**Transconductance Small Signal Model:**

Back in the very straining of the transistor era, an engineer at Jim Early, Bell Labs, predicted that there would be a slope to the I_{C} curves, and that they would all of the project back to the same intersection point on the horizontal axis. Having built that prediction, J_{im} went down into the laboratory, made the measurement, and confirmed his prediction, therefore showing that the theory of transistor behaviour was being correctly understood. The point of intersection of the V_{CE} axis is known as the Early Voltage. As the symbol V_{E}, for the emitter voltage was already taken, they ought to label the Early Voltage V_{A} instead. (Even though the intersection points are on the negative half of the V_{CE }axis, V_{A} is universally quoted as a positive number.)

Generally, the slope is much less than we have illustrated here, and so for any given value of I_{C}, we can just take the slope of the line as I_{C}/V_{A}, and therefore the resistance, which is usually called r_{0} is just V_{A}/Ic. Therefore, we add r_{0} to the small signal model for the bipolar transistor. This is illustrated in Figure (a). In a good quality modern transistor, the Early Voltage, V_{A} shall be on the order of 150-250 Volts. So if we let V_{A} = 200 V, and we suppose that we have our transistor biased at 1 mA, then r0 =200 V/1 mA = 200 k?, which is usually much larger than most of the other resistors you shall encounter in a typical circuit. In most instances, r_{0} can be ignored with no problem.

**(b)**

**Figure: (a) Including r0 in the Small Signal Model; and (b) Transconductance Small Signal Model**

Sometimes this is advantageous to use a mutual transconductance model of Figure (b) rather than a current gain model for the transistor. If we say the input small signal voltage is v, then obviously

i = v_{BE} / r_{π} = v_{BE} / (β/ 40I_{C})

But

i = βi_{b} = β v_{be }/ (β /40Ic) =40Ic v_{be =} g_{m}v_{be}

where g_{m} is called the mutual transconductance of the transistor. Notice down that β has totally cancelled out in the expression for g_{m} and that g_{m} based only upon the bias current, I_{C}, passing through the collector and not on any of the physical properties of the transistor itself.