Taylor made small-signal circuits

1. BJT Ebers-Moll model equation

Ebers-Moll
Figure 1. Ebers-Moll model for NPN

Briefly review the true pn junction diode equation from day09_diode-equation.pdf. Also review the section Tourbook: BJT Circuit Models.

BJT terminal currents for the Ebers-Moll model
\[\begin{align} i_{E} &= & \dfrac{I_S}{\alpha_F} &\left[ \exp\left(\dfrac{v_{BE}}{V_T}\right) - 1 \right] & - I_S &\left[ \exp\left(\dfrac{v_{BC}}{V_T}\right) - 1 \right] \\ i_{C} &= & I_S &\left[ \exp\left(\dfrac{v_{BE}}{V_T}\right) - 1 \right] & - \dfrac{I_S}{\alpha_R} &\left[ \exp\left(\dfrac{v_{BC}}{V_T}\right) - 1 \right] \end{align}\]

Comments on these as reminders:

  • The E-M model has no notion of operating modes — it works for all conditions. This is better for accuracy but obviously a pain for intuitive understanding and hand calculations.

  • The current and voltage variables' capitalization mean this is the total value (DC + AC).

  • \(I_S\) is a transistor parameter in units of amperes, it is not a “real” current.

  • \(V_T = \frac{k_B T}{q}\) depends only on temperature.

  • There are two betas, usually βF ≫ βR, and thus also αF > αR.

Also review the Tourbook section about the Ebers-Moll model and its approximation in forward active mode.

\[i_C = I_S \left[\exp\left(\frac{v_{BE}}{V_T}\right) - 1\right]\label{eq-ic}\]
How and when can we drop the “−1”?

2. Large-signal circuit analysis

ce taylor
Figure 2. Simple CE circuit

It is totally fine to simply use the full transistor equation to find the voltage at node out. For this small circuit, the math isn’t so bad.

\[\begin{align} i_C &= I_S \exp\left(\frac{v_{BE}}{V_T}\right) \\ v_{OUT} &= V_{CC} - R_C i_C \label{e-iC} \\ v_{OUT} &= V_{CC} - R_C I_S \exp\left(\frac{v_{BE}}{V_T}\right) \end{align}\]


Sketch what this function looks like.

ce full math
Figure 3. Full plot of Figure 2 response

2.1. Potential inside the exponential

Substitute \(v_{BE}\) with \((V_{BE} + v_{be})\) in equation \(\eqref{eq-ic}\) and pay close attention to the symbol capitalization.[1]

How IS becomes IC
\[\begin{align} i_C &= I_S \exp\left(\frac{v_{BE}}{V_T}\right) \\ i_C &= I_S \exp\left( \frac{V_{BE} + v_{be}}{V_T}\right) \\ &= \underbrace{I_S \exp\left(\frac{V_{BE}}{V_T}\right)}_{= I_C} \exp\left(\frac{v_{be}}{V_T}\right) \\ i_C &= I_C \exp\left(\frac{v_{be}}{V_T}\right) \label{e-full} \end{align}\]

Notice the sleight of hand that happened! It really wasn’t changing \(I_S \rightarrow I_C\), it was the changing of \(v_{BE}\) to \(v_{be}\). The \(I_C\) is simply another substitution when we saw the DC term showing up after the \(v_{BE}\) expansion.

3. Towards gm

Taylor series definition
\[\hat{f}(x) = \sum\limits_{n=0}^\infty \frac{f^{(n)}(a)}{n!} \left(x - a\right)^n\]

Expand the function \(\eqref{e-full}\) about \(v_{be} = 0\) in a Taylor series.[2]

What does it mean when \(v_{be} = 0\) ?
Table 1. Symbol mapping
Taylor transistor

x

\(v_{be}\)

a

0

\(f(x)\)

\(I_C \exp\left(\frac{v_{be}}{V_T}\right)\)

Work out the first several terms of the series:

Table 2. Series terms (a = 0)
n \(f^{(n)}\) \(f^{(n)}(a)\) \(\dfrac{f^{(n)}(a)}{n!} (x-a)^n\)

0

\(I_C \exp\left(\dfrac{v_{be}}{V_T}\right)\)

\(I_C\)

\(I_C\)

1

\(I_C \exp\left(\dfrac{v_{be}}{V_T}\right) \dfrac{1}{V_T}\)

\(\dfrac{I_C}{V_T}\)

\(\dfrac{I_C}{V_T} v_{be}\)

2

\(I_C \exp\left(\dfrac{v_{be}}{V_T}\right) \left(\dfrac{1}{V_T}\right)^2\)

\(\dfrac{I_C}{V_T^2}\)

\(\dfrac{I_C}{2 V_T^2} v_{be}^2\)

3

\(I_C \exp\left(\dfrac{v_{be}}{V_T}\right) \left(\dfrac{1}{V_T}\right)^3\)

\(\dfrac{I_C}{V_T^3}\)

\(\dfrac{I_C}{6 V_T^3} v_{be}^3\)

…​

…​

…​

…​

Assemble the terms and include only n = 0 and n = 1:

\[\begin{align} i_C &\approx I_C + \frac{I_C}{V_T} v_{be} + \cdots \label{e-iC-approx} \\ I_C + i_c &\approx \\ i_c &\approx \frac{I_C}{V_T} v_{be} + \cdots \label{e-vccs}\\ \end{align}\]

Equation \(\eqref{e-vccs}\) precisely describes the behavior of a voltage-controlled current source with transconductance \(g_m = \dfrac{I_C}{V_T}\).

gm model
Figure 4. VCCS schematic implied by \(\eqref{e-vccs}\)
The Small-Signal Approximation

IFF the AC signal \(v_{be}\) is “small”, then \(v_{be}^2\) is very small, which we will round to zero. We can be specific about this! Look behind the curtain: there is no magic! The English of the first sentence translates to:

\[\begin{align} \dfrac{I_C}{2 V_T^2} v_{be}^2 &\ll \dfrac{I_C}{V_T} v_{be} \\ \dfrac{1}{2 V_T} v_{be} &\ll 1 \\ v_{be} &\ll 2 V_T \end{align}\]

At room temperature the changes in vbe should be smaller than 50 mV for the small-signal approximation to be comfortably valid.

3.1. Substitute back into circuit analysis

Equation \(\eqref{e-full}\) represents the exact shape of the output voltage over the range of vBE corresponding to active mode.

Substitute the iC approximation of \(\eqref{e-iC-approx}\) into \(\eqref{e-iC}\).

\[\begin{align} v_{OUT} &\approx V_{CC} - R_C \left(I_C + \frac{I_C}{V_T} v_{be}\right) \\[10pt] &\approx V_{CC} - R_C I_C - R_C \frac{I_C}{V_T} v_{be} \\ \end{align}\]

Notice, I beg of you, that there are two parts to the output voltage’s total value: the part that Doesn’t Change in response to vbe, and the part that Always Changes. Split them apart.

\[\begin{align} V_{OUT} + v_{out} &\approx \underbrace{V_{CC} - R_C I_C} \underbrace{- R_C \frac{I_C}{V_T} v_{be}} \\ \nonumber \\ V_{OUT} &= V_{CC} - R_C I_C \\[10pt] v_{out} &\approx - R_C \frac{I_C}{V_T} v_{be} \end{align}\]

Take the original circuit of Figure 2.

  • Find the small-signal equivalent circuit and

  • gain \(A_v = v_{out} / v_{be}\).

Set \(g_m = I_C / V_T\) and revel in the wonder that is the idea of small-signal equivalent circuits.













(↑↑↑ space for you to do this yourself)

Small wonders!

You get the exact same answer as if you approximated all non-linear elements with their first-order Taylor series approximation.

Go back and consider what this approximation looks like graphically on Figure 3.

Which would you rather deal with?

4. How does rπ and ro work?

These are for you to do (as acdc03)!

It is the relationship between base current and base-emitter voltage.

5. References

John D. Cook’s Blog: Much less than, Much greater than (PDF archive)

1. recall the idea behind this expansion
2. The a = 0 special case is sometimes called a Maclaurin series. Alas, this footnote is here simply to note that this comment is always there when talking about the more general Taylor series.