Question 6.7: Discuss Fano’s limit to the input and output wideband matchi......

Consider the simplified equivalent circuit of a field-effect or bipolar transistor. The input and output equivalent circuits can be approximated as:

• for the FET, the input as a series RC network (gate plus intrinsic resistance and gatesource capacitance), the output as a parallel RC network (drain-source resistance and capacitance);
• for the bipolar, the input as a parallel RC network (base-emitter resistance and capacitance), the output as a parallel RC network (collector-emitter resistance and capacitance).

In both cases we only need to discuss the possibility to match, over a prescribed bandwidth and to a prescribed reflection coefficient Γ, a series or parallel RC network. According to Fano’s theory [11], the following inequality holds independent of the complexity of the reactive matching network for a series RC network:

\int_0^{\infty} \frac{1}{\omega^2} \log \left|\Gamma^{-1}(\omega)\right| d \omega \leq \pi R C ,

where Γ is the reflection coefficient of the matching network loaded by the series RC load. Similarly, for a parallel RC load we have:

\int_0^{\infty} \log \left|\Gamma^{-1}(\omega)\right| d \omega \leq \frac{\pi}{R C}.

For the sake of definiteness, suppose that the reflection coefficient be constant \left(\Gamma_0\right) on the bandwidth B centered around the centerband frequency f_0, and equal to 1 outside such a bandwidth. In the parallel RC case we obtain:

i.e., in the case where the inverse of the reflection coefficient is maximum (and therefore the reflection coefficient is minimum):

\left|\Gamma_0\right|=\exp \left[-\frac{2 \pi^2 R C}{B}\left(f_0^2-\frac{B^2}{4}\right)\right] .

In the parallel RC case we have instead:

\int_0^{\infty} \log \left|\Gamma^{-1}(\omega)\right| d \omega=-2 \pi B \log \left|\Gamma_0\right| \leq \frac{\pi}{R C},

i.e., in the best case:

\left|\Gamma_0\right|=\exp \left(-\frac{1}{2 B R C}\right).

As an example, suppose that in a HEMT R_G+R_I=5 \; \Omega and C_{G S}=0.2\text{ pF}; we have RC = 5·0.2 \times 10^{-12}=1\text{ ps}. Assuming f_0=10\text{ GHz} the minimum reflection coefficient on a 10 percent bandwidth (B = 1 GHz) is:

\left|\Gamma_0\right|=\exp \left[-\frac{2 \cdot \pi^2 \cdot 1 \times 10^{-12}}{1 \times 10^9}\left(\left(10 \times 10^9\right)^2-\frac{\left(1 \times 10^9\right)^2}{4}\right)\right]=0.14 .

but for a 100 percent bandwidth (B = 10 GHz, i.e., from 5 to 15 GHz) we obtain:

\left|\Gamma_0\right|=\exp \left[-\frac{2 \cdot \pi^2 \cdot 1 \times 10^{-12}}{10 \times 10^9}\left(\left(10 \times 10^9\right)^2-\frac{\left(10 \times 10^9\right)^2}{4}\right)\right]=0.86 .

In other words, in the second case the amplifier will be severely mismatched all over the bandwidth. In the limiting case of a bandwidth from DC to 20 GHz we have B = 20 GHz and the minimum reflection coefficient will be 1. Notice that with a real matching network the input reflection coefficient is not constant, but is typically oscillating; the above conclusions can however be referred to the average reflection coefficient on the band.