Measuring Complex Impedance with Simple Circuit

Measuring Complex Impedance with Simple Circuit

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Hope this article will help you how to measuring Complex Impedance with Simple Circuit? This article shows a method that can be used to measure complex impedance from audio frequencies to very high frequencies (VHF – if tight construction pr...

Hope this article will help you how to measuring Complex Impedance with Simple Circuit? This article shows a method that can be used to measure complex impedance from audio frequencies to very high frequencies (VHF – if tight construction practices are followed and with appropriate diode detectors).  All that is required is a known resistance and three AC voltage measurements – for audio frequencies this could be a standard hand-held DVM.  The only disadvantage of this method is that only the phase magnitude and not the polarity can be measured.  In many cases the phase polarity can be inferred from other known information.
An alternative vector method is also shown that does not require a floating voltage measurement and can resolve phase polarity too.  This method is capable of working well into the UHF range with appropriate equipment.
Mathematical Development
The basic circuit is shown in Figure 1.  The resistor, R, is a known value and would often be around 50 ohms for RF measurements.  For audio frequency measurements it might have to be lowered or raised as needed depending on the magnitude of the impedance being measured.  Impedances in the range of about 0.1 to 10 times R can be accurately measured.
The unknown impedance is modeled as a series circuit consisting of an unknown resistance, Rx, and an unknown reactance, jXx.  The magnitude of the impedance is Zx.


measuring complex impedance with simple circuit 1.jpg


A Simple Circuit for Measuring Complex Impedance
The three voltages that are measured are:
1. VA which is the applied voltage
2. VI which is the voltage across the known resistor and related to the current
3. VZ which is the voltage across the unknown impedance
Although only the magnitudes of these voltages are known, these are actually vectors as shown in Figure 2 which applies to both methods.  The angle, θ, is unknown at the  moment but can be determined.  The vector voltages across the internal resistance and reactance of the impedance are shown for reference but there is no way to directly measure these.  The angle, Φ, is used in the alternative vector method discuss later.


measuring complex impedance with simple circuit 2.jpg


The law of cosines is used to calculate the cosine of the angle, θ.


measuring complex impedance with simple circuit 3.jpg

measuring complex impedance with simple circuit 4.jpg

measuring complex impedance with simple circuit 5.jpg


With real measurements there are two things that can go wrong in Equation 1 that need to be artificially corrected.  As a reality check the result of Eq. 1 should be between 0 and 1.00 but subtle measurement errors can skew that as follows.
1. Because of small measurement errors it is possible that cos(θ) will be negative – probably only by a small amount.  If that happens then use 0.00 as this means that the magnitude of the unknown impedance is purely reactive – in theory this
A Simple Circuit for Measuring Complex Impedance
should never exactly go to zero as the measurement R will cause a small angle shift from 90 degrees.  Repeat the measurement using a larger R.
2. It is possible that cos(θ) will numerically explode if VI is a very small value –  particularly zero.  This can happen if the impedance is very large in comparison to R.  In such a case the proper thing to do is to substitute 1.00 for the result but the accuracy of the subsequent calculations is going to be very poor.  The better solution is to use a larger R so that a definitely measureable voltage across it can be made.
The magnitude of the total impedance (including R) can be calculated as:
Za = R * VA / VI
We note from Figure 1 that the sum of R and Rx can be found by:
R + Rx = Za * cos(θ)
Thus, we can solve for Rx by
Rx = Za * cos(θ) – R
Considering possible measurement errors it is conceivable that Rx could compute to be negative which is not likely to be the real situation.  The proper thing to do if that happens is to take Rx to be zero.  The impedance is purely reactive.
The magnitude of the unknown impedance can be calculated as:
Zx = R * VZ / VI
The magnitude of the unknown reactance can be calculated as
Xx = sqrt(Zx2– Rx2)
Considering possible measurement errors it is conceivable that the square root of a negative number might occur.  If that happens then Xx should be taken to be zero.

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