Chapter 7: DC Machines
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
- Dc machines are characterized by their versatility.
- By means of various combinations of shunt-, series-, and separately-excited field windings they can be designed to display a wide variety of volt-ampere or speed-torque characteristics for both dynamic and steady-state operation.
- Because of the ease with which they can be controlled, systems of dc machines have been frequently used in applications requiring a wide range of motor speeds or precise control of motor output.
§7.1 Introduction
- The essential features of a dc machine are shown schematically in Fig.7.1
- Fig. 7.1(b) shows the circuit representation of the machine.
- The stator has salient poles and is excited by one or more field coils.
- The air-gap flux distribution created by the field windings is symmetric about the center line of the field poles. This axis is called the field axis or direct axis.
- The ac voltage generated in each rotating armature coil is converted to dc in the external armature terminals by means of a rotating commutator and stationary brushes to which the armature leads are connected.
- The commutator-brush combination forms a mechanical rectifier, resulting in a dc armature voltage as well as an armature-mmf wave which is fixed in space.
- The brushes are located so that commutation occurs when the coil sides are in the neutral zone, midway between the field poles.
- The axis of the armature-mmf wave is 90 electrical degrees from the axis of the field poles, i.e., in the quadrature axis.
- The armature-mmf wave is along the brush axis.
Figure 7.1 Schematic representations of a dc machine.
- Recall equation (7.1). Note that the torque is proportional to the product of the magnitudes of the interacting fields and to the sine of the electrical space angle between their magnetic axes.The negative sign indicates that the electromechanical torque acts in a direction to decrease the displacement angle between the fields.
{}T=−π2poles22ΦsrFrsinδrT=−π2poles22ΦsrFrsinδr size 12{T= - { {π} over {2} } left [ { { ital "poles"} over {2} } right ] rSup { size 8{2} } Φ rSub { size 8{ ital "sr"} } F rSub { size 8{r} } "sin"δ rSub { size 8{r} } } {} (7.1)
- For the dc machine, the electromagnetic torque
TmechTmech size 12{T rSub { size 8{ ital "mech"} } } {}can be expressed in terms of the interaction of the direct-axis air-gap per pole
ΦdΦd size 12{Φ rSub { size 8{d} } } {} and the space-fundamental component
Fa1Fa1 size 12{F rSub { size 8{a1} } } {}of the armature-mmf wave, in a form similar to (7.1). Note that
sinδr=1sinδr=1 size 12{"sin"δ rSub { size 8{r} } =1} {}.
Tmech=π2pole22ΦdFalTmech=π2pole22ΦdFal size 12{T rSub { size 8{ ital "mech"} } = { {π} over {2} } left [ { { ital "pole"} over {2} } right ] rSup { size 8{2} } Φ rSub { size 8{d} } F rSub { size 8{ ital "al"} } } {} (7.2)
Tmech=KaΦdiaTmech=KaΦdia size 12{T rSub { size 8{ ital "mech"} } =K rSub { size 8{a} } Φ rSub { size 8{d} } i rSub { size 8{a} } } {} (7.3)
Ka=poles.Ca2πmKa=poles.Ca2πm size 12{K rSub { size 8{a} } = { { ital "poles" "." C rSub { size 8{a} } } over {2πm} } } {} (7.4)
KaKa size 12{K rSub { size 8{a} } } {}: a constant determined by the design of the winding, the winding constant
ia=ia= size 12{i rSub { size 8{a} } ={}} {}current in external armature circuit
Ca=Ca= size 12{C rSub { size 8{a} } ={}} {}total number of conductors in armature winding,
m= number of parallel paths through winding
- The rectified voltage
eaea size 12{e rSub { size 8{a} } } {} between brushes, known also as the speed voltage, is
ea=KaΦdωmea=KaΦdωm size 12{e rSub { size 8{a} } =K rSub { size 8{a} } Φ rSub { size 8{d} } ω rSub { size 8{m} } } {} (7.5)
- The generated voltage as observed from the brushes is the sum of the rectified voltage of all the coils in series between brushes and is shown by the rippling line labeled
eaea size 12{e rSub { size 8{a} } } {} in Fig.7.2.
- With a dozen or so commutator segments per pole, the ripple becomes very small and the average generated voltage observed from the brushes equals the sum of the average values of the rectified coils voltages.
Figure 7.2 Rectified coil voltages and resultant voltage between brushes in a dc machine.
- Note that the electric power equals the mechanical power.
eaia=Tmechωmeaia=Tmechωm size 12{e rSub { size 8{a} } i rSub { size 8{a} } =T rSub { size 8{ ital "mech"} } ω rSub { size 8{m} } } {} (7.6)
- The flux-mmf characteristic is referred to as the magnetization curve.
- The direct-axis air-gap flux is produced by the combined mmf
∑Nfif∑Nfif size 12{ Sum {N rSub { size 8{f} } } i rSub { size 8{f} } } {} of the field winding.
- The form of a typical magnetization curve is shown in Fig.7.3(a).
- The dashed straight line through the origin coinciding with the straight portion of the magnetization curves is called the air-gap line.
- It is assumed that the armature mmf has no effect on the direct-axis flux because the axis of the armature-mmf wave is along the quadrature axis and hence perpendicular to the field axis. (This assumption needs reexamining!)
- Note the residual magnetism in the figure.The magnetic material of the field does not fully demagnetize when the net field mmf is reduced to zero.
- It is usually more convenient to express the magnetization curve in terms of the armature emf
ea0ea0 size 12{e rSub { size 8{a0} } } {}at a constant speed
ωm0ωm0 size 12{ω rSub { size 8{m0} } } {}as shown in Fig.7.3(b).
eaωm=KaΦd=ea0ωm0eaωm=KaΦd=ea0ωm0 size 12{ { {e rSub { size 8{a} } } over {ω rSub { size 8{m} } } } =K rSub { size 8{a} } Φ rSub { size 8{d} } = { {e rSub { size 8{a0} } } over {ω rSub { size 8{m0} } } } } {} (7.7)
ea=(ωmωm0)ea0ea=(ωmωm0)ea0 size 12{e rSub { size 8{a} } = \( { {ω rSub { size 8{m} } } over {ω rSub { size 8{m0} } } } \) e rSub { size 8{a0} } } {} (7.8)
ea=(nn0)ea0ea=(nn0)ea0 size 12{e rSub { size 8{a} } = \( { {n} over {n rSub { size 8{0} } } } \) e rSub { size 8{a0} } } {} (7.9)
- Fig. 7.3(c) shows the magnetization curve with only one field winding excited. This curve can easily be obtained by test methods.
Figure 7.3 Typical form of magnetization curves of a dc machine.
- Various methods of excitation of the field windings are shown in Fig.7.4.
Figure 7.4 Field-circuit connections of dc machines:
(a) separate excitation, (b) series, (c) shunt, (d) compound.
Consider first dc generators.
- Separately-excited generators.
- Self-excited generators: series generators, shunt generators, compound generators.
- With self-excited generators, residual magnetism must be present in the machine iron to get the self-excitation process started.
- N.B.: long- and short-shunt, cumulatively and differentially compound.
- Typical steady-state volt-ampere characteristics are shown in Fig.7.5, constant-speed operation being assumed.
- The relation between the steady-state generated emf Ea and the armature terminal voltage
VaVa size 12{V rSub { size 8{a} } } {} is
Va=Ea−IaRaVa=Ea−IaRa size 12{V rSub { size 8{a} } =E rSub { size 8{a} } - I rSub { size 8{a} } R rSub { size 8{a} } } {} (7.10)
Figure 7.5 Volt-ampere characteristics of dc generators.
Any of the methods of excitation used for generators can also be used for motors.
- Typical steady-state dc-motor speed-torque characteristics are shown in Fig.7.6, in which it is assumed that the motor terminals are supplied from a constant-voltage source.
- In a motor the relation between the emf
EaEa size 12{E rSub { size 8{a} } } {} generated in the armature and and the armature terminal voltage
VaVa size 12{V rSub { size 8{a} } } {} is
Va=Ea+IaRaVa=Ea+IaRa size 12{V rSub { size 8{a} } =E rSub { size 8{a} } +I rSub { size 8{a} } R rSub { size 8{a} } } {} (7.11)
Ia=Va−EaRaIa=Va−EaRa size 12{I rSub { size 8{a} } = { {V rSub { size 8{a} } - E rSub { size 8{a} } } over {R rSub { size 8{a} } } } } {} (7.12)
- The application advantages of dc machines lie in the variety of performance characteristics offered by the possibilities of shunt, series, and compound excitation.
Figure 7.6 Speed-torque characteristics of dc motors.
§7.4 Analytical Fundamentals: Electric-Circuit Aspects
- Analysis of dc machines: electric-circuit and magnetic-circuit aspects
The electromagnetic torque
TmechTmech size 12{T rSub { size 8{ ital "mech"} } } {}
Tmech=KaΦdIaTmech=KaΦdIa size 12{T rSub { size 8{ ital "mech"} } =K rSub { size 8{a} } Φ rSub { size 8{d} } I rSub { size 8{a} } } {} (7.13)
The generated voltage
EaEa size 12{E rSub { size 8{a} } } {}
Ea=KaΦdωmEa=KaΦdωm size 12{E rSub { size 8{a} } =K rSub { size 8{a} } Φ rSub { size 8{d} } ω rSub { size 8{m} } } {} (7.14)
Ka=poles.Ca2πmKa=poles.Ca2πm size 12{K rSub { size 8{a} } = { { ital "poles" "." C rSub { size 8{a} } } over {2πm} } } {} (7.15)
EaIaEaIa size 12{E rSub { size 8{a} } I rSub { size 8{a} } } {} : electromagnetic power
Tmech=EaIaωm=KaΦdIaTmech=EaIaωm=KaΦdIa size 12{T rSub { size 8{ ital "mech"} } = { {E rSub { size 8{a} } I rSub { size 8{a} } } over {ω rSub { size 8{m} } } } =K rSub { size 8{a} } Φ rSub { size 8{d} } I rSub { size 8{a} } } {} (7.16)
Note that the electromagnetic power differs from the mechanical power at the machine shaft by the rotational losses and differs from the electric power at the machine terminals by the shunt-field and armature
I2RI2R size 12{I rSup { size 8{2} } R} {} losses.
- Voltage and current (Refer to Fig.7.7.):
VaVa size 12{V rSub { size 8{a} } } {}: the terminal voltage of the armature winding
VtVt size 12{V rSub { size 8{t} } } {}: the terminal voltage of the dc machine, including the voltage drop across the series connected field winding
Va=VtVa=Vt size 12{V rSub { size 8{a} } =V rSub { size 8{t} } } {} if there is no series field winding
RaRa size 12{R rSub { size 8{a} } } {}: the resistance of armature,
RsRs size 12{R rSub { size 8{s} } } {}: the resistance of the series field
Va=Ea±IaRaVa=Ea±IaRa size 12{V rSub { size 8{a} } =E rSub { size 8{a} } +- I rSub { size 8{a} } R rSub { size 8{a} } } {} (7.17)
Vt=Ea±Ia(Ra+Rs)Vt=Ea±Ia(Ra+Rs) size 12{V rSub { size 8{t} } =E rSub { size 8{a} } +- I rSub { size 8{a} } \( R rSub { size 8{a} } +R rSub { size 8{s} } \) } {} (7.18)
IL=Ia±IfIL=Ia±If size 12{I rSub { size 8{L} } =I rSub { size 8{a} } +- I rSub { size 8{f} } } {} (7.19)
Figure 7.7 Motor or generator connection diagram with current directions.
- For compound machines, Fig.7.7 shows a long-shunt connection and the short-shunt connection is illustrated in Fig.7.8.
Figure 7.8 Short-shunt compound-generator connections.
§7.5 Analytical Fundamentals: Magnetic-Circuit Aspects
- The net flux per pole is that resulting from the combined mmf’s of the field and armature windings.
- First we consider the mmf intentionally placed on the stator main poles to create the working flux, i.e., the main-field mmf, and then we include armature-reaction effects.
§7.5.1 Armature Reaction Neglected
- With no load on the machine or with armature-reaction effects ignored, the resultant mmf is the algebraic sum of the mmf’s acting on the main or direct axis.
Main−field mmf=NfIf±NsIsMain−field mmf=NfIf±NsIs size 12{"Main" - "field mmf"=N rSub { size 8{f} } I rSub { size 8{f} } +- N rSub { size 8{s} } I rSub { size 8{s} } } {} (7.20)
Gross mmf=If+NsNfIs equivalent shunt−field amperesGross mmf=If+NsNfIs equivalent shunt−field amperes size 12{"Gross mmf"=I rSub { size 8{f} } + left [ { {N rSub { size 8{s} } } over {N rSub { size 8{f} } } } right ]I rSub { size 8{s} } " equivalent shunt" - "field amperes"} {} (7.21)
- An example of a no-load magnetization characteristic is given by the curve for
Ia=0Ia=0 size 12{I rSub { size 8{a} } =0} {} in Fig.7.9.
- The generated voltage
EaEa size 12{E rSub { size 8{a} } } {} at any speed
ωmωm size 12{ω rSub { size 8{m} } } {} is given by
Ea=ωmωm0Ea0Ea=ωmωm0Ea0 size 12{E rSub { size 8{a} } = left [ { {ω rSub { size 8{m} } } over {ω rSub { size 8{m0} } } } right ]E rSub { size 8{a0} } } {} (7.22)
Ea=nn0Ea0Ea=nn0Ea0 size 12{E rSub { size 8{a} } = left [ { {n} over {n rSub { size 8{0} } } } right ]E rSub { size 8{a0} } } {} (7.23)
Figure 7.9 Magnetization curves for a 100-kW, 250-V, 1200-r/min dc machine.
Also shown are field-resistance lines for the discussion of self-excitation in §7.6.1.
§7.5.2 Effects of Armature Reaction Included
- Current in the armature winding gives rise to a demagnetizing effect caused by a cross-magnetizing armature reaction.
One common approach is to base analyses on the measured performance of the machine.
- Data are taken with both the field and armature excited, and the tests are conducted so that the effects on generated emf of varying both the main-field excitation and armature mmf can be noted.
- Refer to Fig.7.9. The inclusion of armature reaction becomes simply a matter of using the magnetization curve corresponding to the armature current involved.
- The load-saturation curves are displaced to the right of the no-load curve by an amount which is a function of
IaIa size 12{I rSub { size 8{a} } } {}.
- The effect of armature reaction is approximately the same as a demagnetizing mmf
FarFar size 12{F rSub { size 8{ ital "ar"} } } {} acting on the main-field axis.
Net mmf=gross mmf−Far=NfIf+NsIs−ARNet mmf=gross mmf−Far=NfIf+NsIs−AR size 12{"Net mmf"="gross mmf" - F rSub { size 8{ ital "ar"} } =N rSub { size 8{f} } I rSub { size 8{f} } +N rSub { size 8{s} } I rSub { size 8{s} } - ital "AR"} {} (7.24)
- Over the normal operating range, the demagnetizing effect of armature reaction may be assumed to be approximately proportional to the armature current.
- The amount of armature of armature reaction present in Fig.7.9 is definitely more than one would expect to find in a normal, well-designed machine operating at normal currents.
§7.6 Analysis of Steady-State Performance
- Generator operation and motor operation
- For a generator, the speed is usually fixed by the prime mover, and problems often encountered are to determine the terminal voltage corresponding to a specified load and excitation or to find the excitation required for a specified load and terminal voltage.
- For a motor, problems frequently encountered are to determine the speed corresponding to a specific load and excitation or to find the excitation required for specific load and speed conditions; terminal voltage is often fixed at the value of the available source.
§7.6.1 Generator Analysis
- Analysis is based on the type of field connection.
- Separately-excited generators are the simplest to analyze.
- Its main-field current is independent of the generator voltage.
- For a given load, the equivalent main-field excitation is given by (7.21) and the associated armature-generated voltage
EaEa size 12{E rSub { size 8{a} } } {} is determined by the appropriate magnetization curve.
- The voltage
EaEa size 12{E rSub { size 8{a} } } {} , together with (7.17) or (7.18), fixes the terminal voltage.
- Shunt-excited generators will be found to self-excite under properly chosen operating condition under which the generated voltage will build up spontaneously.
- The process is typically initiated by the presence of a small amount of residual magnetism in the field structure and the shunt-field excitation depends on the terminal voltage. Consider the field-resistance line, the line 0a in Fig. 7.9.
- The tendency of a shunt-connected generator to self-excite can be observed by examining the buildup of voltage for an unloaded shunt generator.
–Buildup continues until the volt-ampere relations represented by the magnetization curve and the field-resistance line are simultaneously satisfied.
Figure 7.10 Equivalent circuit for analysis of voltage buildup in a self-excited dc generator.
Note that in Fig.7.10,
(La+Lf)difdt=ea−(Ra+Rf)if(La+Lf)difdt=ea−(Ra+Rf)if size 12{ \( L rSub { size 8{a} } +L rSub { size 8{f} } \) { { ital "di" rSub { size 8{f} } } over { ital "dt"} } =e rSub { size 8{a} } - \( R rSub { size 8{a} } +R rSub { size 8{f} } \) i rSub { size 8{f} } } {} (7.25)
- The field resistance line should also include the armature resistance.
- Notice that if the field resistance is too high, as shown by line 0b in Fig.7.9, voltage buildup will not be achieved.
- The critical field resistance, corresponding to the slope of the field-resistance line tangent to the magnetization curve, is the resistance above which buildup will not be obtained.
§7.6.2 Motor Analysis
- The terminal voltage of a motor is usually held substantially constant or controlled to a specific value. Motor analysis is most nearly resembles that for separately-excited generators.
- Speed is an important variable and often the one whose value is to be found.
Va=Ea±IaRaVa=Ea±IaRa size 12{V rSub { size 8{a} } =E rSub { size 8{a} } +- I rSub { size 8{a} } R rSub { size 8{a} } } {} (7.26)
Vt=Ea±Ia(Ra+Rs)Vt=Ea±Ia(Ra+Rs) size 12{V rSub { size 8{t} } =E rSub { size 8{a} } +- I rSub { size 8{a} } \( R rSub { size 8{a} } +R rSub { size 8{s} } \) } {} (7.27)
Gross mmf=If+NsNfIs equivalent shunt−field amperesGross mmf=If+NsNfIs equivalent shunt−field amperes size 12{"Gross mmf"=I rSub { size 8{f} } + left [ { {N rSub { size 8{s} } } over {N rSub { size 8{f} } } } right ]" "I rSub { size 8{s} } " equivalent shunt" - "field amperes"} {} (7.28)
Tmech=KaΦdIaTmech=KaΦdIa size 12{T rSub { size 8{ ital "mech"} } =K rSub { size 8{a} } Φ rSub { size 8{d} } I rSub { size 8{a} } } {} (7.29)
Ea=KaΦdωmEa=KaΦdωm size 12{E rSub { size 8{a} } =K rSub { size 8{a} } Φ rSub { size 8{d} } ω rSub { size 8{m} } } {} (7.30)
Ea=ωmωm0Ea0Ea=ωmωm0Ea0 size 12{E rSub { size 8{a} } = left [ { {ω rSub { size 8{m} } } over {ω rSub { size 8{m0} } } } right ]E rSub { size 8{a0} } } {} (7.31)
Ea=nn0Ea0Ea=nn0Ea0 size 12{E rSub { size 8{a} } = left [ { {n} over {n rSub { size 8{0} } } } right ]E rSub { size 8{a0} } } {} (7.32)
