V_{G S, P}\right)^2}{4 R_g}\left(\frac{V_{G S, P}^{\prime}}{V_{G S, P}}\right)^2=P_{ sat , i n}\left[\frac{\cos \left(\alpha_M / 2\right)}{\cos (\alpha / 2)}\right]^2,

where P_{\text {sat, in }} is the input power corresponding to the maximum output power condition; we thus have that the circulation angle depends on the input available power as:

\cos (\alpha / 2)=\sqrt{\frac{P_{ \text{sat} , i n}}{P_{ \text{av} , i n}}} \cos \left(\alpha_M / 2\right). \hspace{30 pt} \text{(8.28)}

Therefore, the output power will be:

P_{o u t}=G_{ t }(\alpha) P_{ \text{av} , \text { in }}=G_{ t , A} \frac{[\alpha-\sin (\alpha)][1-\cos (\alpha / 2)]}{4 \pi} P_{ \text{av} , \text {in}} .

In other words, the output power generally is a nonlinear function of the input power, since \alpha=\alpha\left(P_{ \text{av} , i n}\right), see (8.28). Notice that, when decreasing the input power, the amplifier is back in class A operation; this happens when α = 2π, i.e., when:

V_{G S, P}^{\prime} \leq \frac{\cos \left(\alpha_M / 2\right)}{1-\cos \left(\alpha_M / 2\right)}\left|V_T\right|.

When class A operation is reached, the amplifier ultimately remains in class A also for lower power. Thus, the class \text{AB }P_{\text {in }}-P_{\text {out}} is a straight line with slope 1 (in log scale) for low power, where the device operates in class A; for large enough input power the device enters class AB and the circulation angle increases with increasing power, leading to a sublinear characteristic, as shown in Fig. 8.23. Note that this behavior is typical of class AB (and also of class C), while both in class A and B the circulation angle does not depend on the input power.