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Chapter 5

Q. 5.5

A broadband amplifier stage with G B W_{S}=100 \mathrm{GHz} is realized in a SiGe BiCMOS process with SiGe HBT f_{M A X}of 300 \mathrm{GHz}. Find the optimal number of cascaded identical stages that leads to the highest bandwidth amplifier with 40 \mathrm{~dB} gain.

Step-by-Step

Verified Solution

The gain of 40 \mathrm{~dB} corresponds to A_{t o t}=100. From (5.55), we find the optimal number of stages to be n_{O P T}=9.2. Indeed, if we use n=9, we obtain

\frac{G B W_{t o t}}{G B W_{S}}=100^{1-1 / 9} \times \sqrt{2^{1 / 9}-1}=16.95
If n=10, we obtain:
\frac{G B W_{t o t}}{G B W_{S}}=100^{0.9} \times \sqrt{2^{0.1}-1}=16.90
and, if n=5
\frac{G B W_{\text {tot }}}{G B W_{S}}=100^{0.8} \times \sqrt{2^{0.2}-1}=15.34

These results indicate that the best overall G B W_{\text {tot }} is 1.695 \mathrm{THz} (much higher than the f_{M A X} of the transistor in this BiCMOS process), corresponding to an amplifier bandwidth of 17 \mathrm{GHz}. Each of the 9 stages has a gain \mathrm{A}_{0}=1.668(4.44 \mathrm{~dB}) and a 3 \mathrm{~dB} bandwidth of 60 \mathrm{GHz} . However, 9 stages will occupy a large die area and the solution may prove uneconomical. Could we design an amplifier with a smaller number of stages and acceptable bandwidth? It turns out that even if we use only 5 stages, G B W_{t o t} is 1.534 \mathrm{THz}, only 10 \% smaller than the maximum possible. The amplifier will have a bandwidth of 15.34 \mathrm{GHz}, and each of the 5 stages would have a gain of 2.51 (or 8 \mathrm{~dB} ) and a bandwidth of 39.8 \mathrm{GHz}.

Equations (5.54) and (5.55) are strictly valid for single-pole gain stages. Similar expressions can be derived for amplifier stages with second- or higher-order frequency response functions. For the frequently encountered case of a chain of second-order Butterworth stages with Q=\sqrt{2} and no zeros, we obtain [9]

\frac{G B W_{t o t}}{G B W_{S}}=A_{t o t}^{1-1 / n}\left(2^{1 / n}-1\right)^{1 / 4}                                    (5.56)

Finally, it should be noted that the analysis above also applies to the gain-bandwidth product of a chain of cascaded tuned amplifier stages.

*Equations (5.54)\frac{G B W_{t o t}}{G B W_{S}}=A_{t o t}^{1-1 / n}\left(2^{1 / n}-1\right)^{1 / 4}

*Equations (5.55) n_{o p t} \approx 2 \times \ln A_{\text {tot }}