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## 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.

## 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) $n_{o p t} \approx 2 \times \ln A_{\text {tot }}$