Measuring an impedance by the oscilloscope. - dirac

Measuring an impedance by the
oscilloscope.
In order to measure an impedance Ẑ one in principle should know the complex
amplitude Û of the voltage across the impedance for a given complex amplitude
Î of the current through the impedance. However one is only able to measure
voltages by the oscilloscope. This problem can be circumvented since it is
possible two measure two voltages concurrently relative to a common zero
(ground). The unknown impedance is placed in series with a known auxiliary
resistor R (or more generally: a known impedance) as shown in the figure 1.
The voltage Û across the combination of the two components as well as the
voltage ˆV across the resistor only is measured on channel 1 and channel 2 of
the oscilloscope respectively. It is the same current Î that flows through the
unknown impedance and the resistor and so ˆV = RÎ and Û = (R + Ẑ)Î. If Â
is defined as the complex ratio between the voltages we have
 = ˆV
Û =
R
(R + Ẑ) (1)
The voltages are divided in the same ratios as the impedances and for that
reason the simple circuit is called a voltage divider. Solving the equation with
respect to Ẑ gives Ẑ = ( 1 − 1)R (2)
Â
Modulus of the voltage divider ratio, |Â| can be found from reading out the
real amplitudes on the oscilloscope. Likewise one can readout the phase shift,
φ A = φ V − φ U .
The oscilloscope has an 8 bit analog/digital converter, i.e. the relative
accuracy of a voltage measurement cannot exceed 0, 4% due to discretization
error. On the other hand the voltage can be amplified switching the range.

Figur 1 Voltage divider for impedance measurement.
One should of course choose a range for U(t) and V (t) respectively in order
the signals fill out the screen without being “cut”. It is not so important for
high accuracy that ˆV becomes small. However it will give large errors if ˆV
approaches Û since one can’t amplify the difference between the those signals
before the discretization. The method is thus going to be uncertain for |Z|

Figur 2 Alternative voltage divider for measuring an impedance.
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