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# Measuring the Parameters of a Transformer Equivalent Circuit Diagram

Used in many different applications, the transformer is one of the most important components in alternating current technology. It is used in electrical energy technology to transform between different voltage levels. To ensure efficient transmission of energy in this case, good efficiency and optimum utilization are required.

Despite the wide prevalence of power electronic circuits, the transformer is also still required for small power supplies to allow for the required galvanic isolation. It is used in measurement technology to convert measured quantities. Transformers must meet different requirements depending on the intended use. Adaptations to these requirements can be made through the selection of the core material that is used and by varying the geometry of the core. The individual properties of a transformer can be represented by a simple equivalent circuit diagram. This can be used to evaluate how suitable a transformer is for a proposed application and its behavior at various load points. In this article the equivalent circuit diagram of the transformer is first derived and explained. Next measurements and calculation methods for determining the equivalent circuit diagram and the loss of iron in the transformer core are presented. The measurements and calculations are performed with the HBM Gen3i data recorder. The appendix contains all the necessary formulas and they can be imported into Perception.

## 1. Equivalent circuit diagram of the transformer

Figure 1 shows the operating principle of a transformer with two windings that are magnetically connected with a ferrite core. Due to the high permeability of the ferrite core in comparison to the air, the flux is directed through it. Nonetheless, slight leakage fluxes and do occur. Resistances and simulate the ohmic part of the windings. To describe the operating behavior of the transformer, an equivalent circuit diagram is derived from this model as shown in Figure 2. This diagram shows the transmission ratio between the primary and secondary side for an ideal transformer. The other effects that occur are represented by passive components. The magnetic fluxes are described by the leakage inductances and, and also by the main inductance. The resistor is connected in parallel to the main inductance and serves to simulate the iron losses in the core material. These consist of eddy current losses and hysteresis losses.

Eddy current losses arise due to a current flow in the ferrite core which is caused by induced voltages. In accordance with Lenz's law, this current opposes the change that caused it. To minimize the current flow, the ferrite core is made up of plates that are isolated from each other. The hysteresis losses are caused by the periodic remagnetization of the ferrite core, since energy is required to align the molecular magnets in the iron (Weiss domains). Since both the main inductance L_{µ} and the iron loss resistance R_{Fe} are dependent on the core material with non-linear permeability µ_{Fe}, both follow a non-linear course.

R_{K }= R_{1 }+ R_{2} ü² (1)

L_{K }= L_{1σ }+ L_{2σ} ü² (2)

The leakage inductances may be considered as linear, since their field lines run primarily through the air, which exhibits constant permeability. For further considerations the equivalent circuit diagram from Figure 2 is simplified still further (Figure 3). The voltage drop on R_{1} and L_{1σ} is negligibly small in comparison to the voltage drop due to the iron loss resistance R_{Fe} and the main inductance L_{µ} in normal operation. This makes it possible to contact the iron loss resistance R_{Fe} and main inductance L_{µ} directly with the input terminals. [1] In equations (1) and (2) the ohmic resistance R_{2} and the leakage inductance L_{2σ} of the secondary side are converted to the primary side and combined to form R_{K} and L_{K}. The measurements and calculations performed below refer to the equivalent circuit diagram simplified in this manner. Quantities I'_{2}, U'_{2} and Z'_{ load} have been converted from the secondary side to the primary side taking into consideration the transmission ratio.

## 2. Measurements in no-load

The values of the iron loss resistance R_{Fe} and main inductance L_{µ} can be determined by a no-load test as shown in Figure 4. Since these values exhibit non-linear behavior, the unloaded transformer is supplied with a variable transformer as a sinusoidal voltage source with variable amplitude. This makes it possible to approach and measure different load points with differently linked magnetic flux Ψ. The magnetic flux is calculated from the applied voltage as follows:

Ψ = ∫▒ u ̂⋅ sin(2πft)dt (3)

Ψ = -u ̂/2πf ⋅ cos(2πft) (4)

The metrologically acquired quantities are the primary voltage u_{1} (t), primary current i_{1} (t) and secondary voltage u_{2} (t). To determine the iron loss resistance R_{Fe} and main inductance L_{µ}, first the root mean square value of the primary voltage U_{1}, the primary-side active power P_{1} and the reactive power Q_{1} are determined. The calculations are performed on a cyclical basis. The component values and transmission ratio ü can be calculated with formulas (5) (6) and (7).

R_{Fe }= U_{1}^{2}/P_{1 }(5)

L_{µ }= 1/(2 π f) ⋅ (U_{1}²/Q_{1}) (6)

ü = U_{1}/U_{2 }(7)

As can be seen in Figure 5, the component values are not constant due to dependence on the magnetic flux. The calculated component values are an average over a sine wave.

The measured values are examined over the course of time to allow for further examination. The distortion of the current over time (red curve) is clearly seen in Figure 6. The core material runs into saturation. The correlation between the flux density B and the magnetic field strength H is illustrated most vividly by the hysteresis curve. If the core geometry is known, the flux density and field strength can be determined from the measured quantities with:

B = Ψ/A_{Fe} (8)

H = I/l_{Fe} (9)

Due to the unknown core geometry of the test specimen measured here, the hysteresis curve is realized in Figure 7 as a ΨI characteristic curve. The new curve and a number of load points are also shown in Figure 8. The new curve is determined by approaching different load points in which the linked flux and current are acquired in the voltage zero crossing. It is produced when a field strength is first applied to an unmagnetized core and is the characteristic curve of the main inductance L_{µ}. The flux density increases slowly at first. As the field strength increases, the flux density increases faster and faster until the core goes into saturation and the flux density hardly rises at all. Now if the field strength is reduced, the flux density does not return on the new curve. Instead it follows the hysteresis curve. When the field strength is equal to zero a residual magnetism remains, referred to as remanence. The field strength required to eliminate the residual magnetism is called the coercive field strength. [2]

Another method for determining the expected iron losses is the Steinmetz formula (10).

P_{Fe }= k ⋅ f^{a }⋅ Ψ^{b } (10)

The Steinmetz formula is based on the fact that the surface enclosed by the hysteresis curve is equal to the iron losses. The precondition for applying the Steinmetz formula is a sinusoidal input voltage. The iron losses for differently linked flux calculated from the measured values can be used to determine the unknown coefficients a and b from formula (10) by curve fitting (Figure 9). The curve produced in this manner can then be used to estimate iron losses for other load points in advance.

## 3. Measurements in the short circuit

In the short-circuit test the secondary side is short-circuited by a low-ohm impedance Z_{load} (Figure 11). The current is set to the nominal (rated) current by a variable transformer.

The current through the main inductance and iron loss resistance is negligible in this operating state. The quantities acquired metrologically are the primary voltage u_{1} (t), primary current i_{1} (t), secondary current i_{2} (t) and the voltage u_{2} (t) over the load. First the voltage drop is calculated with R_{K} and L_{K}.

u_{K} (t) = u_{1} (t) - ((u_{2} (t))/ü) (11)

Then u_{K} (t) and i'_{2} (t) can be used to calculate the power transformed on R_{K} and L_{K} and the component values can be calculated.

R_{K }= P_{K }/ I'_{22} (12)

L_{K }= 1/2π f ⋅ (Q_{K }/ I'_{2}²) (13)

## 4. Sources

[1] J. Teigelkötter, Energieeffiziente elektrische Antriebe, Springer Vieweg Verlag, 2013.

[2] M. S. Hering, Physik für Ingenieure (9.Auflage), Berlin, Heidelberg, New York: Springer, 2004.