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.

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)

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)

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.

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)

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).

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.