Chapter 2: Transformers1.12007/11/08 04:53:41.988 US/Central2007/12/02 08:12:43.692 US/CentralNGUYENHuu Phucnhphuc@hcmut.edu.vnNGUYENHuu Phucnhphuc@hcmut.edu.vnTransformersChapter 2: TransformersThis lecture note is based on the textbook # 1. Electric Machinery - A.E. Fitzgerald, Charles Kingsley, Jr., Stephen D. Umans- 6th edition- Mc Graw Hill series in Electrical Engineering. Power and EnergyThis chapter is to discuss certain aspects of the theory of magnetically-coupled circuits, with emphasis on transformer action.The static transformer is not an energy conversion device, but an indispensable component in many energy conversion systems.It is a significant component in ac power systems:Electric generation at the most economical generator voltagePower transfer at the most economical transmission voltagePower utilization at the most voltage for the particular utilization deviceIt is widely used in low-power, low-current electronic and control circuits:Matching the impedances of a source and its load for maximum power transferIsolating one circuit from anotherIsolating direct current while maintaining ac continuity between two circuitsThe transformer is one of the simpler devices comprising two or more electric circuits coupled by a common magnetic circuit.Its analysis involves many of the principles essential to the study of electric machinery.§2.1 Introduction to TransformersEssentially, a transformer consists of two or more windings coupled by mutual magnetic flux.One of these windings, the primary, is connected to an alternating-voltage.An alternating flux will be produced whose magnitude will depend on the primary voltage, the frequency of the applied voltage, and the number of turns.The mutual flux will link the other winding, the secondary, and will induce a voltage in it whose value will depend on the number of secondary turns as well as the magnitude of the mutual flux and the frequency.By properly proportioning the number of primary and secondary turns, almost any desired voltage ratio, or ratio of transformation, can be obtained.The essence of transformer action requires only the existence of time-varying mutual flux linking two windings.Iron-core transformer: coupling between the windings can be made much more effectively using a core of iron or other ferromagnetic material.The magnetic circuit usually consists of a stack of thin laminations.Silicon steel has the desirable properties of low cost, low core loss, and high permeability at high flux densities (1.0 to 1.5 T).Silicon-steel laminations 0.014 in thick are generally used for transformers operating at frequencies below a few hundred hertz.Two common types of construction: core type and shell type (Fig. 2.1).Figure 2.1 Schematic views of (a) core-type and (b) shell-type transformers.Most of the flux is confined to the core and therefore links both windings.Leakage flux links one winding without linking the other.Leakage flux is a small fraction of the total flux.Leakage flux is reduced by subdividing the windings into sections and by placing them as close together as possible.§2.2 No-Load ConditionsFigure 2.4 shows in schematic form a transformer with its secondary circuit open and an alternating voltage
v1 size 12{v rSub { size 8{1} } } {} applied to its primary terminals.Figure 2.4 Transformer with open secondary.The primary and secondary windings are actually interleaved in practice.A small steady-state current
iϕ size 12{i rSub { size 8{ϕ} } } {}(the exciting current) flows in the primary and establishes an alternating flux in the magnetic current.e1 size 12{e rSub { size 8{1} } } {} = emf induced in the primary (counter emf)λ1 size 12{λ rSub { size 8{1} } } {} = flux linkage of the primary windingϕ size 12{ϕ} {}= flux in the core linking both windingsN1 size 12{N rSub { size 8{1} } } {}= number of turns in the primary windingThe induced emf (counter emf) leads the flux by
90o size 12{"90" rSup { size 8{o} } } {}e1=dλ1dt=N1dϕdt size 12{e rSub { size 8{1} } = { {dλ rSub { size 8{1} } } over { ital "dt"} } =N rSub { size 8{1} } { {dϕ} over { ital "dt"} } } {} (2.1)v1=R1iϕ+e1 size 12{v rSub { size 8{1} } =R rSub { size 8{1} } i rSub { size 8{ϕ} } +e rSub { size 8{1} } } {} (2.2)e1≈v1 size 12{e rSub { size 8{1} } approx v rSub { size 8{1} } } {} if the no-load resistance drop is very small and the waveforms of voltage and flux are very nearly sinusoidal.ϕ=φmaxsinωt size 12{ϕ=φ rSub { size 8{"max"} } "sin"ωt} {} (2.3)e1=N1dϕdt=ωφmaxcosωt size 12{e rSub { size 8{1} } =N rSub { size 8{1} } { {dϕ} over { ital "dt"} } = ital "ωφ" rSub { size 8{"max"} } "cos"ωt} {} (2.4)E1=2π2fN1φmax=2πfN1φmax size 12{E rSub { size 8{1} } = { {2π} over { sqrt {2} } } ital "fN" rSub { size 8{1} } φ rSub { size 8{"max"} } = sqrt {2} π ital "fN" rSub { size 8{1} } φ rSub { size 8{"max"} } } {} (2.5)φmax=V12πfN1 size 12{φ rSub { size 8{"max"} } = { {V rSub { size 8{1} } } over { sqrt {2} π ital "fN" rSub { size 8{1} } } } } {} (2.6)The core flux is determined by the applied voltage, its frequency, and the number of turns in the winding. The core flux is fixed by the applied voltage, and the required exciting current is determined by the magnetic properties of the core; the exciting current must adjust itself so as to produce the mmf required to create the flux demanded by (2.6).A curve of the exciting current as a function of time can be found graphically from the ac hysteresis loop as shown in Fig. 2.5.Figure 2.5 Excitation phenomena. (a) Voltage, flux, and exciting current;(b) corresponding hysteresis loop.If the exciting current is analyzed by Fourier-series methods, its is found to consist of a fundamental component and a series of odd harmonics.The fundamental component can, in turn, be resolved into two components, one in phase with the counter emf and the other lagging the counter emf by 90o.Core-loss component: the in-phase component supplies the power absorbed by hysteresis and eddy-current losses in the core.Magnetizing current: It comprises a fundamental component lagging the counter emf by
90o size 12{"90" rSup { size 8{o} } } {} , together with all the harmonics, of which the principal is the third (typically 40%).The peculiarities of the exciting-current waveform usually need not by taken into account, because the exciting current itself is small, especially in large transformers. (typically about 1 to 2 percent of full-load current)Phasor diagram in Fig. 2.6.Eˆ1= size 12{ { hat {E}} rSub { size 8{1} } ={}} {}the rms value of the induced emfΦˆ= size 12{ { hat {Φ}}={}} {}the rms value of the fluxIˆϕ= size 12{ { hat {I}} rSub { size 8{ϕ} } ={}} {}the rms value of the equivalent sinusoidal exciting currentIϕ size 12{I rSub { size 8{ϕ} } } {} lags
E1 size 12{E rSub { size 8{1} } } {} by a phase angle
θc size 12{θ rSub { size 8{c} } } {}.Figure 2.6 No-load phasor diagram.The core loss
Pc size 12{P rSub { size 8{c} } } {} equals the product of the in-phase components of the
Eˆ1 size 12{ { hat {E}} rSub { size 8{1} } } {}and
Iϕ size 12{I rSub { size 8{ϕ} } } {} :Pc=E1Iϕcosθc size 12{P rSub { size 8{c} } =E rSub { size 8{1} } I rSub { size 8{ϕ} } "cos"θ rSub { size 8{c} } } {} (2.7)Ic= size 12{I rSub { size 8{c} } ={}} {} core-loss current,
Im= size 12{I rSub { size 8{m} } ={}} {} magnetizing current§2.3 Effect of Secondary Current; Ideal TransformerFigure 2.7 Ideal transformer and load.Ideal Transformer (Fig. 2.7)Assumptions:Winding resistances are negligible.Leakage flux is assumed negligible.There are no losses in the core.Only a negligible mmf is required to establish the flux in the core.The impressed voltage, the counter emf, the induced emf, and the terminal voltage:v1=e1=N1dϕdt size 12{v rSub { size 8{1} } =e rSub { size 8{1} } =N rSub { size 8{1} } { {dϕ} over { ital "dt"} } } {} (2.8)v2=e2=N2dϕdt size 12{v rSub { size 8{2} } =e rSub { size 8{2} } =N rSub { size 8{2} } { {dϕ} over { ital "dt"} } } {} (2.9)v1v2=N1N2 size 12{ { {v rSub { size 8{1} } } over {v rSub { size 8{2} } } } = { {N rSub { size 8{1} } } over {N rSub { size 8{2} } } } } {} (2.10)An ideal transformer transforms voltages in the direct ratio of the turns in its windings.Let a load be connected to the secondary.N1i1−N2i2=0 size 12{N rSub { size 8{1} } i rSub { size 8{1} } - N rSub { size 8{2} } i rSub { size 8{2} } =0} {} (2.11)N1i1=N2i2 size 12{N rSub { size 8{1} } i rSub { size 8{1} } =N rSub { size 8{2} } i rSub { size 8{2} } } {} (2.12)i1i2=N2N1 size 12{ { {i rSub { size 8{1} } } over {i rSub { size 8{2} } } } = { {N rSub { size 8{2} } } over {N rSub { size 8{1} } } } } {} (2.13)An ideal transformer transforms currents in the inverse ratio of the turns in its windings.From (2.10) and (2.13),v1i1=v2i2 size 12{v rSub { size 8{1} } i rSub { size 8{1} } =v rSub { size 8{2} } i rSub { size 8{2} } } {} (2.14)Instantaneous power input to the primary equals the instantaneous power output from the secondary.Impedance transformation properties: Fig. 2.8.Figure 2.8 Three circuits which are identical at terminals ab when the transformer is ideal. vˆ1=N1N2vˆ2 and {vˆ2=N2N1vˆ1 size 12{ { hat {v}} rSub { size 8{1} } = { {N rSub { size 8{1} } } over {N rSub { size 8{2} } } } { hat {v}} rSub { size 8{2} } " and {" hat ital {v}} rSub { size 8{2} } = { {N rSub { size 8{2} } } over {N rSub { size 8{1} } } } { hat {v}} rSub { size 8{1} } } {} (2.15)Iˆ1=N1N2Iˆ2 and {Iˆ2=N2N1Iˆ1 size 12{ { hat {I}} rSub { size 8{1} } = { {N rSub { size 8{1} } } over {N rSub { size 8{2} } } } { hat {I}} rSub { size 8{2} } " and {" hat ital {I}} rSub { size 8{2} } = { {N rSub { size 8{2} } } over {N rSub { size 8{1} } } } { hat {I}} rSub { size 8{1} } } {} (2.16)Vˆ1Iˆ1=N1N22Vˆ2Iˆ2 size 12{ { { { hat {V}} rSub { size 8{1} } } over { { hat {I}} rSub { size 8{1} } } } = left ( { {N rSub { size 8{1} } } over {N rSub { size 8{2} } } } right ) rSup { size 8{2} } { { { hat {V}} rSub { size 8{2} } } over { { hat {I}} rSub { size 8{2} } } } } {}{} (2.17)Z2=Vˆ2Iˆ2 size 12{Z rSub { size 8{2} } = { { { hat {V}} rSub { size 8{2} } } over { { hat {I}} rSub { size 8{2} } } } } {} (2.18)Z1=N1N22Z2 size 12{Z rSub { size 8{1} } = left ( { {N rSub { size 8{1} } } over {N rSub { size 8{2} } } } right ) rSup { size 8{2} } Z rSub { size 8{2} } } {} (2.19)Transferring an impedance from one side to the other is called “referring the impedance to the other side.” Impedances transform as the square of the turns ratio.Summary for the ideal transformer:Voltages are transformed in the direct ratio of turns.Currents are transformed in the inverse ratio of turns.Impedances are transformed in the direct ratio squared.Power and voltamperes are unchanged.§2.4 Transformer Reactances and Equivalent CircuitsA more complete model must take into account the effects of winding resistances, leakage fluxes, and finite exciting current due to the finite and nonlinear permeability of the core.Note that the capacitances of the windings will be neglected.Method of the equivalent circuit technique is adopted for analysis.Development of the transformer equivalent circuitLeakage flux: Fig. 2.9.Figure 2.9 Schematic view of mutual and leakage fluxes in a transformer.L11 size 12{L rSub { size 8{1 rSub { size 6{1} } } } } {}= primary leakage inductance,
X11 size 12{X rSub { size 8{1 rSub { size 6{1} } } } } {} = primary leakage reactanceX11=2πfL11 size 12{X rSub { size 8{1 rSub { size 6{1} } } } =2π ital "fL" rSub {1 rSub { size 6{1} } } } {} (2.20)Effect of the primary winding resistance:
R1 size 12{R rSub { size 8{1} } } {}Effect of the exciting current:N1Iˆϕ=N1Iˆ1−N2Iˆ2=N1(Iˆϕ+Iˆ2')−N2Iˆ2alignl { stack {
size 12{N rSub { size 8{1} } { hat {I}} rSub { size 8{ϕ} } =N rSub { size 8{1} } { hat {I}} rSub { size 8{1} } - N rSub { size 8{2} } { hat {I}} rSub { size 8{2} } } {} #
" "=N rSub { size 8{1} } \( { hat {I}} rSub { size 8{ϕ} } + { hat {I}} sup { ' } rSub { size 8{2} } \) - N rSub { size 8{2} } { hat {I}} rSub { size 8{2} } {}
} } {} (2.21)- (2.22)Iˆ2'=N2N1Iˆ2 size 12{ { hat {I}} sup { ' } rSub { size 8{2} } = { {N rSub { size 8{2} } } over {N rSub { size 8{1} } } } { hat {I}} rSub { size 8{2} } } {} (2.23) Lm= size 12{L rSub { size 8{m} } ={}} {} magnetizing inductance,
Xm= size 12{X rSub { size 8{m} } ={}} {} magnetizing reactanceXm=2πfLm size 12{X rSub { size 8{m} } =2π ital "fL" rSub { size 8{m} } } {} (2.24)Ideal transformer:Eˆ1Eˆ2=N1N2 size 12{ { { { hat {E}} rSub { size 8{1} } } over { { hat {E}} rSub { size 8{2} } } } = { {N rSub { size 8{1} } } over {N rSub { size 8{2} } } } } {} (2.25)Secondary resistance, secondary leakage reactanceEquivalent-T circuit for a transformer:Xˆ12=N1N22X12 , {R2'=N1N22R2 , {V2'=N1N22V2 size 12{ { hat {X}} rSub { size 8{1 rSub { size 6{2} } } } = left ( { {N rSub {1} } over { size 12{N rSub {2} } } } right ) rSup {2} size 12{X rSub {1 rSub { size 6{2} } } } size 12{" , {" ital {R}} sup { ' } rSub {2} } size 12{ {}= left ( { {N rSub {1} } over { size 12{N rSub {2} } } } right ) rSup {2} } size 12{R rSub {2} } size 12{" , {" ital {V}} sup { ' } rSub {2} } size 12{ {}= left ( { {N rSub {1} } over { size 12{N rSub {2} } } } right ) rSup {2} } size 12{V rSub {2} }} {} (2.26)Steps in the development of the transformer equivalent circuit: Fig.2.10The actual transformer can be seen to be equivalent to an ideal transformer plus external impedancesRefer to the assumptions for an ideal transformer to understand the definitions and meanings of these resistances and reactances.Figure 2.10 Steps in the development of the transformer equivalent circuit.§2.5 Engineering Aspects of Transformer AnalysisApproximate forms of the equivalent circuit:Figure 2.11 Approximate transformer equivalent circuits.Two tests serve to determine the parameters of the equivalent circuits of Figs. 2.10 and 2.11.Short-circuit test and open-circuit testShort-Circuit TestThe test is used to find the equivalent series impedance
Req+jXeq size 12{R rSub { size 8{ ital "eq"} } + ital "jX" rSub { size 8{ ital "eq"} } } {} .The high voltage side is usually taken as the primary to which voltage is applied.The short circuit is applied to the secondaryTypically an applied voltage on the order of 10 to 15 % or less of the rated value will result in rated current.See Fig. 2.12.Note that
Zϕ=Rc//jXm size 12{Z rSub { size 8{ϕ} } =R rSub { size 8{c} } "//" ital "jX" rSub { size 8{m} } } {}.Figure 2.12 Equivalent circuit with short-circuited secondary. (a) Complete equivalent circuit.(b) Cantilever equivalent circuit with the exciting branch at the transformer secondary.Zsc=R1+jX11+Zϕ(R2+jX12)Zϕ+R2+jX12 size 12{Z rSub { size 8{ ital "sc"} } =R rSub { size 8{1} } + ital "jX" rSub { size 8{1 rSub { size 6{1} } } } + { {Z rSub {ϕ} size 12{ \( R rSub {2} } size 12{+ ital "jX" rSub {1 rSub { size 6{2} } } } size 12{ \) }} over {Z rSub {ϕ} size 12{+R rSub {2} } size 12{+ ital "jX" rSub {1 rSub { size 6{2} } } }} } } {} (2.27)Zsc≈R1+jX11+R2+jX12=Req+jXeq size 12{Z rSub { size 8{ ital "sc"} } approx R rSub { size 8{1} } + ital "jX" rSub { size 8{1 rSub { size 6{1} } } } +R rSub {2} size 12{+ ital "jX" rSub {1 rSub { size 6{2} } } } size 12{ {}=R rSub { ital "eq"} } size 12{+ ital "jX" rSub { ital "eq"} }} {} (2.28)Typically the instrumentation will measure the rms magnitude of the applied voltage
Vsc size 12{V rSub { size 8{ ital "sc"} } } {} , the short-circuit current
Isc size 12{I rSub { size 8{ ital "sc"} } } {} , and the power
Psc size 12{P rSub { size 8{ ital "sc"} } } {}. The circuit parameters (referred to the primary) can be found as (2.29)-(2.31).∣Zeq∣=∣Zsc∣=VscIsc size 12{ \lline Z rSub { size 8{ ital "eq"} } \lline = \lline Z rSub { size 8{ ital "sc"} } \lline = { {V rSub { size 8{ ital "sc"} } } over {I rSub { size 8{ ital "sc"} } } } } {} (2.29)Req=Rsc=PscIsc2 size 12{R rSub { size 8{ ital "eq"} } =R rSub { size 8{ ital "sc"} } = { {P rSub { size 8{ ital "sc"} } } over {I rSub { size 8{ ital "sc"} } rSup { size 8{2} } } } } {} (2.30)Xeq=Xsc=∣Zsc∣2−Rsc2 size 12{X rSub { size 8{ ital "eq"} } =X rSub { size 8{ ital "sc"} } = sqrt { \lline Z rSub { size 8{ ital "sc"} } \lline rSup { size 8{2} } - R rSub { size 8{ ital "sc"} } rSup { size 8{2} } } } {} (2.31)The equivalent impedance can be referred from one side to the other.Approximate values of the individual primary and secondary resistances and leakage reactances can be obtained by assuming that
R1=R2=0.5Req size 12{R rSub { size 8{1} } =R rSub { size 8{2} } =0 "." 5R rSub { size 8{ ital "eq"} } } {} and
Xl1=Xl2=0.5Xeq size 12{X rSub { size 8{l rSub { size 6{1} } } } =X rSub {l rSub { size 6{2} } } size 12{ {}=0 "." 5X rSub { ital "eq"} }} {} when all impedances are referred to the same side.Note that it is possible to measure
R1 size 12{R rSub { size 8{1} } } {}and
R2 size 12{R rSub { size 8{2} } } {} directly by a dc resistance measurement on each winding. However, no such simple test exists for
Xl1 size 12{X rSub { size 8{l rSub { size 6{1} } } } } {} and
Xl2 size 12{X rSub { size 8{l rSub { size 6{2} } } } } {}.Open-Circuit TestThe test is used to find the equivalent shunt impedance
Rc//jXm size 12{R rSub { size 8{c} } "//" ital "jX" rSub { size 8{m} } } {} .The test is performed with the secondary open-circuited and rated voltage impressed on the primary. If the transformer is to be used at other than its rated voltage, the test should be done at that voltage.An exciting current of a few percent of full-load current is obtained.See Fig. 2.16. Note that
Zϕ=Rc//jXm size 12{Z rSub { size 8{ϕ} } =R rSub { size 8{c} } "//" ital "jX" rSub { size 8{m} } } {} .Figure 2.13 Equivalent circuit with open-circuited secondary. (a) Complete equivalent circuit.(b) Cantilever equivalent circuit with the exciting branch at the transformer primary.Zoc=R1+jX11+Zϕ=R1+jX11+Rc(jXm)Rc+jXm size 12{Z rSub { size 8{ ital "oc"} } =R rSub { size 8{1} } + ital "jX" rSub { size 8{1 rSub { size 6{1} } } } +Z rSub {ϕ} size 12{ {}=R rSub {1} } size 12{+ ital "jX" rSub {1 rSub { size 6{1} } } } size 12{+ { {R rSub {c} size 12{ \( ital "jX" rSub {m} } size 12{ \) }} over {R rSub {c} size 12{+ ital "jX" rSub {m} }} } }} {} (2.32)Zoc≈Zϕ=Rc(jXm)Rc+jXm size 12{Z rSub { size 8{ ital "oc"} } approx Z rSub { size 8{ϕ} } = { {R rSub { size 8{c} } \( ital "jX" rSub { size 8{m} } \) } over {R rSub { size 8{c} } + ital "jX" rSub { size 8{m} } } } } {} (2.33)Typically the instrumentation will measure the rms magnitude of the applied voltage
Voc size 12{V rSub { size 8{ ital "oc"} } } {}, the open-circuit current
Ioc size 12{I rSub { size 8{ ital "oc"} } } {} , and the power
Poc size 12{P rSub { size 8{ ital "oc"} } } {} . The circuit parameters (referred to the primary) can be found as (2.34)-(2.36).Rc=Voc2Poc size 12{R rSub { size 8{c} } = { {V rSub { size 8{ ital "oc"} } rSup { size 8{2} } } over {P rSub { size 8{ ital "oc"} } } } } {} (2.34)∣Zϕ∣=VocPoc size 12{ \lline Z rSub { size 8{ϕ} } \lline = { {V rSub { size 8{ ital "oc"} } } over {P rSub { size 8{ ital "oc"} } } } } {} (2.35)Xm=1(1/∣Zϕ∣)2−(1/Rc)2 size 12{X rSub { size 8{m} } = { {1} over { sqrt { \( 1/ \lline Z rSub { size 8{ϕ} } \lline \) rSup { size 8{2} } - \( 1/R rSub { size 8{c} } \) rSup { size 8{2} } } } } } {} (2.36)The open-circuit test can be used to obtain the core loss for efficiency computations and to check the magnitude of the exciting current.Note the term “Voltage Regulation” which is to be discussed in Example 2.6.§2.6 Autotransformers; Multiwinding TransformersTwo-winding Other winding configurations.§2.6.1 AutotransformersAutotransformer connection: Fig. 2.14.Figure 2.14 (a) Two-winding transformer. (b) Connection as an autotransformer.The windings of the two-winding transformer are electrically isolated whereas those of the autotransformer are connected directly together.In the transformer connection, winding ab must be provided with extra insulation.Autotransformer have lower leakage reactances, lower losses, and smaller exciting current and cost less than two-winding transformers when the voltage ration does not differ too greatly from 1:1.The rated voltages of the transformer can be expressed in terms of those of the two-winding transformer asVLrated=V1rated size 12{V rSub { size 8{L rSub { size 6{ ital "rated"} } } } =V rSub {1 rSub { size 6{ ital "rated"} } } } {} (2.37)VHrated=V1rated+V2rated=N1+N2N1VLrated size 12{V rSub { size 8{H rSub { size 6{ ital "rated"} } } } =V rSub {1 rSub { size 6{ ital "rated"} } } size 12{+V rSub {2 rSub { size 6{ ital "rated"} } } } size 12{ {}= left ( { {N rSub {1} size 12{+N rSub {2} }} over { size 12{N rSub {1} } } } right )} size 12{V rSub {L rSub { size 6{ ital "rated"} } } }} {} (2.38)The effective turns ratio of the autotransformer is thus
(N1+N2)/N1 size 12{ \( N rSub { size 8{1} } +N rSub { size 8{2} } \) /N rSub { size 8{1} } } {} . The power rating of the autotransformer is equal to
(N1+N2)/N2 size 12{ \( N rSub { size 8{1} } +N rSub { size 8{2} } \) /N rSub { size 8{2} } } {} times that of the two winding transformer.§2.6.2 Multiwinding TransformersTransformers having three or more windings, known as multiwinding or multicircuit transformers, are often used to interconnect three or more circuits which may have different voltages.Transformers having a primary and multiple secondaries are frequently found in multiple-output dc power supplies.Distribution transformers used to supply power for domestic purposes usually have two 120V secondaries connected in series.The three-phase transformer banks used to interconnect two transmission system of different voltages often have a third, or tertiary, set of windings to provide voltage for auxiliary power purposes in substation or to supply a local distribution system.Static capacitors or synchronous condensers may be connected to the tertiary windings for power factor correction or voltage regulation.Sometimes
Δ size 12{Δ} {}-connected tertiary windings are put on three-phase banks to provide a low-impedance path for third harmonic components of the exciting current to reduce third-harmonic components of the neutral voltage.§2.7 Transformers in Three-Phase CircuitsThree single-phase transformers can be connected to form a three-phase transformer bank in any of the four ways shown in Fig. 2.15. Note that
a=N1/N2 size 12{a=N rSub { size 8{1} } /N rSub { size 8{2} } } {}.Figure 2.15 Common three-phase transformer connections;the transformer windings are indicated by the heavy lines.The
Y−Δ size 12{Y - Δ} {} connection is commonly used in stepping down from a high voltage to a medium or low voltage.The
Δ−Y size 12{Δ - Y} {}connection is commonly used for stepping up to a high voltage.The
Δ−Δ size 12{Δ - Δ} {} connection has the advantage that one transformer can be removed for repair or maintenance while the remaining two continue to function as a three-phase bank with the rating reduced to 58 percent of that of the original bank.(Open-delta, or V, connection)The Y-Y connection is seldom used because of difficulties with exciting-current phenomenon.Because there is no neutral connection to carry harmonics of the exciting current and harmonic voltages are produced which significantly distort the transformer voltages.A three-phase bank may consist of one three-phase transformer having all six windings on a common multi-legged core and contained in a single tank.They cost less, weigh less, require less floor space, and have somewhat higher efficiency.It is usually convenient to carry out circuit computations involving three-phase transformer banks under balanced conditions on a single-phase (per-phase-Y, line-to-neutral) basis.Y−Δ size 12{Y - Δ} {},
Δ−Y size 12{Δ - Y} {}, and
Δ−Δ size 12{Δ - Δ} {}connections equivalent Y-Y connectionsA balanced
Δ size 12{Δ} {}-connected circuit of
ZΔΩ/phase size 12{Z rSub { size 8{Δ} } " " %OMEGA / ital "phase"} {} is equivalent to a balanced Y-connected circuit of
ZYΩ/phase size 12{Z rSub { size 8{Y} } " " %OMEGA / ital "phase"} {} ifZY=13ZΔ size 12{Z rSub { size 8{Y} } = { {1} over {3} } Z rSub { size 8{Δ} } } {} (2.39)§2.8 Voltage and Current TransformersTransformers are often used in instrumentation applications to match the magnitude of a voltage or current to the range of a meter or other instrumentation.Most 60-Hz power-systems’ instrumentation is based upon voltages in the range of 0-120V rms and currents in the range of 0-5 A rms.Power system voltages range up to 765-kV line-to-line and currents can be 10’s of kA.Some method of supplying an accurate, low-level representation of these signals to the instrumentation is required.Potential Transformer (PT) and Current Transformer (CT), also referred to as Instrumentation Transformer, are designed to approximate the ideal transformer as closely as is practically possible.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 SystemComputations 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 , YQuantityinperunit=ActualquantityBasevalueofquantity 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:Pbase, Qbase, VAbase=VbaseIbase 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)Rbase, Xbase, Zbase=VbaseIbase 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
VAbase size 12{ ital "VA" rSub { size 8{ ital "base"} } } {} and
Vbase size 12{V rSub { size 8{ ital "base"} } } {} and are chosen first; values of
Ibase 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
VAbase 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
Vbase 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 BasesWhen 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,
Iϕ=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 base1VAbase1VAbase2 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(Vbase1)2VAbase2(Vbase2)2VAbase1 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)Vpu on base2=Vpu on base1Vbase1Vbase2 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)Ipu on base2=Ipu on base1Vbase2VAbase1Vbase1VAbase2 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:(Pbase,Qbase,VAbase)3−phase=3VAbase, 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
(VAbase,3−phase) size 12{ \( ital "VA" rSub { size 8{ ital "base",3 - ital "phase"} } \) } {} and the line-to-line voltage base
(Vbase,3−phase=Vbase,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 isVbase,1−n=13Vbase,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.Ibase,3−phase=Ibase,perphase=VAbase,3−phase3Vbase,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.Zbase,3−phase=Zbase,perphase=Vbase,1−nIbase,perphase=Vbase,3−phase3Ibase,3−phase=(Vbase,3−phase)2VAbase,3−phasealignl { 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.