Output Impedance of a Transistor for the Emitter Follower (Common Collector)

The output impedance seen by the load (R_E in this example) is defined as:

Z_out = - dV_out / dI_out

The minus sign in the derivative comes from the fact the output impedance has the effect of decreasing V_out. The output current I_E, which is related to the base current.

V_in = I_BR_S + V_BE +V_E [V_out = V_E]

V_out = V_E = V_in – I_BR_S – V_BE [I_out = I_E and I_B = I_E/(β + 1)]

V_out = V_in – {I_E/(β + 1)}R_S – V_BE = V_in – {I_out/( β + 1)}R_S – V_BE

= - I_out{R_S/(β + 1)} + (V_in – V_BE)

So, Z_out = - dV_out/dI_out = -d/dI_out [ - I_out{R_S/(β + 1)} + (V_in – V_BE) ]

Z_out = R_S/(β + 1)

Thus we obtain the result that the impedance of the source, as viewed by the load, is reduced by the factor ~ 1/ β.

Zout = {RS/( β + 1) ≈ RS/β

Input Impedance of a Transistor

Impedance is defined as Z = V/I. In linear circuits (with resistors, capacitors, inductors, batteries, etc.) this ratio is the reciprocal of the slope of the I versus V graph. In circuits with nonlinear elements such as a transistor, the input impedance of the resistor is defined as the reciprocal of the slope of the I versus V graph. This is simply the derivative of V_in with respect to I_in-

Z_in = dV_in / dI_in

We can easily find Z_in from what we know already of the behavior of the transistor. We know that the sum of V_BE and the IR drop across R_E must equal V_in.

V_in = V_B = V_BE +V_E = V_BE + I_ER_E [I_R = I_C + I_B = βI_B + I_B = (β + 1)I_B]

V_in =V_BE + I_ER_E = I_B(β + 1)R_E [I_B = I_in]

V_in = V_BE + I_in (β + 1) R_E

Taking the derivative of V_in with respect to I_in, remembering that V_BE is a constant, we get the result:

Z_in = dV_in/dI_in = d/dI_in(V_BE + I_in (β + 1)R_E) = (β + 1)R_E

Zin = (β + 1)RE ≈ βRE

Because I_E = I_B (β + 1). The IR drop across R_E is greater then it would be for I_B alone. The amplification of the base current causes R_E to appear larger to a source looking into the input by (β + 1).

Common Nomenclature (npn Example)

Ohmmeters view of the BJT

Clearly a transistor cannot be made on the bench by combining two diodes. (Why is that?) Most ohmmeters not only measure resistance, but also measure the forward voltage drop across a diode. From this perspective you can identify the base and the type of transistor based on the following equivalent circuits.