Input and Output Impedance of a BJT

Question

Hi I'm new to electronics and I would like to know how to measure experimentally the input and output resistance of the following common emitter configuration. I have access to a voltmeter, signal generator, power supply and an oscilloscope as well.

Answer 1

Measuring input/output impedance is little more than calculating the resistors in a voltage divider. Consider the following two situations:

^{simulate this circuit – Schematic created using CircuitLab}

Important is that the whole circuit is not clipping, so the voltage \$U_i\$ is small in relation to the supply voltage. There should be no distortion. Also it is essential that you measure only the AC component, so in case of the output impedance you do your measurements at the right end of the output capacitor.

For the output impedance you have to do two measurements at a given frequency. For audio 1kHz is a good start, but as Andy states, you might be interested in the impedance at various frequencies.

With a known input amplitude \$U_i\$ do two measurements:

1. Measure the voltage amplitude \$U_{o}\$ with \$R_l\$ removed (left situation);
2. Measure the voltage amplitude \$U_{o}\$ with a known \$R_l\$ in place (right situation);

\$(1) R_o = \dfrac{\Delta U_{R_o}}{\Delta I} \$

\$(2) R_o = \dfrac{U_i-U_o}{\frac{U_i}{R_o+R_l}} \$

\$(3) R_o = \dfrac{R_o+R_l}{U_i} (U_i - U_0)\$

\$(4) R_o = R_o + R_l - (R_o+R_l)\dfrac{U_o}{U_i}\$

\$(5) R_l = (R_o+R_l)\dfrac{U_o}{U_i}\$

\$(6) R_l(1-\dfrac{U_o}{U_i}) = R_o\dfrac{U_o}{U_i}\$

\$(7) R_o = R_l(1-\dfrac{U_o}{U_i})\dfrac{U_i}{U_o}\$

So the resulting formula to calculate the output impedance is:

\$(8) R_o = R_l(\dfrac{U_i}{U_o}-1)\$

Calculating output impedance follows the same method as calculating input impedance, you are just calculating the other resistor in the divider.

The input impedance is identical to \$R_l\$ in (6) where \$R_o\$ is the output impedance of your signal source (optionally increased by an extra series resistor. So:

\$R_l = R_{amp,in} , U_o = U_{amp,in} , U_i = U_{generator,out} , R_o = R_{out,generator}\$

\$(6) R_l(1-\dfrac{U_o}{U_i}) = R_o\dfrac{U_o}{U_i}\$

\$(9) R_{amp,in}(1-\dfrac{U_{amp,in}}{U_{generator,out}}) = R_{generator,out}\dfrac{U_{amp,input}}{U_{generator,out}}\$

\$(10) R_{amp,in} = R_{generator,out} \cdot \dfrac{U_{amp,in}}{U_{generator,out}} \cdot \dfrac{1}{(1-\dfrac{U_{amp,in}}{U_{generator,out}})}\$

\$(11) R_{amp,in} = R_{generator,out} \cdot \dfrac{U_{amp,in}}{U_{generator,out}-U_{amp,input}}\$

Answer 2

The input impedance. First, choose what frequency you are interested in. If it's audio then maybe 1kHz is a good single frequency to use. However, you may be interested in the frequency range from 10Hz to 100kHz and if you are, choose several points to measure. Conventionally logarithmic steps are used such as: -

10Hz, 33Hz, 100Hz, 330Hz, 1kHz, 3.3kHz, 10kHz, 33kHz, 100kHz - but tighter steps can be used depending on how precise you need things to be. Generally speaking though, a simple amp (as you've shown in your circuit) would be ok at those frequencies.

Measurement: A voltmeter probably won't be good enough unless it is a true RMS voltmeter with a decent frequency range. If you can get access to one of these or an oscilloscope then that is the way to proceed. Connect the scope to the output and look for the signal generated by V1 through the amplifier. Make sure it is a sinewave and make sure it looks clean. If necessary inject a smaller signal. I'm assuming R1 on your circuit is a variable resistor or decade resistance box because without one you can't determine impedance easily.

Make sure R1 is set to zero ohms.

Once you have the signal on the oscilloscope, increase R1 until the signal level halves on the scope. The value of R1 can be regarded as the input impedance of your circuit. For the simple circuit like you have shown it is reasonable to use R1 as the input impedance but on other circuits like RF amplifiers a more detailed value would be required; one which measures the complex impedance.

For output impedance, you have to load the output and I note you have conveniently drawn RL. This should be a variable resistor or decade resistance box. Start with RL at open circuit and look at the signal on the scope. Start lowering RL until the scope signal halves in amplitude - this is your impedance.

Again, I add that these are simple impedance measurements and this method won't necessarily be suitable for high frequency amplifier or measurments where complex impedances are required. This method may not be suitable for higher power systems because of the loading current when measuring output impedance.