Ideal Transistor

The basic solid state amplification devices are bipolar, JFET and MOSFET transistors.  All of them can be considered as transconductance device, which is a current source controlled by a voltage.

A bipolar transistor has a diode input structure: the input voltage between the base and the emitter sees a small signal resistance of $V_T/I_b$, where $V_T= kT/q$ is the thermal voltage, about 26mV at the room temperature and the $I_b$ is the base bias current, usually in the 10$\mu$A range, so the resistance $r_b$ is about $2.6K \Omega$.   A voltage $v_{be}$, that generates $i_b = v_{be}/r_b$, would cause a collector-emitter current $i_c = \beta i_b = \beta v_{be}/r_b$; so the transconductance is $g_m = \beta/r_b$. With $\beta$ around 100, the transductance is about 40mA/V. Another way to look at it is an ideal transistor with $r_e = 1/g_m$ at its emitter. An ideal transistor is to have an infinite transconductance, which implies that $v_{be}$ approaches zero.  Let's use this model for the three basic amplifier configurations.  The gain of the common-emitter amplifier with emitter degeneration, $- R_C/(r_e + R_E)$. $r_e$ is about $25 \Omega$, which is generally much less than $R_E$, so the gain simplifies to $-R_C/R_E$, independent of the transistor parameters.  For the common-collection amplifier or the follower, the gain is a simple voltage divider, $R_E/(r_e + R_E) = 1/(1 + r_e/R_E) \approx 1$.  For the common-base amplifier, the gain is $r_e/R_C$.

Now we have raised the concept of an ideal transistor, let's be more clear.  The transconductance is infinite.  Because we have been using the small-signal model, which is a linearized model around the bias point, we have ignored the biasing.  An ideal transistor would require no biasing, no input current into the base  and the emitter current is bi-directional, flowing both in and out the emitter.   Such an ideal transistor has been approximated by an integrated circuit that is called an operational transconductance amplifier (OTA) or a diamond transistor.   One example is Burr-Brown (TI)'s OPA860.  It has a high impedance input as the base and a bi-directional emitter and requires no bias.  The transconductance is adjustable, but is around 100 mA/V or equivalently $r_e = 10 \Omega$.  So the transconductance gain is not quite infinite, but $r_e$ is small enough compared to $R_E$.