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Chapter 2: Transformers

Module by: NGUYEN Phuc

Chapter 2: Transformers
This 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 Energy
  • This 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 voltage
  • Power transfer at the most economical transmission voltage
  • Power utilization at the most voltage for the particular utilization device
  • It is widely used in low-power, low-current electronic and control circuits:
    • Matching the impedances of a source and its load for maximum power transfer
    • Isolating one circuit from another
    • Isolating direct current while maintaining ac continuity between two circuits
  • The 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 Transformers
  • Essentially, 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 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 Conditions
  • Figure 2.4 shows in schematic form a transformer with its secondary circuit open and an alternating voltage v1v1 size 12{v rSub { size 8{1} } } {} applied to its primary terminals.
Figure 2
Figure 2.4 Transformer with open secondary.
  • The primary and secondary windings are actually interleaved in practice.
  • A small steady-state current iϕiϕ size 12{i rSub { size 8{ϕ} } } {}(the exciting current) flows in the primary and establishes an alternating flux in the magnetic current.
  • e1e1 size 12{e rSub { size 8{1} } } {} = emf induced in the primary (counter emf)
λ1λ1 size 12{λ rSub { size 8{1} } } {} = flux linkage of the primary winding
ϕϕ size 12{ϕ} {}= flux in the core linking both windings
N1N1 size 12{N rSub { size 8{1} } } {}= number of turns in the primary winding
  • The induced emf (counter emf) leads the flux by 90o90o size 12{"90" rSup { size 8{o} } } {}
e1=dλ1dt=N1dϕdte1=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+e1v1=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≈v1e1≈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=φmaxsinωtϕ=φmaxsinωt size 12{ϕ=φ rSub { size 8{"max"} } "sin"ωt} {} (2.3)
e1=N1dϕdt=ωφmaxcosωte1=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=2πfN1φmaxE1=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=V12πfN1φmax=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 3
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 90o90o 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=Eˆ1= size 12{ { hat {E}} rSub { size 8{1} } ={}} {}the rms value of the induced emf
Φˆ=Φˆ= size 12{ { hat {Φ}}={}} {}the rms value of the flux
Iˆϕ=Iˆϕ= size 12{ { hat {I}} rSub { size 8{ϕ} } ={}} {}the rms value of the equivalent sinusoidal exciting current
  • IϕIϕ size 12{I rSub { size 8{ϕ} } } {} lags E1E1 size 12{E rSub { size 8{1} } } {} by a phase angle θcθc size 12{θ rSub { size 8{c} } } {}.
Figure 4
Figure 2.6 No-load phasor diagram.
  • The core loss PcPc size 12{P rSub { size 8{c} } } {} equals the product of the in-phase components of the Eˆ1Eˆ1 size 12{ { hat {E}} rSub { size 8{1} } } {}and IϕIϕ size 12{I rSub { size 8{ϕ} } } {} :
Pc=E1IϕcosθcPc=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=Ic= size 12{I rSub { size 8{c} } ={}} {} core-loss current, Im=Im= size 12{I rSub { size 8{m} } ={}} {} magnetizing current
§2.3 Effect of Secondary Current; Ideal Transformer
Figure 5
Figure 2.7 Ideal transformer and load.
  • Ideal Transformer (Fig. 2.7)
  • Assumptions:
  1. Winding resistances are negligible.
  2. Leakage flux is assumed negligible.
  3. There are no losses in the core.
  4. 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=N1dϕdtv1=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=N2dϕdtv2=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=N1N2v1v2=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=0N1i1−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=N2i2N1i1=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=N2N1i1i2=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=v2i2v1i1=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 6
Figure 2.8 Three circuits which are identical at terminals ab when the transformer is ideal.
vˆ1=N1N2vˆ2 and {vˆ2=N2N1vˆ1vˆ1=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=N2N1Iˆ1Iˆ1=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=N1N22Vˆ2Iˆ2Vˆ1Iˆ1=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=Vˆ2Iˆ2Z2=Vˆ2Iˆ2 size 12{Z rSub { size 8{2} } = { { { hat {V}} rSub { size 8{2} } } over { { hat {I}} rSub { size 8{2} } } } } {} (2.18)
Z1=N1N22Z2Z1=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 Circuits
  • A 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 circuit
  • Leakage flux: Fig. 2.9.
Figure 7
Figure 2.9 Schematic view of mutual and leakage fluxes in a transformer.
  • L11L11 size 12{L rSub { size 8{1 rSub { size 6{1} } } } } {}= primary leakage inductance, X11X11 size 12{X rSub { size 8{1 rSub { size 6{1} } } } } {} = primary leakage reactance
X11=2πfL11X11=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: R1R1 size 12{R rSub { size 8{1} } } {}
  • Effect of the exciting current:
N1Iˆϕ=N1Iˆ1−N2Iˆ2=N1(Iˆϕ+Iˆ2')−N2Iˆ2N1Iˆϕ=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'=N2N1Iˆ2Iˆ2'=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=Lm= size 12{L rSub { size 8{m} } ={}} {} magnetizing inductance, Xm=Xm= size 12{X rSub { size 8{m} } ={}} {} magnetizing reactance
Xm=2πfLmXm=2πfLm size 12{X rSub { size 8{m} } =2π ital "fL" rSub { size 8{m} } } {} (2.24)
  • Ideal transformer:
Eˆ1Eˆ2=N1N2Eˆ1Eˆ2=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 reactance
  • Equivalent-T circuit for a transformer:
Xˆ12=N1N22X12 , {R2'=N1N22R2 , {V2'=N1N22V2Xˆ12=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.2.10
  • The actual transformer can be seen to be equivalent to an ideal transformer plus external impedances
  • Refer to the assumptions for an ideal transformer to understand the definitions and meanings of these resistances and reactances.
Figure 8
Figure 9
Figure 10
Figure 2.10 Steps in the development of the transformer equivalent circuit.
§2.5 Engineering Aspects of Transformer Analysis
  • Approximate forms of the equivalent circuit:
Figure 11
Figure 12
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 test
  • Short-Circuit Test
  • The test is used to find the equivalent series impedance Req+jXeqReq+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 secondary
  • Typically 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//jXmZϕ=Rc//jXm size 12{Z rSub { size 8{ϕ} } =R rSub { size 8{c} } "//" ital "jX" rSub { size 8{m} } } {}.
Figure 13
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+jX12Zsc=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+jXeqZsc≈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 VscVsc size 12{V rSub { size 8{ ital "sc"} } } {} , the short-circuit current IscIsc size 12{I rSub { size 8{ ital "sc"} } } {} , and the power PscPsc 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∣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=PscIsc2Req=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−Rsc2Xeq=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=0.5ReqR1=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=0.5XeqXl1=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 R1R1 size 12{R rSub { size 8{1} } } {}and R2R2 size 12{R rSub { size 8{2} } } {} directly by a dc resistance measurement on each winding. However, no such simple test exists for Xl1Xl1 size 12{X rSub { size 8{l rSub { size 6{1} } } } } {} and Xl2Xl2 size 12{X rSub { size 8{l rSub { size 6{2} } } } } {}.
  • Open-Circuit Test
  • The test is used to find the equivalent shunt impedance Rc//jXmRc//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//jXmZϕ