# Parallel of transformers. Is R/X condition necessary?

Let say I've got two transformer T1 and T2 with both the same voltage ratio. Each transformer has an internal impedance Z1=R1+jX1 and Z2=R2+jX2 respectivley. Also each transformer has a power capacity S1 and S2 [kVA]. Then I connect both them in parallel (parallel windings to the same supply line, and secondaries to the same load) simulate this circuit – Schematic created using CircuitLab

1. What conditions should Z1 and Z2 meet in order to achive an evenly distribution of load betwen the two transformers?

I'm pretty much sure that the only condition that should be met is that: $$|Z_1|\cdot|I_1| = |Z_2|\cdot|I_2|$$ where I1, and I2 are the nominal currents of each transformer.

Also, I have read about the condition: $$\frac{R_1}{X_1}=\frac{R_2}{X_2}$$ I've been thinking about this relation and I think it's only necessary to achive the same current phase in both transformers, but it is not necesary to accomplish an evenly distribution of load betwen the two transformers.

1. Can you provide any evidence that the last relation is useful in any way other than the one alredy explained (3.b. is the one explained useful)?
2. Is it really necessary to met the last reltion to connect two transformers in parallel?

...this issue is treated in most textbooks. Though, I found that some sources state that the ratio R/X is a necessary condition, and I don't think so.

• An even distribution of load is acheived by distributing the secondary loads evenly and has very little to do with the transformers - planning how you attach loads is what you need to do. – Andy aka Oct 26 '19 at 10:09
• @Andyaka: Does the question make sense for loads that can be distributed? Can the transformers be said to be connected in parallel if the secondaries are not connected together? – Charles Cowie Oct 26 '19 at 12:36
• The question doesn’t make sense without further detail @CharlesCowie. – Andy aka Oct 26 '19 at 14:26
• Parallel transformers is mentioned in the title and in the last line of the question. Connecting both the primary windings and the secondary windings of two transformers together for parallel operation is generally not advisable. However there are reasons for knowing something about doing that. If that is what the question is about, the question should be revised to make it more explicit. – Charles Cowie Oct 26 '19 at 14:43
• Thank you. I edited the question in order to make it a bit more explicit. – cuba.original Oct 26 '19 at 15:02

## 1 Answer

Looking at the equivalent circuit as being constructed from ideal components, it is not necessary for X/R to be equal. In addition/ X/R can only be equal if R1 = R2 and X1 = X2. Considering factors such as temperature coefficient of resistance, the transformers will not share the current equally unless they are identical in every way.

Consider that the drop across the parallel combination of R1 + jX1 and R2 + jX2 is V1. The current I1 = V1/Z1@a and I2 = V1/Z2@b where a and b are phase angles determine by X/R. If Z1 an Z2 have equal values but different phase angles, the currents I1 and I2 will also have equal values but different phase angles.