Block D Transformers

Objectives

Observe and sketch the excitation current waveform of a transformer.
Observe and sketch the secondary voltage waveform of a transformer.
Use an RC integrator and an oscilloscope to demonstrate the hysteresis curve of a transformer.
Measure the regulation of a transformer.

Background

A single-phase transformer consists of two (or more) coils of copper wire wound on an iron framework, as shown in Fig. 1. The primary winding is connected to the AC source V_p, while the secondary supplies power to a load at a voltage V_s. The number of turns in the primary winding is N₁, and N₂ in the secondary. If N₂ is greater than N₁, then V_s is greater than V_p, and we call the transformer a step-up transformer. The turns ratio is defined as

..........(1)

If the transformer is ideal, then the voltages are directly proportional to the number of turns, and the currents are inversely proportional to the number of turns.

..........(2)

..........(3)

A circuit model for a real transformer is given in Fig. 2. R₁ and R₂ are the primary and secondary resistances, respectively. X₁ and X₂ are the primary and secondary reactances. R_m is the loss resistance which represents the transformer losses due to hysteresis and eddy currents. These losses are nonlinear, so R_m changes with voltage. Transformers are usually operated at a single rated voltage, so this is not a problem in analysis. X_m is the magnetizing reactance. It allows a magnetizing current to flow in the primary. We shall see in this experiment that the nonlinearity of the iron causes the primary current to be nonsinusoidal even when the primary voltage is sinusoidal.

An ideal transformer is embedded in the middle of the model of the real transformer. It is desirable to remove this transformer to make the circuit easier to solve. It is also desirable to move R_m and X_m to the input rather than toward the center of the model. This is an engineering approximation that is quite acceptable in most cases. The circuit that is actually solved is shown in Fig. 3. We introduce a new output voltage and current given by

V^'₂ = a V₂ ..........(4)

I^'₂ = I₂ /a..........(5)

This process is called referring all quantities to the primary side. We solve a new circuit using V^'₂ and I^'₂, and if we need the actual output voltage and current, we just multiply or divide by the factor "a" as shown in (4) and (5). Deciding where the "a" goes can be confusing to the student. Remember that the power dissipated in the secondary resistance is the same regardless of the model. We get the same value using I₂²R₂ as by using (I^'₂)²(a²R₂) = (I₂/a)²(a²R₂). Also remember that on the side of the transformer with the greater number of turns we have more voltage and less current.

This approximate circuit model is used to determine the voltage regulation, which is defined as the change in load voltage when the load is removed. The load voltage changes from V₂ to V_oc, the open circuit value.

..........(6)

The normalization process means that we get the same regulation by solving either Fig. 2. or Fig. 3. Figure 3 is easier because the open circuit voltage is just V₁, since there is no voltage drop in the series impedance if there is no current through it.

Regulation is a measure of the effect observed in a house when an electrical load is switched on, i.e. a dimming of the lights. If the voltage drops too far, other electrical loads can be affected. Computers are relatively sensitive to sharp dips in voltage, hence they demand a small value of voltage regulation.

One of the activities in this experiment is the measurement of B and H, or at least quantities proportional to B and H in the transformer. B is the magnetic flux density with units of Tesla, and H is the magnetic field intensity with units of (Amp turns)/m. The basic circuit we use is shown in Fig. 4. It can be shown that H is directly proportional to the current i_m flowing in the primary. The scope only measures voltage, but we can easily convert from current to voltage by putting a one Ohm resistor in series with the current and reading the voltage across this resistance. We would need to know the number of turns in the primary and the effective length of the flux path in the transformer steel before we could determine the actual value of H. Neither of these quantities are available unless we contact the manufacturer, so we will just measure a quantity proportional to H in this experiment.

Measurement of B is somewhat more challenging. We know by Faraday's Law that the time-varying voltage across a coil of N turns is v = N(d /dt) where is the total magnetic flux in Webers. We also know that the flux density is the flux divided by the cross-sectional area, B = /A. But to get we have to integrate the voltage v, as shown in Fig. 4., to produce another voltage to measure on the scope. This integrated voltage will be in phase with the current, so we can observe the proper phase relationship between B and H.

There are several ways of accomplishing an integration of this type. One is to use an operational amplifier with a capacitor in the feedback path. This is a good way to integrate voltages. There is another method, however, which is satisfactory in cases like this, where a sinusoid needs to be integrated and the magnitude is not important since we do not know the number of turns or the cross-sectional area anyhow. This is the resistor and capacitor combination shown in Fig. 5. Analysis of this circuit will be left for the homework.

Discussion and Calculations

Compute V_out in Fig. 5. using ordinary phasor analysis (don't integrate) if the input voltage in phasor form is 6 < 0^o volts and f = 60 Hz. Compare the phase difference between input and output for this circuit and for the case where a sine wave is passed through a perfect integrator. (The amplitude of the outputs will be different between the two circuits, but we are only interested in phase.)
Compute the voltage regulation of the transformer in Fig. 3 for R₁ + a² R₂ = 0.5 , X₁ + a² X₂ = 1.5 , and a 25 resistor connected to the terminal marked V^'₂. Take a = 1, V₁ = 120 volts. Work the problem using phasor analysis, but use only the amplitudes in calculating the voltage regulation.

Instructional Activity in Class

First we want to plot B and H versus time. Assemble the circuit in Fig. 6., using one of the GE transformers. Observe the dot markings on the transformer to insure the proper phase relationships. Use one of the large block mounted one Ohm resistors with the terminal connected to the transformer also connected to CH1. Switch the variable autotransformer to 120 V and turn it all the way up. Sketch the waveforms of the excitation current and the flux on the same set of axes, making sure to note the peak amplitude of the excitation current. The excitation current is not sinusoidal, so be careful to note the exact shape.

Next we want to plot B versus H, with time as a parameter (the traditional BH curve). We use the same circuit as before. Select the XY Trigger mode (not in the Trigger menu). Note that CH1 is the X (horizontal axis) and CH2 is the Y (vertical axis). Adjust the CH1 and CH2 Volts/div scales so that the complete waveform is displayed on the screen.

Sketch the BH curve for several settings of the variable autotransformer, say 25%, 50%, 75%, and 100%, in your notebook. You can put all four curves on the same set of axes so the relative sizes can be easily compared.

This particular transformer is rated at a higher voltage than it was designed for. Properly rated transformers have a hysteresis curve that extends past the knee of the curve only slightly. The peak value of the excitation current should be less than 10 percent of the peak rated current for good design. What is the rms voltage across the nominal 120 volt winding for which the excitation current is 10 per cent of the peak rated current of 5 A?

Now we want to measure the voltage regulation of this transformer. Assemble the circuit in Fig. 7. Use the digital multimeter to measure input and output voltage of the transformer by placing the leads first across the input terminals and then across the output terminals. Use the variable autotransformer to increase the load from near zero to about 5A. The load is located next to the variable autotransformer in the lower right hand panel of the bench. Do not use one of the variable resistors on the upper panels because you will exceed the current rating. Record input and output voltage for 0, 1, 2, 3, 4, and 5 A currents and compute the regulation. Use the input voltage under each load condition to represent the no load (open circuit) voltage.
Look ahead to the questions in the Conclusion to make sure everyone in your group knows the answers before you leave the lab. If not, ask the instructor.

Conclusion

In Lab Activity 2 you found the variation of flux (proportional to B) and current (proportional to H) with respect to each other. In the hysteresis curve time is a parameter which causes the curve to be traced in a counterclockwise direction as time increases. The two curves of Lab Activity 1 with time as the independent variable contain the same information as the single curve of Lab Activity 2 with time as a parameter.

Sketch the two sets of curves, label some common points, and discuss how one can map the hysteresis curve into the curves for flux and current as a function of time.

When we use the input voltage as the open circuit voltage in calculating the voltage regulation (Lab Activity 3), which transformer model (complete or approximate) are we assuming? Why?

Can the lab power supply be considered an ideal voltage source? Discuss why the transformer input voltage decreases with increasing load.
The capacitor in the integrator is rated at 400 V (peak) and the resistor is rated at 1 W. The maximum voltage available on the bench is 208 V rms.

Is there any way the integrator can be connected wrongly and overheat the resistor or breakdown the capacitor? Would an operational amplifier integrator circuit built of integrated circuit chips rated at ± 15 V have the same ruggedness? (This sort of ruggedness is sometimes called bullet proof. It is highly desirable whenever it can be done at acceptable cost.)