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If you search the internet for “transformer equivalent circuits,” you’ll get five pages of over 100 small circuit diagrams. *Figures 1 and 2* show two typical circuits, one that’s “referred to the primary” and the other that’s “referred to the secondary.” The *Figure 2* circuit also has the mutual impedance in the primary. Most other circuits are variations of these.

*1. This is a typical transformer equivalent circuit. (Source: Wikipedia)*

*2. This is anothed transformer equivalent circuit. ^{1} (Source: Magnetic Circuits and Transformers)*

While all of these equivalent circuits are useful for many applications, they all have major problems, particularly when modeling inductive voltage dividers (IVDs) or transformer-ratio-arm (TRA) bridges.

A more useful circuit is shown in *Figure 3*. The source of this circuit is unknown to the author—he used it used it in a chapter of a 1971 book^{2} and a 1972 paper^{3} (he would like to know the original reference). It doesn’t include capacitance (three capacitances could be added between terminals) and there’s no isolation (an ideal 1:1 transformer could be added).

*3. The main benefit of this circuit is its use in modeling a voltage divider.*

This circuit may look a little strange. It’s obvious what most of the components are, but there are two strange components with factors (N_{1}/N_{2 }± 1)_{ }and (N_{2}/N_{1} ± 1). Here, the upper sign indicates the non-inverting connection; the lower is the inverting.

The calculated open-circuit input impedance using this circuit is z_{1} + N_{1}^{2}z_{12}, as one would expect. As is the open-circuit transfer voltage, Z_{12}/(z_{1 }+ Z_{12}N_{1}/N_{2)}, which is closely equal to N_{2}/N_{1}. These and other calculations using this circuit also give the expected results indicating the circuit is valid.

A main benefit of this circuit is its use in modeling a voltage divider. If one uses the inverting connection (lower signs), rotates the circuit CW, and applies a voltage, the calculated voltage ratio across the two windings, E_{1}/E_{2}, is:

This is very close to N_{1}/N_{2} if z_{1} and z_{2} are small compared to Z_{12}, and that ratio can be very small. The winding resistances, r1 and r2, would closely cancel if the same-sized wire was used for both windings; the leakage inductances, l1 and l2, would also tend to cancel. Moreover, if the windings are twisted together, the leakage inductances can be extremely small. The mutual impedance, Z_{12}, can be large.

Toroidal cores with a permeability of up to about 100 are usually used. As a result, ratios accurate to 1 ppm or better are possible. To get a 10:1 ratio, 11 wires can be twisted together in a cable and connected in series. If 22 wires are used, one can get ratios of 1:1, 10:1 and 1:10.

**TRA Bridge**

A main application of precision transformers is as ratio arms in low-frequency comparison (TRA) bridges. Such ratio arms are not only accurate, their accuracy is permanent except for catastrophic damage.

TRA bridges have been in use for many years,^{4,5,6} but they became particularly important some years ago when Thompson-Lampard-type calculable air capacitors were used in the determination of the standard of impedance. These capacitors had a capacitance of about 1 pF or less and thus had to be scaled up in value to be of practical use. A group of experts was assembled at the National Bureau of Standards (NBS), which is now the National Institute of Standards and Technology (NIST), to design a bridge for that task.^{7} Their method was adapted for a popular commercial capacitance bridge.^{8}

Another important advantage of TRA bridges is their immunity from loading capacitance while yet presenting a high input impedance to the source voltage, Z_{12}(N_{1} + N_{2})^{2}/N_{1}N_{2}, which reduces to 4Z_{12} if N_{1} equals N_{2.}. However, if one winding is shunted, the error depends only on the winding resistance and leakage inductance. The shunt impedance is usually capacitive, the capacitance of shielded cables or the capacitances to guard when measuring three-terminal capacitors. With a capacitance C shunting winding number one, the calculated ratio is approximately:

And if N_{1} = N_{2},_{ }this simplifies to 1/[1 +jwC(z_{1} + z_{2})/2]. If resistive ratio arms were used, they would have to be of very low resistance to get this immunity from loading, and, if so, would be subject to connection errors and would require a powerful input source.

This equivalent circuit can be useful for calculations on all types of two-winding transformers, voltage and current transformers, power transformers—even RF transformers. Calculations on them would be difficult using the other equivalent circuits. Calculations on transformer voltage dividers and bridges would be close to impossible and *just plain wrong* using the other equivalent circuits.

*Henry P. Hall, a Life Fellow of the IEEE, worked 40 years at General Radio Co./GenRad Corp. designing impedance bridges, meters, and standards. He holds a BA from Williams College and a BS and MSEE from MIT.*

**References**

1. M.I.T. Staff “*Magnetic Circuits and Transformers,*” p. 338, M.I.T. Press, 1943.

2. H.P. Hall, “Impedance Measurement” chapter in *“Electronic Measurements and Instrumentation,” *pp. 264-318, McGraw-Hill, 1971.

3. H. P. Hall, “Achieving Fractional-ppm Ratio Precision with Two-Stage Transformers,*”* *Electronic Instrumentation*, pp. 30-34, Feb. 1972.

4. A. Elais, “Ueber Widerstandsmessungen mit dem Differentialinductor,” *Ann*. *dem Phys*, Vol. 335, pp. 828-833 (1888).

5. A.D. Blumlein, “Alternating Bridge Circuits,” British Patent No. 323,027, 1938.

6. C.W. Oatley and J.G. Yates, “Bridges with Coupled Inductive Arms as Precision Instruments for the Comparison of Laboratory Standards of Resistance and Capacitance,” *Proc. IEE,* March 1954, Vol. 110, No. 2, p. 453.

7. M. C. McGregor, J. F. Hersh, R. D. Cutkosky, F. K. Harris, and F. R. Kotter*, *“New Apparatus at the National Bureau of Standards for Absolute Capacitance Measurements,” *IRE Trans. on I&M*, pp. 253-261, Dec. 1958.

8. J.H. Hersh*, *“Accuracy, Precision, and Convenience for Capacitance* *Measurements,” *General Radio* *Experimenter*, Aug.-Sept. 1962.