Hello, Transformer Fans:

In this blog I discuss how to use Ansys Maxwell to compute self-inductance, mutual-inductance, and leakage- inductances in a three phase transformer. There are multiple methods and expressions to use to compute these inductances.

FUNDAMENTAL INDUCTANCE DEFINITIONS

Inductance has units of the Henry, H, which in base units is expressed as Wb/A, where Wb is the unit for magnetic flux, and A is the unit for amps. So the units show us that inductance is a measure of how much magnetic flux can be produced in a material per unit of amps applied to a winding. Ferromagnetic materials applied in electromagnetic machines have relative permeability greater than one, and the higher the permeability is, the higher is the inductance, the more flux can be produced in a material per unit amp.

By definition leakage-inductance, mutual inductance, and coupling coefficient are terms associated with pairs of windings. Tests can be done to determine the inductances of windings by considering two windings at a time.

Self inductance of a winding: all of the flux produced by a winding divided by the current producing the flux in the winding with all other windings Open Circuited. Also, self-inductance can be computed as an apparent inductance (operating flux linkage divided by winding operating current), or can be computed as a differential inductance (differential flux linkage divided by differential winding operating current). In a linear material the apparent inductance is equal to the differential inductance.

Leakage inductance of a winding: the flux produced by an excited winding that does not link another winding (Short Circuited) divided by the current producing the flux in the excited winding.

Mutual inductance of a winding: the partial flux produced by an excited winding that does link with another winding (Open Circuited) divided by the current producing the flux in the excited winding.

For an N winding system there will be N self-inductances, and the number of possible mutual-inductances and the number of possible leakage-inductances are equal to

There are six windings in this three phase transformer example and considering two windings at a time permutation shows there are 30 possible mutual inductances, 30 possible leakage-inductances, and there are six self inductances. The NxN matrix below shows the self-inductances and the possible mutual-inductances in this six winding example. However, there are practically 18 inductances that are relevant in a three-phase transformer circuit model. Each winding is represented by one self-inductance, one leakage-inductance, and one mutual-inductance, and there are six windings (3*6 = 18). The self-inductances are highlighted in yellow and are along the diagonal, while the six relevant mutual inductances are highlighted in green. Also, there only six practical leakage inductances computed from the relevant self-inductance and mutual-inductance in this three-phase transformer example.

(See the Calculations section below for more information).

MAXWELL 3D FEA MODEL

EXCITATION

In this example the low voltage phase A winding (LVA) will be excited and the high voltage phase A winding will be open circuited to determine the self inductance of winding LVA and the mutual inductance of winding HVA and LVA due to the current in winding LVA. All windings have the same number of turns and the value of the excitation current does not matter for a linear magnetic material, and 20mA is used arbitrarily. The Eddy Current solver (AC, frequency domain) is used in this example. Similar methods and expressions can be used in the Magnetostatic and Magnetic Transient solver.

Open Circuit Test: Self inductance of the LVA winding is determined by applying excitation to the LVA winding, and having all other windings be Open Circuited. Mutual inductances are also determined with this test.

Short Circuit Test: The leakage inductance of winding LVA w/r to winding HVA is determined by applying excitation to winding LVA and having winding HVA be Short Circuited.

Similar tests can be done to determine the inductances of other windings by considering two windings at a time.

OPEN CIRCUIT TEST: Determining Self-Inductance and Mutual-Inductance of Winding LVA

SHORT CIRCUIT TEST: Determining Leakage-Inductance w/r to Winding LVA and HVA

CALCULATIONS

The inductance values for a winding are computed using the following expressions.

RESULTS

The results show that the inductances computed using the various expressions described above match well with each other, almost perfectly.

OPEN CIRCUIT TEST: Determining Self-Inductance and Mutual-Inductance of Winding LVA

SHORT CIRCUIT TEST: Determining Leakage-Inductance w/r to Winding LVA and HVA

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February 16, 2024