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  • The load on an instrumentation transformer is frequently referred to as the burden on that transformer.
  • A potential transformer should ideally accurately measure voltage while appearing as an open circuit to the system under measurement, i.e. drawing negligible current and power.
  • Its load impedance should be “large” in some sense.
  • An ideal current transformer would accurately measure current while appearing as a short circuit to the system under measurement, i.e. developing negligible voltage drop and drawing negligible power.
  • Its load impedance should be “small” in some sense.

§2.9 The Per-Unit System

  • Computations relating to machines, transformers, and systems of machines are often carried out in per-unit system.
  • All pertinent quantities are expressed as decimal fractions of appropriately chose base values.
  • All the usual computations are then carried out in these per unit values instead of the familiar volts, amperes, ohms, and so on.
  • Advantages:
  • The parameter values typically fall in a reasonably narrow numerical range when expressed in a per-unit system based upon their rating.
  • When transformer equivalent-circuit parameters are converted to their per-unit values, the ideal transformer turns ratio becomes 1:1 and hence the ideal transformer can be eliminated.
  • Actual quantities: V , I , P , Q ,VA , R , X , Z , G , B , Y

Quantity in per unit = Actual quantity Base value of quantity size 12{ ital "Quantity" ital "in" ital "per" ital "unit"= { { ital "Actual" ital "quantity"} over { ital "Base" ital "value" ital "of" ital "quantity"} } } {} (2.40)

  • To a certain extent, base values can be chosen arbitrarily, but certain relations between them must be observed. For a single-phase system:

P base , Q base , VA base = V base I base size 12{P rSub { size 8{ ital "base"} } ", "Q rSub { size 8{ ital "base"} } ", " ital "VA" rSub { size 8{ ital "base"} } =V rSub { size 8{ ital "base"} } I rSub { size 8{ ital "base"} } } {} (2.41)

R base , X base , Z base = V base I base size 12{R rSub { size 8{ ital "base"} } ", "X rSub { size 8{ ital "base"} } ", "Z rSub { size 8{ ital "base"} } = { {V rSub { size 8{ ital "base"} } } over {I rSub { size 8{ ital "base"} } } } } {} (2.42)

  • Only two independent base quantities can be chose arbitrarily; the remaining quantities are determined by (2.48) and (2.49).
  • In typical usage, values of VA base size 12{ ital "VA" rSub { size 8{ ital "base"} } } {} and V base size 12{V rSub { size 8{ ital "base"} } } {} and are chosen first; values of I base size 12{I rSub { size 8{ ital "base"} } } {} and all other quantities in (2.48) and (2.49) are then uniquely established.
  • The value of VA base size 12{ ital "VA" rSub { size 8{ ital "base"} } } {} must be the same over the entire system under analysis.
  • When a transformer is encountered, the values of V base size 12{V rSub { size 8{ ital "base"} } } {} differ on each side and should be chosen in the same ratio as the turns ratio of the transformer.
  • The per-unit ideal transformer will have a unity turns ratio and hence can be eliminated.
  • Usually the rated or nominal voltages of the respective sides are chosen.
  • The procedure for performing system analyses in per-unit is summarized as follows:
    • Select a VA base and a base voltage at some point in the system.
    • Convert all quantites toper unit on the chosen VA base and with a voltage base that transforms as the turns ratio of any transformer which is encountered as one moves through the system.
    • Perform a standard electrical analysis with all quantities in per unit.
    • When the analysis is completed, all quantities can be converted back to real unit(e.g., volts, amperes, watts, etc.) by multiplying their per-unit values by their corresponding base values.
  • Machine Ratings as Bases

When expressed in per-unit form on their rating as a base, the per-unit values of machine parameters fall within a relatively narrow range.

  • The physics behind each type of device is the same and, in a crude sense, they can each be considered to be simply scaled versions of the same basic device.
  • When normalized to their own rating, the effect of the scaling is eliminated and the result is a set of per-unit parameter values which is quite similar over the whole size range of that device. For power and distribution transformers, I ϕ = 0 . 02 size 12{I rSub { size 8{ϕ} } =0 "." "02"} {} - 0.06pu , R  0.005 - 0.02pu , and X  0.015 - 0.10pu.
  • Manufacturers often supply device parameters in per unit on the device base.
  • When performing a system analysis, it may be necessary to convert the supplied per-unit parameter values to per-unit values on the base chosen for the analysis.

( P , Q , VA ) pu on base2 = ( P , Q , VA ) pu on base1 VA base 1 VA base 2 size 12{ \( P,Q, ital "VA" \) rSub { size 8{"pu on base2"} } = \( P,Q, ital "VA" \) rSub { size 8{"pu on base1"} } left [ { { ital "VA" rSub { size 8{ ital "base"1} } } over { ital "VA" rSub { size 8{ ital "base"2} } } } right ]} {} (2.43)

( P , X , Z ) pu on base2 = ( P , X , Z ) pu on base1 ( V base 1 ) 2 VA base 2 ( V base 2 ) 2 VA base 1 size 12{ \( P,X,Z \) rSub { size 8{"pu on base2"} } = \( P,X,Z \) rSub { size 8{"pu on base1"} } left [ { { \( V rSub { size 8{ ital "base"1} } \) rSup { size 8{2} } ital "VA" rSub { size 8{ ital "base"2} } } over { \( V rSub { size 8{ ital "base"2} } \) rSup { size 8{2} } ital "VA" rSub { size 8{ ital "base"1} } } } right ]} {} (2.44)

V pu on base2 = V pu on base1 V base 1 V base 2 size 12{V rSub { size 8{"pu on base2"} } =V rSub { size 8{"pu on base1"} } left [ { {V rSub { size 8{ ital "base"1} } } over {V rSub { size 8{ ital "base"2} } } } right ]} {} (2.45)

I pu on base2 = I pu on base1 V base 2 VA base 1 V base 1 VA base 2 size 12{I rSub { size 8{"pu on base2"} } =I rSub { size 8{"pu on base1"} } left [ { {V rSub { size 8{ ital "base"2} } ital "VA" rSub { size 8{ ital "base"1} } } over {V rSub { size 8{ ital "base"1} } ital "VA" rSub { size 8{ ital "base"2} } } } right ]} {} (2.46)

Balanced Three-Phase System:

  • Relations for base values:

( P base , Q base , VA base ) 3 phase = 3 VA base, per phase size 12{ \( P rSub { size 8{ ital "base"} } ,Q rSub { size 8{ ital "base"} } , ital "VA" rSub { size 8{ ital "base"} } \) rSub { size 8{3 - ital "phase"} } =3 ital "VA" rSub { size 8{"base, per phase"} } } {} (2.47)

  • The three-phase volt-ampere base ( VA base , 3 phase ) size 12{ \( ital "VA" rSub { size 8{ ital "base",3 - ital "phase"} } \) } {} and the line-to-line voltage base ( V base , 3 phase = V base , 1 1 ) size 12{ \( V rSub { size 8{ ital "base",3 - ital "phase"} } =V rSub { size 8{ ital "base",1 - 1} } \) } {} are usually chosen first.
  • The base values for the phase (line-to-neutral) voltage then is

V base , 1 n = 1 3 V base , 1 1 size 12{V rSub { size 8{ ital "base",1 - n} } = { {1} over { sqrt {3} } } V rSub { size 8{ ital "base",1 - 1} } } {} (2.48)

  • The base current for three-phase system is equal to the phase current, which is the same as the base current for a single-phase (per-phase) analysis.

I base , 3 phase = I base , per phase = VA base , 3 phase 3 V base , 3 phase size 12{I rSub { size 8{ ital "base",3 - ital "phase"} } =I rSub { size 8{ ital "base", ital "per" ital "phase"} } = { { ital "VA" rSub { size 8{ ital "base",3 - ital "phase"} } } over { sqrt {3} V rSub { size 8{ ital "base",3 - ital "phase"} } } } } {} (2.49)

  • The three-phase base impedance is chosen to be the single-phase base impedance.

Z base , 3 phase = Z base , per phase = V base , 1 n I base , per phase = V base , 3 phase 3 I base , 3 phase = ( V base , 3 phase ) 2 VA base , 3 phase alignl { stack { size 12{Z rSub { size 8{ ital "base",3 - ital "phase"} } =Z rSub { size 8{ ital "base", ital "per" ital "phase"} } } {} #" "= { {V rSub { size 8{ ital "base",1 - n} } } over {I rSub { size 8{ ital "base", ital "per" ital "phase"} } } } {} # " "= { {V rSub { size 8{ ital "base",3 - ital "phase"} } } over { sqrt {3} I rSub { size 8{ ital "base",3 - ital "phase"} } } } {} #" "= { { \( V rSub { size 8{ ital "base",3 - ital "phase"} } \) rSup { size 8{2} } } over { ital "VA" rSub { size 8{ ital "base",3 - ital "phase"} } } } {} } } {} ((2.50); (2.51); (2.52);(2.53)

Note that the factors of 3 and 3 are automatically taken care of in per unit by the base values. Three-phase problems can thus be solved in per unit as if they were single-phase problems.

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Source:  OpenStax, Electrical machines. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10767/1.1
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