# Why is $r_o$ related to $V_A$ instead of $V_T$?

We can learn that $$\r_o \approx \frac{V_A}{I_{C}}\$$ , and $$\r_{\pi} \approx \frac{V_T}{I_{B}}\$$ in BJT, where $$\V_A\$$ is Early voltage and $$\V_T\$$ is thermal voltage.

I'm curious about why $$\r_o\$$ is related to $$\V_A\$$ instead of $$\V_T\$$?

Also, why is $$\r_\pi\$$ related to $$\V_T\$$ instead of $$\V_A\$$?

• $r_\pi$ is directly related to $r_e^{\:'}$ which itself is directly related to the local slope of the diode equation (which is related to $V_T$.) Have you looked at it? $r_o$ is a similar concept -- another local slope calculation. How comfortable are you with taking derivatives (partials with respect to a single parameter) to find slopes? Most of what you need to know can be readily found on your own if you understand the basic concepts behind derivatives/partials. Commented Apr 25, 2023 at 13:49
• Vt is a physician/semiconductor "constant". Va on the other hand is an "artificially" created parameter to model the Early effect. linear modeling. And it gives the "false impression" that the Early effect is linear.
– G36
Commented Apr 25, 2023 at 14:02

There's a simplification used that relates the collector current to the base-to-emitter junction voltage in an active-mode BJT:

$$I_{_\text{C}}=I_{_\text{SAT}}\cdot\left[\exp\left(\frac{V_{_\text{BE}}}{V_T}\right)-1\right]$$

[The above is a simplification from one of the three equivalent Ebers-Moll DC models -- the large-scale, non-linear hybrid-$$\\pi\$$ version (and that is not the linearized hybrid-$$\\pi\$$ often used for small signal analysis.) Sometimes, the above equation also includes an ideality factor (emission coefficient.) But I'm leaving it out for now. Also, while I'm sure you already understand that $$\V_T\$$ is based on temperature, it's also the case that $$\I_{_\text{SAT}}\$$ is also highly dependent on temperature (Boltzmann factor found in basic thermodynamics.)]

If you ignore the $$\-1\$$ term there and just solve the above equation for $$\V_{_\text{BE}}\$$ and take its partial derivative with respect to $$\I_{_\text{C}}\$$ then you will readily find a resistance value that when multiplied by $$\\beta\$$ is $$\r_\pi\$$:

\begin{align*} I_{_\text{C}}&\approx I_{_\text{SAT}}\cdot\exp\left(\frac{V_{_\text{BE}}}{V_T}\right)&\therefore &&V_{_\text{BE}}&\approx V_T\cdot\ln\left(\frac{I_{_\text{C}}}{I_{_\text{SAT}}}\right)\\\\ &&&&\text{d}V_{_\text{BE}}&\approx V_T\cdot\frac{\text{d}\,I_{_\text{C}}}{I_{_\text{C}}}\\\\ &&&&\frac{\text{d}V_{_\text{BE}}}{\text{d}\,I_{_\text{C}}}&\approx \frac{V_T}{I_{_\text{C}}} \end{align*}

That's a local slope value that can also be seen as a so-called dynamic resistance. (Keep in mind this is all about small signal analysis, which takes everything as very tiny changes about an operating point.) As seen from the base, you'd multiply this by $$\\beta\$$ to get $$\r_\pi\$$. I don't want to go into a long description it. You either can see why, or not. I'll leave it there, for now.

So $$\r_\pi=\frac{\partial}{\partial I_{_\text{B}}}V_{_\text{BE}}=\beta\cdot\frac{V_T}{I_{_\text{C}}}\$$.

In the case of $$\r_o\$$, it's based on a different approximation (simplification) of the hybrid-$$\\pi\$$ DC model. Here we have $$\r_o=\frac{\partial}{\partial I_{_\text{C}}}V_{_\text{BC}}\$$. In later changes to the Ebers-Moll models, the 3rd revision where base-width modulation was added, the following simplification used $$\V_A\$$ in approximately this way:

\begin{align*} I_{_\text{C}}^{\:'}&\approx I_{_\text{C}}\cdot\left(1+\frac{V_{_\text{BC}}}{V_A}\right) \end{align*}

The $$\\partial I_{_\text{C}}^{\:'}\$$ with respect to $$\\partial V_{_\text{BC}}\$$ is $$\\frac{I_{_\text{C}}}{V_A}\$$ so therefore $$\r_o\approx \frac{V_A}{I_{_\text{C}}}\$$.

There are more details, of course. But this gets the gist across.