# Single input impedance of long-tailed pair

I have following circuit: and I'd like to know input resistance of single input Ui. I tried to calculate it like this: if I increase Ui by little, there's some voltage drop on base-emitter junction (which determines my emitter current), and also some drop on both R1 and R2. Here, I don't know what to do with the other transistor - from the point of view of input, it can be represented as emitter-base resistance of Q3 and also some voltage drop. But as base of Q3 is always on +2V, voltage drop is equal to Ube of Q4. So, if I'm correct:

$$r_{in} = \frac{U_i}{I_{b_{Q4}}} = \frac{U_{be_{Q4}} + U_{R1} + U_{R2} + U_{be_{Q3}}}{\frac{U_{be_{Q4}}}{r_{be_{Q4}}}} \\ U_{be_{Q4}} = U_{be_{Q3}} \\ U_{R1} = U_{R2} = (I_b + I_e)R_1 = \frac{U_{be_{Q4}}}{r_{be_{Q4}}}(1+\beta)R_1\\ \\ r_{in} = \frac{2U_{be_{Q4}} + 2\frac{U_{be_{Q4}}}{r_{be_{Q4}}}(1+\beta)R_1}{\frac{U_{be_{Q4}}}{r_{be_{Q4}}}} = 2r_{be_{Q4}} + 2(1+\beta)R_1$$

So, finally, it's equal to two times input resistance of common-emmiter stage (with emitter degeneration)? Thanks for help.

## 1 Answer

I use R_In = Beta * (2 * 0.026 / (0.5 * Itail))

Thus 1mA, split into two paths, produces 0.5 ma each side, with 'reac' of 0.026 / 0.0005 = 52 ohms as the derivative of the volts/amps of the emitter diodes.

Double that, because one stage serves as CommonBase load for the other CommonEmitter. Becomes 104 ohms.

Scale up by beta, to 10,400 ohms.

The "52 ohms" is highly dependent upon the emitter current, and for inputs or 20 or 20 milliVolts PeakPeak will show 2:1 or 3:1 changes in the emitter currents and thus large changes in the "input impedance"; one bipolar will have much lower input R, and the other (nearly starved for current) will have much higher input R; the total R is what you want.

Somewhere in there is the Rout of the long-tail-pair.

• I guess I've missed that you can treat the other transistor as common base load. Thank you!:) – Kacper Banasik May 2 '17 at 16:04
• Courtesy of "Electronics World" magazine article, long defunct. – analogsystemsrf May 2 '17 at 17:27