2/18: Impedances of Test Instruments and Op Amps

in which Julia investigates instrument impedances and begins op amps

Lab 3-8: Measuring the oscilloscope's impedance with a 100 Hz sine wave

Circuit to measure the impedance of the oscilloscope.

4.04 V is the maximum amplitude for the oscilloscope in series with 1 MΩ resistor.
2.08 V is the maximum amplitude for the oscilloscope without resistor.
The low-frequency (100 Hz input signal) attenuation is Vout/Vin = 0.52.

1.16 V is the maximum amplitude for the oscilloscope without resistor.
4.08 V is the maximum amplitude for the oscilloscope in series with 1 MΩ resistor.
Thus the high-frequency (10 kHz) attenuation is 0.28.

The oscilloscope output (purple) for the low-frequency input signal (yellow); similar results were obtained for a high-frequency input signal.

Modeling the oscilloscope as a resistor in parallel with a capacitor:

Zcap = -i/(ω*C)

Zr = R

Zscope = 1/[1/R + 1/(ω*C)]

This is the impedance for low frequencies; for high frequencies, the capacitor's impedance is small, so the scope's impedance is approximately equal to its internal resistance.  For low frequencies, the capacitor's impedance is large, so the scope's impedance is:

In this situation, we know both Vout  and Vin, so we can solve for Zscope using Ohm's Law and the above formula:

V = I*R so Vin = I*Zscope; Vout = I*(Zscope + 1 MΩ)

Vin/Zscope = I

Vout = (Vin/Zscope)*(Zscope + 1 MΩ)

Zscope*(Vout/Vin) = (Zscope + 1 MΩ)

Zscope*(Vout/Vin - 1) = 1 MΩ

Zscope = (1 MΩ)/(Vout/Vin - 1)

For low frequencies (ω = 100 Hz), Zscope = (1 MΩ)/(3.57 - 1) = (1 MΩ)/(2.57) ≈  390 kΩ

Zscope = 1/[1/R + 1/(ω*C)] = 390 kΩ
1/(Zscope) = 1/R + 1/(ω*C)
 1/R = 1/(Zscope) - 1/(ω*C)
R = 1/[1/(Zscope) - 1/(ω*C)]
R = 1/[(ω*C - Zscope)/(ω*C*Zscope)]

R = (100*C*390 kΩ)/(100*C - 390 kΩ)
R = (C*39 MΩ)/(100*C - .39 MΩ)

For high frequencies, the attenuation (Vout/Vin) = 0.52, so Zscope ≈ 1 MΩ (the same resistance as is externally in series with the scope).  Thus R = (C*100 MΩ)/(100*C - 1 MΩ).

Note: the input impedance of an oscilloscope is frequency dependent, getting smaller as frequency of input signal increases.  Use a 10x scope probe (as we discovered during the first few labs) to minimize the effects of the measurement on the circuit; be sure to impedance-match this by adjusting capacitance before testing.

The Operational Amplifier ("Op Amp")

Key points:
The output impedance of a device can be thought of as its ability to provide current to a load.
When connecting to a "load" or a "source", the output impedance of the source must be small compared to the input impedance of the load: Zout < (1/10)Zin

To avoid "pickup", measure voltage difference (to cancel out 60 Hz signal).  The differential mode gain of amplifier, Gdiff, is found by the following formula:

Vout = Gdiff*(V+ - V-)

Generally, a large Gdiff is desired, for zero common mode gain: if the same input voltage is fed through V+ (the "non-inverting input") and V- (the "inverting input"), Vout should be 0; thus, as usual, a large Zin and small Zout is desired.  Op amps meet this criterion fairly well.

Lab 8-1: Open Loop Test Circuit

The op amp open loop test circuit.

Differential mode gain, Gdiff, is also known as "open loop gain"--it can be observed in an open loop op amp circuit.  Here we used a potentiometer of resistance 0 - 10kΩ, and attempted to adjust it so that the voltage divider would yield an input voltage of 0 V for the op amp.  After several attempts, during which we saw Vout switch between +12V and -12V just as Vin changed sign, we considered this op amp's gain (200 V/mV) and realized we were unlikely to ever see a 0 V output.

The oscilloscope switches to -12 V just as Vin changes sign.

It was impossible to exactly set the potentiometer to cause V+ to exactly equal V-; the gain of 200 V/mV means that a voltage difference as small as 6 x 10^-5 V would be magnified by a maximum of 2 x 10^5 and thus appear as about 12 V.

Next lab: More with op amps!