EGR 325, ELECTROMECHANICS

TECHNICAL PAPER #2

by

David W. Johnson

THE TRANSFORMER EQUIVALENT CIRCUIT

from

EXPERIMENT #9

APRIL 17, 2000

**0.0 INTRODUCTION**

In electrical engineering, it is often useful to use an equivalent circuit model to describe the non-ideal operation of a device such as a transformer. While an ideal model may be well suited for rough approximations, the non-ideal parameters are needed for careful transformer circuit designs. Knowing the non-ideal parameters allows the engineer to optimize a design using equations rather than inefficiently spending time testing physical implementations in the lab.

If all dimensions and material properties of a transformer are known, the non-ideal parameters can be directly calculated. However, this is usually not the case, and a simple technique for obtaining the parameters can be used. A method for determining the parameters of the equivalent circuit model using two simple tests is described. Expressions for calculating the parameters are derived in terms of laboratory measurements. The procedure is performed in the lab for a transformer. As an example of the usefulness of the non-ideal equivalent circuit, the parameters found in the lab are used to calculate one important transformer characteristic, maximum efficiency.

**1.0 ANALYSIS**

**1.1 MODEL**

The equivalent circuit model for the non-ideal transformer is shown
in Figure 1. An ideal transformer with resistors and inductors in parallel
and series replaces the non-ideal transformer. This model is called the
high side equivalent circuit model because all parameters have been moved
to the primary side of the ideal transformer. The series resistance, R_{eq},
is the resistance of the copper winding. The series inductance, X_{eq},
accounts for the flux leakage. That is, a small amount of flux travels
through the air outside the magnetic core path. The parallel resistance,
R_{m}, represents the core loss of the magnetic core material due
to hysteresis. The parallel inductance, X_{m}, called the magnetizing
inductance, accounts for the finite permeability of the magnetic core.

**Figure 1. **High side transformer equivalent circuit model.

It is easy to see how each parameter of the equivalent circuit model could be adjusted by changing the transformer design. For example, increasing the diameter of the wire in the windings decreases the series resistance. Therefore, the equivalent circuit model parameters can be used as a way to evaluate a transformer, or compare transformers.

The parameters can be found in the same way that Thevenin equivalent
circuit parameters are found: open circuit and short circuit tests. The
parallel parameter values are found with no load connected to the secondary
(open circuit) and the series parameter values are found with the secondary
terminals shorted (short circuit). It is possible, for convenience in the
lab, to make the tests on either the primary or the secondary. Figure 2
shows the equivalents circuits for the two tests. For the open circuit
test, the series parameters are neglected for convenience. This is reasonable
since the voltage drops are across R_{eq} and X_{eq} are
normally small.

**Figure 2. **Equivalent circuits for tests. (a) Open circuit. (b)
Short circuit.

Expressions for the non-ideal transformer parameters are derived from the equivalent circuits shown in Figure 2. The results are Equations (1), (2), (3), and (4). All parameters are expressed in terms of quantities measured in the open circuit and short circuit tests.

(1)

(2)

(3)

(4)

**1.2 SAMPLE CALCULATIONS**

For open circuit measurements of V_{oc}=114.81 VAC, i_{oc}=0.24
A, and P_{oc}=6.4 W, the parallel parameters of the transformer
are calculated in Equations (5) and (6).

(5)

(6)

For short circuit measurements of V_{sc}=11.14 VAC, i_{sc}=3.88
A, P_{sc}=6.1 W, the series parameters of the transformer are calculated
in Equations (7) and (8).

(7)

(8)

**2.0 EXPERIMENT**

**2.1 Description of THE EXPERIMENT AND SETUP**

A 1:1 transformer was tested in the lab to determine its non-ideal parameter
values. Figure 3 shows the wiring diagram used to make the open circuit
test. With the secondary open, the primary voltage was increased from zero
to rated voltage, where the rated voltage is the name plate stamp. A digital
multimeter was used as an ammeter to measure the open circuit current.
A wattmeter was used to measure the open circuit power. The power measured
was the power dissipated in R_{m}, the core losses.

**Figure 3. **Wiring diagram for open circuit test.

The short circuit wiring diagram is shown in Figure 4. With the secondary terminals shorted, the primary voltage was increased from zero until the rated current was reached in the primary. At this point the primary voltage was measured. It was much less than rated voltage. Again, the power and current were measured.

**Figure 4. **Wiring diagram for short circuit test.

**2.2 PRESENTATION OF MEASURED DATA**

Using the parameters of the non-ideal transformer equivalent circuit
model, the peak efficiency of the transformer can be calculated. For the
transformer tested in the lab, the results are shown in are Equations (5),
(6), (7), and (8). The values for R_{eq} and R_{m} can
be used to find the minimum current, I_{F}, and the maximum current,
I_{M}.

(9)

(10)

The maximum efficiency is calculated in Equation (11).

(11)

**3.0 RESULTS AND COMPARISON**

The experimental results obtained from the open circuit and short circuit tests were not evaluated. It would be possible to test the maximum efficiency of the transformer by setting the load so that the transformer is operating at maximum efficiency. The actual efficiency of the transformer could be found by dividing the power out by the power in. This value should be close to the value found in Equation (11).

**4.0 CONCLUSIONS**

The procedure used to find the parameter values of the non-ideal transformer
equivalent circuit model allows the engineer to more efficiently design
transformer circuits. Modeling and simulation are more accurate when the
non-ideal parameters are used. This means that designs can be optimized
prior to implementation.