# Coupling Transformers in a Bus Line

I'm trying to figure out how to model a transformer in order to estimate the number of transformers on a bus line. Each transformer is connected in parallel and may or may not have a connected load. I've been able to collect some empirical data from an available device, but I'm having trouble formulating an equation to satisfy that data.

I would assume that, ignoring hysteresis and eddy currents for simplification, calculating a bus like this would be a simple matter of combining the primary-side resistance/reactance with any referred secondary-side resistance/reactance and then adding these resistances in parallel with all other transformers connected in the bus. But when I perform actual measurements, my calculated numbers are off. When testing a circuit with (transformers are a simple 1:1 ratio), the measurement is closer to 75 ohms rather than approximately 71-72 ohms (adding 13.52k + 5k + 3k + 75 in parallel). The measurement will also continue to stay relatively close to 75 ohms rather than dropping to the low 70s/high 60s that I would expect when introducing lower load values.

What exactly am I missing here?

Just examining the 75 ohm and transformer connected to the 3k load, the combined impedance (assuming perfect transformers) is: -

75 ohms // (3k + 53.6 + 53.6) = 73.23 ohms

Clearly as you add more parallel transformers this number is going to drop so how do you account for the actual reading being a little higher: -

• The transformer dc winding resistances will add a little
• The tranformer leakage inductances will add a little impedance

Given what you have shown there can be no other reason other than your measurement accuracy is poor.

• This makes sense. What I'm really wondering about though, is if there is a simple way to ballpark a measurement of x number of transformers in parallel. As a another example, I sequentially kept adding 1 more transformer in parallel to the next (exactly like the above setup, but without the 75 ohm terminating resistor) and got the following input impedances: 1 transformer - 6.57kOhm, 2 transformers - 2.72kOhm, 3 transformers - 1.6kOhm, 4 transformers - 1.12kOhm. Assuming all inductances are 27.6mH, I'm wondering how these values are formulated (since these relationships aren't linear).
– Alex
Oct 4, 2013 at 19:03
• Try doing the same trick with adding transformers but in reverse order. It's either inconsistencies in the transformer manufacture or the easy-to-say-yet-hard-to-explain something else. Oct 4, 2013 at 19:16
• So let's say that I'm looking at just one transformer with no load connected at all. Would I be correct in saying the measured impedance should just be (ideally) Z = angular frequency * magnetic inductance?
– Alex
Oct 4, 2013 at 19:54
• @Alex correct . It's just the inductance of the side that is connected. Oct 4, 2013 at 20:06
• Is there a book or resource that shows how to calculate the combined impedance through the transformer(s)? For example, in your calculation why is the 3K ohm resistor in series with the two 53.6 ohm resistors when they're across a transformer? Does the number of turns matter? Sorry for the simple questions, it's been a while since I've analyzed circuits and am rusty. Sep 23, 2019 at 23:12

In my opinion, you missed the leakage inductance of the transformers, that may have significant impedance on this frequency.

This way, you have to measure the phase between the current and voltage in order to determine the proper impedance and to separate the active from the inductive parts.

BTW, for electronics, the difference between 72ohms and 75ohms (only 4%) is not so big and should not affect the work of the circuit. So, the computation accuracy is acceptable in this example.

• How does the leakage inductance affect the input impedance? Is it an effect on both the primary and secondary sides of the transformer?
– Alex
Oct 4, 2013 at 19:06
• I am not very sure, but how you measure the impedance? Oct 4, 2013 at 19:12
• The actual impedance I measured with a device that has that capability.
– Alex
Oct 4, 2013 at 19:18
• Yes, very informative. :P Oct 4, 2013 at 19:20
• Leakage inductance will affect both sides, yes.
– user16324
Oct 5, 2013 at 12:05