Transformer Impedance

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Positive Sequence Impedance

The transformer positive sequence impedance is also called the short circuit impedance (because it is measured from a short circuit test) or the impedance voltage (because a voltage is measured from the short circuit test - more details below).

Two Winding Transformer

Impedance Measurement

Positive sequence impedance test circuit for three-phase 2-winding transformer

The positive sequence impedance is measured from a standard short circuit test (see figure right). In this test, one set of windings is shorted (typically the LV windings) and a three-phase voltage source is applied to the other set of windings. The voltage is steadily increased until the rated phase current is measured.

The voltage at this point is called the impedance voltage. When the impedance voltage is expressed as a per-unit value (on the transformer rated line-to-line voltage), the impedance voltage is equivalent to the per-unit impedance, i.e.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{V_{z}}{V_{n}} = \frac{I_{n}}{I_{n}} \frac{Z_{1}}{Z_{b}} = Z_{1,pu} \, }

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{z} \,} is the measured impedance voltage (V)

is the rated transformer line-to-line voltage / base voltage (V)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_{n} \,} is the rated transformer current (A)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_{1} \,} is the positive sequence impedance (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega} )
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_{b} \,} is the transformer base impedance (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega} )
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_{1,pu} \,} is the positive sequence impedance (pu)

The impedance voltage is also often expressed as a percentage (Z%).

Copper Losses

Full load copper (or iron) losses are measured in the same test as the impedance voltage by taking the power reading (from the wattmeter in the test circuit). The copper losses represent the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I^{2}R} losses from the full load current in the phase windings, as well as stray losses due to induced eddy currents from leakage fluxes in the windings, core clamps, tank and other magnetic pathways.

Calculation of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_{1}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_{1}}

The resistive and reactive components of the positive sequence transformer impedance can be estimated from the two short circuit test measurements - 1) impedance voltage, and 2) full load copper losses. The expressions below calculate the resistance and reactance in per-unit quanitities.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_{1,pu} = \frac{P_{c}}{1000 S_{n}} \, }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_{1,pu} = \sqrt{Z_{1,pu}^{2} - R_{1,pu}^{2}} \, }

Where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_{1,pu} \, } is the positive sequence resistance (pu)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_{1,pu} \, } is the positive sequence reactance (pu)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_{1,pu} \, } is the positive sequence impedance (pu)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_{c} \, } are the full load copper losses (W)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_{n} \, } is the transformer rated power (kVA)

Application in Transformer Models

The positive sequence impedance measured and calculated above is most often applied in the standard positive sequence two-winding transformer model as representing the transformer leakage impedance. This representation assumes that the parallel magnetising and core loss branch in the T-circuit is a negligible part of the measured positive sequence impedance (note that the magnetising reactance and core losses are estimated separately in no-load (open circuit) tests).

You will notice that the standard positive sequence transformer model has leakage impedances on both the primary and secondar sides of the transformer. However in our measurements, we only calculate a single impedance. It is common practice to distribute the impedance across both sides (for example, 50%-50% if no other information is available).

Winding Connections

It is worth noting that the transformer impedance voltage (expressed in percent or per-unit) is independent of the winding connection. This is because each winding is tested separately and normally quoted as a per-unit (or percent) impedance voltage based on the transformer's rated line-to-line voltage (see IEEE Std C57.12.90). If the winding connections are changed, then the line-to-line voltage would also have to change. But since it is a per-unit (or percent) value, the impedance voltage would remain the same.

Zero Sequence Impedance

Two Winding Transformer

Impedance Measurement

Zero sequence impedance test circuit for three-phase 2-winding transformer (with accessible neutral)

In the zero sequence test, a single-phase voltage source is applied between the three phase terminals (connected together) and an externally available neutral terminal (see the test circuit in the figure right). Like in the positive sequence test, the voltage is steadily increased until the rated current is measured.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{V_{z}}{V_{n}} = \frac{1}{3} \frac{I_{n}}{I_{n}} \frac{Z_{0}}{Z_{b}} = \frac{1}{3} Z_{0,pu} \, }

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{z} \,} is the measured zero sequence impedance voltage (V)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{n} \,} is the rated transformer line-to-line voltage / base voltage (V)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_{n} \,} is the rated transformer current (A)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_{0} \,} is the zero sequence impedance (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega} )
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_{b} \,} is the transformer base impedance (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega} )
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_{0,pu} \,} is the zero sequence impedance (pu)

Note that the measured impedance voltage is actually equivalent to a third of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_{0}} . This is because the three phase windings are connected together and the current flowing through each individual winding is a third of the measured rated current.

Tleis [2] also suggests that for earthed star-star (YNyn) transformers with a three-limbed core construction, zero sequence flux can circulate around a path through the core, tank, insulating oil and air. Therefore this magnetic path acts like a "virtual" delta winding, and the zero sequence impedance can be lower than the standard measured value. Tleis suggests treating these transformers like three-winding transformers and performing at least three test measurements.

Calculation of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_{0}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_{0}}

Occasionally the zero sequence copper losses (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_{c,0}} ) are also measured as part of the zero sequence test. When this is the case, calculations similar to that of the positive sequence impedance can be applied:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_{0,pu} = \frac{P_{c,0}}{1000 S_{n}} \, }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_{0,pu} = \sqrt{Z_{0,pu}^{2} - R_{0,pu}^{2}} \, }

Where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_{0,pu} \, } is the zero sequence resistance (pu)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_{0,pu} \, } is the zero sequence reactance (pu)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_{0,pu} \, } is the zero sequence impedance (pu)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_{c,0} \, } are the zero sequence copper losses (W)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_{n} \, } is the transformer rated power (kVA)

However, when the zero sequence copper losses are not known, it is common to use the same X/R ratio as in the positive sequence, i.e.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_{0,pu} = \frac{Z_{0,pu}}{\sqrt{1 + \left( \frac{X_{1}}{R_{1}} \right)^{2}}} \, }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_{0,pu} = \left( \frac{X_{1}}{R_{1}} \right) R_{0,pu} \, }

Where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{X_{1}}{R_{1}} \, } is the positive sequence X/R ratio

References

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