A common source of confusion for hobbyists is determining the impedance of the windings of an audio transformer. It is often believed that the resistance measured with a DMM's ohmmeter function (the DC resistance) is the impedance of a transformer winding. This is not true. Let's make some measurements.
I'm going to be discussing some measurements on a transformer with various resistive loads connected. I will draw some conclusions that might not be always true if, for example, unusual loads were connected such as a negative impedance or a capacitor. I'm not considering such loads; only resistances.
I have a small audio transformer with two windings; it's rated for 100k ohms on one winding and 1k ohms on the other. I'll call the 100k ohm winding the primary. First of all, the DC resistance of the primary is 2012 ohms, and the DC resistance of the secondary is 71 ohms. These resistances are far from 100k and 1k ohms, so apparently the rated impedance is not the DC resistance.
Setting the impedance analyzer to sweep from 100 Hz to 5 MHz, and plotting impedance versus frequency, we obtain the plot shown in the first attached image when measuring the 100 k ohm rated winding. There are two curves shown; the top one is taken with the secondary open circuited and the bottom one with the secondary short circuited. Now with a little thought we see that the impedance measured at the primary as shown can't be less than the bottom curve no matter what resistance we connect to the secondary (we can't apply less than zero ohms). Likewise, the impedance at the primary can't be more than the top curve no matter what load is connected to the secondary (we can't apply more than infinite ohms). So we see that the impedance measured at the primary must fall between the two curves shown no matter what resistance we connect to the secondary.
The second image shows a family of impedance curves (I've limited the high end of the frequency sweep) with various resistances connected to the secondary. The bottom curve is taken with zero ohms applied to the secondary, and then successively increasing resistances are applied, namely 100, 300, 1000, 3000, 10000 ohms and finally an open circuit (infinite ohms?).
The rated impedance of the transformer primary is 100k ohms, but the measured impedance varies from about 800 ohms to over a megohm at 5 kHz. Why don't we measure an impedance of 100k ohms, the rated impedance of the primary winding? Because a transformer doesn't have a fixed, intrinsic impedance. The impedance measured at a winding depends strongly on what is connected to the other winding.
Notice, however, that the middle curve is just about 100k ohms over a fairly wide frequency range. That is the curve we got with 1k ohms connected to the secondary. The rated impedance of the secondary is 1k ohms. It looks like we get the rated impedance at the primary is we apply the rated load resistance to the secondary. Further notice that the middle curve is about halfway between the top curve (secondary open) and the bottom curve (secondary shorted) in the midband region around 5 kHz.
This suggests a method to determine the rated impedances of an unknown transformer. Plot two curves for one winding; one with the other winding open and one with the other winding shorted. Draw an additional curve half way between those two curves and that will be very close to the rated impedance of the winding.
To find the rated impedance of the other winding, apply various resistances and find the value that gives a swept impedance halfway between the open and short impedances of the first winding. Or, we could simply reverse the windings and plot two curves for the open and short condition of the primary while measuring the secondary. The rated impedance will be halfway between those two curves.
These curves are plotted with a logarithmic scaling of the vertical. A property of log plots is that at a particular frequency we will have a value for the open circuit impedance and another value for the short circuit impedance. A point on the plot halfway between those two values will have a value equal to the square root of the product of the open and short circuit values. We also know that that halfway point has a value essentially equal to the rated impedance of the winding. This gives us another way to determine the rated impedance.
A transformer with two windings can be treated as a two-port. A property of a two port known since the early days of Bell Labs, is that if you measure the impedances at one port with the other port successively open circuited and short circuited, and take the square root of the product of those two values, you will have a value known as the image impedance.
At a given frequency, measure the impedance at a winding with the other winding open and then short circuited. Multiply those values and take the square root; that is the image impedance and it is the rated impedance at that frequency. Because real transformers have parasitics (non-zero winding DC resistances, distributed capacitances), the image impedance is not constant with frequency. We see that the transformer I've used has only a finite band over which its behavior is useful. The impedance of the primary with a 1k ohm load should be 100k ohms; it satisfies this requirement only over a limited bandwidth, from a few hundred Hz to a few kHz.
Plotting the impedances over a band of frequencies allows us to not only see what the image impedance (halfway between the open and short circuit impedances) is, but to see over what frequency range it is nearly constant.
The third image is the same as the second, but with only the rated load applied to obtain the middle curve. That curve should follow the 100k ohm line at all frequencies if the transformer were perfect.
Edit: After all this, hobbyists aren't going to have an impedance analyzer, but we all know how much we want an LCR meter. Here's another use for an LCR meter. In the audio band, 1 kHz is nearly midband. If you want to get a fairly good estimate of the rated impedances of an unknown transformer, just determine the image impedance of each winding (square root of the product of the open and short circuit impedances) at the single frequency of 1 kHz, and consider those impedances to be the rated impedances. Also doing this at 10 kHz will give you an idea of the bandwidth of the transformer; the image impedances at 10 kHz ideally should be the same as at 1 kHz.
