Determine, to a first approximation, the frequency at which the small-signal current gain decreases to $1 / \sqrt{2}$ of its low-frequency value.

Consider the simplified hybrid-pi circuit shown in Figure 12.41. We are ignoring $C_{\mu}, C_{s}$ , $r_{\mu}, C_{j e}, r_{0}$ , and the series resistances. We must emphasize that this is a first-order calculation and that $C_{\mu}$ normally cannot be neglected.

Question

Determine, to a first approximation, the frequency at which the small-signal current gain decreases to [latex]1 / \sqrt{2}[/latex] of its low-frequency value.

Consider the simplified hybrid-pi circuit shown in Figure 12.41. We are ignoring [latex]C_{\mu}, C_{s}[/latex], [latex]r_{\mu}, C_{j e}, r_{0}[/latex], and the series resistances. We must emphasize that this is a first-order calculation and that [latex]C_{\mu}[/latex] normally cannot be neglected.

Accepted Answer

At very low frequency, we may neglect [latex]C_{\pi}[/latex] so that

[latex]V_{b e}=I_{b} r_{\pi} \quad ext { and } \quad I_{c}=g_{m} V_{b e}=g_{m} r_{\pi} I_{b}[/latex]

We can then write

[latex]h_{f e 0}=\frac{I_{c}}{I_{b}}=g_{m} r_{\pi}[/latex]

where [latex]h_{f e 0}[/latex] is the low-frequency, small-signal common-emitter current gain.

Taking into account [latex]C_{\pi}[/latex], we have

[latex]V_{b e}=I_{b}\left(\frac{r_{\pi}}{1+j \omega r_{\pi} C_{\pi}} ight)[/latex]

Then

[latex]I_{c}=g_{m} V_{b e}=I_{b}\left(\frac{h_{f e 0}}{1+j \omega r_{\pi} C_{\pi}} ight)[/latex]

or the small-signal current gain can be written as

[latex]A_{i}=\frac{I_{c}}{I_{b}}=\left(\frac{h_{f 0}}{1+j \omega r_{\pi} C_{\pi}} ight)[/latex]

The magnitude of the current gain is then

[latex]\left|A_{i} ight|=\left|\frac{I_{c}}{I_{b}} ight|=\frac{h_{f e 0}}{\sqrt{1+\left(\omega r_{\pi} C_{\pi} ight)^{2}}}=\frac{h_{f e 0}}{\sqrt{1+\left(2 \pi f r_{\pi} C_{\pi} ight)^{2}}}[/latex]

The magnitude of the current gain drops to [latex]1 / \sqrt{2}[/latex] of its low-frequency value at [latex]f=1 / 2 \pi r_{\pi} C_{\pi}[/latex].

If, for example, [latex]r_{\pi}=2.6 \mathrm{k} \Omega[/latex] and [latex]C_{\pi}=4 \mathrm{pF}[/latex], then

[latex]f=15.3 \mathrm{MHz}[/latex]

Comment

High-frequency transistors must have small-diffusion capacitances, implying the use of small devices.

Question 12.13: Determine, to a first approximation, the frequency at which ...

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