2.3 Application of the general theory onto DC machines
2.3.1 Separately excited DC machine
The field winding of the separately excited DC machine is fed by external source of DC voltage and is not connected to the armature (seeFigure 12). Let us shortly explain how the directions of voltages, currents, speed, and torques are drawn: The arrowhead of the induced voltage is moving to harmonize with the direction of the magnetic flux in the field circuit. The direction of this movement means the direc- tion of rotation and of developed electromagnetic (internal) torque. The load torque and the loss torque (the torque covering losses) are in opposite directions. The source of voltage is on the terminals and current flows in the opposite direction. On the armature there are arrowheads of voltage and current in coincidence, because the armature is a consumer.
For the rotor winding in the d-axis, Eq. (75) multiplied by iddt will be used:
udiddt¼Ri2ddtþiddψd�ωψqiddt: (77) For the rotor winding in the q-axis, Eq. (74) multiplied by iqdt will be used:
uqiqdt¼Ri2qdtþiqdψqþωψdiqdt: (78) Now the left sides and right sides of these equations are summed, and the result is an equation in which energy components can be identified:
Σuidt¼ΣRi2dtþΣidψþω ψdiq�ψqid
dt: (79)
The expression on the left side presents a rise of the delivered energy during the time dt:Σuidt.
The first expression on the right side presents rise of the energy of the Joule’s losses in the windings:ΣRi2dt.
The second expression on the right side is an increase of the field energy:Σidψ.
The last expression means a rise of the energy conversion from electrical to mechanical form in the case of motor or from mechanical to electrical form in the case of generator:ω ψ diq�ψqid
dt.
The instantaneous value of the electromagnetic power of the converted energy can be gained if the expression for energy conversion will be divided by time dt:
pe¼
ω ψ diq�ψqid dt
dt ¼ω ψdiq�ψqid
¼pΩ ψdiq�ψqid
, (80)
where p is the number of pole pairs. The subscript “e” is used to express
“electromagnetic power pe,” i.e., air gap power, where also the development of instantaneous value of electromagnetic torque teis investigated:
pe¼teΩ: (81)
If left and right sides of Eqs. (80) and (81) are put equal, an expression for the instantaneous value of the electromagnetic torque in general theory of electrical machines yields:
te¼pψdiq�ψqid
: (82)
If the motoring operation is analyzed, it can be seen that at known values of the terminal voltages (six equations) and known parameters of the windings, there are seven unknown variables, because except six currents in six windings there is also angular rotating speed, which is an unknown variable. Therefore, further equation must be added to the system. It is the equation for mechanical variables:
me¼JdΩ
dt þtL, (83)
in which it is expressed that developed electromagnetic torque given by Eq. (82) covers not only the energy of the rotating masses JdΩdt with the moment of inertia J but also the load torque tL.
Therefore, from the last two equations, the time varying of the mechanical angular speed can be calculated:
dΩ dt ¼1
Jðte�tLÞ ¼1
J p ψdiq�ψqid
� �
�tL
h i
: (84)
For the time varying of the electrical angular speed, which is directly linked with the voltage equations, we get:
dω dt ¼p
J p�ψdiq�ψqid�
�tL
h i
: (85)
These equations will be simulated if transients of electrical machines are investigated.
2.3 Application of the general theory onto DC machines
If the equivalent circuit of the universal machine and equivalent circuits of the DC machines are compared in great detail, it can be seen that the basic principle of the winding arrangement in two perpendicular axes is very well kept in DC
machines. It is possible to find a coincidence between generally defined windings f, D, Q, g, d, and q and concrete windings of DC machines, e.g., in this way:
The winding “f” represents field winding of DC machine.
The winding “D” either can represent series field winding in the case of com- pound machines, whereby the winding “f” is its shunt field winding, or, if it is short circuited, can represent damping effects during transients of the massive iron material of the machines. However, it is true that to investigate the parameters of such winding is very difficult [1].
The windings “g” and “Q” can represent stator windings, which are connected in series with the armature winding, if they exist in the machine. They can be com- mutating pole winding and compensating winding.
Windings “d” and “q” are the winding of the armature, but in the case of the classical construction of DC machine, where there is only one pair of terminals, and eventually one pair of the brushes in a two-pole machine, only q-winding and terminals with terminal voltage uqwill be taken into account. The winding in d-axis will be omitted, and by this way also terminals in d-axis, its voltage ud, and current idwill be cancelled.
The modified equivalent circuit of the universal electrical machine applied to DC machine is inFigure 11.
2.3.1 Separately excited DC machine
The field winding of the separately excited DC machine is fed by external source of DC voltage and is not connected to the armature (seeFigure 12). Let us shortly explain how the directions of voltages, currents, speed, and torques are drawn: The arrowhead of the induced voltage is moving to harmonize with the direction of the magnetic flux in the field circuit. The direction of this movement means the direc- tion of rotation and of developed electromagnetic (internal) torque. The load torque and the loss torque (the torque covering losses) are in opposite directions. The source of voltage is on the terminals and current flows in the opposite direction. On the armature there are arrowheads of voltage and current in coincidence, because the armature is a consumer.
To solve transients’ phenomena, a system of the voltage equations of all wind- ings is needed (for simplification g-winding is omitted):
uf ¼Rfifþdψf
dt , (86)
Figure 12.
Equivalent circuit of DC machine with separate excitation in motoring operation.
Figure 11.
Modified equivalent circuit of the universal machine applied on DC machine.
uD¼RDiDþdψD
dt , (87)
uQ ¼RQiQþdψQ
dt , (88)
uq¼Rqiqþdψq
dt þωψd, (89)
where:
ψf¼LffifþLfDiD, (90) ψD¼LDDiDþLDfif, (91) ψQ ¼LQQiQþLQqiq, (92) ψq¼LqqiqþLqQiQ, (93) ψd¼LddidþLdfifþLdDiD¼LdfifþLdDiD, (94) because the current in “d”-winding is zero, seeing that d-winding is omitted. In addition, the fact that mutual inductance of two perpendicular windings is zero was considered.
Then equation for electromagnetic torque is needed. This equation shows that electromagnetic torque in DC machine is developed in the form (again the member with the current idis cancelled):
te¼p ψdiq�ψqid
¼pψdiq, (95) and that it covers the energy of the rotating mass given by moment of inertia, time varying of the mechanical angular speed, and load torque:
te¼JdΩ
dt þtL: (96)
A checking of equation for electromagnetic torque of DC machines for steady-state conditions will be done if forψd, Eq. (72) is applied to the d-axis:
ψd¼N 2a 2
πΦd (97)
is introduced to Eq. (95) for the torque, whereby for the current the subscript
“a” is employed and used for the armature winding and the number of the conductors z is taken as double number of the turns N:
Te¼pψdiq¼pN 2a 2
πΦdIa¼p a
z
2πΦdIa¼CΦdIa: (98) If the damping winding D, neither the windings in the quadrature axis Q, g, are not taken into account and respecting Eq. (90) for linkage magnetic flux, we get equations as they are presented below. The simplest system of the voltage equations is as follows:
uf¼RfifþLfdif
dt, (99)
To solve transients’ phenomena, a system of the voltage equations of all wind- ings is needed (for simplification g-winding is omitted):
uf¼Rfifþdψf
dt , (86)
Figure 12.
Equivalent circuit of DC machine with separate excitation in motoring operation.
Figure 11.
Modified equivalent circuit of the universal machine applied on DC machine.
uD¼RDiDþdψD
dt , (87)
uQ ¼RQiQ þdψQ
dt , (88)
uq¼Rqiqþdψq
dt þωψd, (89)
where:
ψf¼LffifþLfDiD, (90) ψD¼LDDiDþLDfif, (91) ψQ ¼LQQiQþLQqiq, (92) ψq¼LqqiqþLqQiQ, (93) ψd¼LddidþLdfifþLdDiD¼LdfifþLdDiD, (94) because the current in “d”-winding is zero, seeing that d-winding is omitted. In addition, the fact that mutual inductance of two perpendicular windings is zero was considered.
Then equation for electromagnetic torque is needed. This equation shows that electromagnetic torque in DC machine is developed in the form (again the member with the current idis cancelled):
te¼p ψdiq�ψqid
¼pψdiq, (95) and that it covers the energy of the rotating mass given by moment of inertia, time varying of the mechanical angular speed, and load torque:
te¼JdΩ
dt þtL: (96)
A checking of equation for electromagnetic torque of DC machines for steady-state conditions will be done if forψd, Eq. (72) is applied to the d-axis:
ψd¼ N 2a 2
πΦd (97)
is introduced to Eq. (95) for the torque, whereby for the current the subscript
“a” is employed and used for the armature winding and the number of the conductors z is taken as double number of the turns N:
Te¼pψdiq¼pN 2a 2
πΦdIa¼p a
z
2πΦdIa¼CΦdIa: (98) If the damping winding D, neither the windings in the quadrature axis Q, g, are not taken into account and respecting Eq. (90) for linkage magnetic flux, we get equations as they are presented below. The simplest system of the voltage equations is as follows:
uf¼RfifþLfdif
dt, (99)
uq¼RqiqþLqdiq
dt þωψd¼RqiqþLqdiq
dt þωLdfif: (100) The expressionψd¼Ldfif shows that the linkage fluxψdin Eq. (94) is created by the mutual inductance Ldfbetween the field winding and armature winding (by that winding which exists there and is brought to the terminals through the brushes in the q-axis).
We get from Eq. (96) the equation for calculation of the time varying of the mechanical angular speed:
dΩ dt ¼1
Jpψdiq�tL
¼1
JpLdfifiq�tL
, (101)
and the electrical angular speedω, which appears in the voltage equations, is valid:
dω dt ¼p
Jpψdiq�tL
¼p
JpLdfifiq�tL
: (102)
In this way, a system of three equations (Eqs. (99), (100), and (102)), describing the smallest number of windings (three), was created. The solution of these equations brings time waveforms of the unknown variables (iq= f(t), if= f(t), andω= f(t)).
2.3.1.1 Separately excited DC motor
If a DC machine is in motoring operation, the known variables are terminal voltages, moment of inertia, load torque, and parameters of the motor, i.e., resis- tances and inductances of the windings.
Unknown variables are currents, electromagnetic torque, and angular speed.
Therefore, Eqs. (99) and (100) must be adjusted for the calculation of the currents:
dif
dt ¼ 1
Lfðuf�RfifÞ, (103) diq
dt ¼ 1
Lquq�Rqiq�ωψd
: (104)
The third equation is Eq. (102). It is necessary to solve these three equations, Eqs. (102)–(104), to get time waveforms of the unknown field current, armature current, and electrical angular speed, which can be recalculated to the mechanical angular speed or revolutions per minute:Ω¼ω=p or n¼60Ω=2πmin�1, at the known terminal voltages and parameters of the motor.
As it was seen, a very important part of the transients’ simulations is determina- tion of the machine parameters, mainly resistances and inductances but also moment of inertia. The parameters can be calculated in the process of the design of electrical machine, as it was shown in Chapter 1. However, the parameters can be also measured if the machine is fabricated. A guide how to do it is given in [8]. The gained parameters are introduced in equations, and by means of simulation pro- grams, the time waveforms are received. After the decay of the transients, the variables are stabilized; it means a steady-state condition occurs. The simulated waveforms during the transients can be verified by an oscilloscope and steady-state conditions also by classical measurements in steady state.
Designers in the process of the machine design can calculate parameters on the basis of geometrical dimensions, details of construction, and material properties. If they use the above derived equations, they can predestine the properties of the designed machine in transients and steady-state conditions. This is a very good method on how to optimize machine construction in a prefabricated period. When the machine is manufactured, it is possible to verify the parameters and properties by measurements and confirm them or to make some corrections.
2.3.1.2 Simulations of the concrete separately excited DC motor
The derived equations were applied to a concrete motor, the data of which are in Table 1. The fact that the motor must be fully excited before or simultaneously with applying the voltage to the armature must be taken into account. Demonstra- tion of the simulation outputs is inFigure 13. InFigure 13a–d, time waveforms of the simulated variables if= f(t), iq= f(t), n = f(t), and te= f(t) are shown after the voltage is applied to the terminals of the field winding in the instant of t = 0.1 s.
After the field current ifis stabilized, at the instant t = 0.6 s, the voltage was applied to the armature terminals. After the starting up, the no-load condition happened, and the rated load was applied at the instant t = 1 s.
InFigure 13e–g, basic characteristics of n = f(Te) are shown for the steady-state conditions. They illustrate methods on how the steady-state speed can be controlled:
by controlling the armature terminal voltage Uq, by resistance in the armature circuit Rq, as well as by varying the field current if.
Figure 13hpoints to the fact that value of the armature current Iqkdoes not depend on the value of the field current ifand also shows the typical feature of the motor with the rigid mechanical curve that the feeding armature current Iqis very high if motor is stationary; it means such motor has a high short circuit current. This is the reason why the speed control is suitable to check value of the feeding current, which can be ensured by the current control loop.
2.3.1.3 Separately excited generator
Equations for universal machine are derived in general; therefore, they can be used also for generating operation. If the prime mover is taken as a source of stiff speed, then the time changing of the speed can be neglected, i.e., dΩ=dt¼0, andΩ= const is taken. In addition, the current in the armature will be reversed, because now the induced voltage in the armature is a source for the whole circuit (seeFigure 14). According to Eq. (96), equilibrium occurs between the driving torque of prime mover Thnand electromagnetic torque Te, which act against each other, i.e., the prime mover is loaded by the developed electromagnetic torque. If
UqN= UfN= 84 V (in motoring) Rq= 0.033Ω
IqN= 220 A Lq= 0.324 mH
IfN= 6.4 A Rf= 13.2Ω
nN= 3200 min�1 Lf= 1.5246 H
TN= 48 Nm Lqf= 0.0353 H
PN= 16 kW J = 0.04 kg m2
p = 1 Te0= 0.2 Nm
Table 1.
Nameplate and parameters of the simulated separately excited DC motor.
uq¼RqiqþLqdiq
dt þωψd¼RqiqþLqdiq
dt þωLdfif: (100) The expressionψd¼Ldfif shows that the linkage fluxψdin Eq. (94) is created by the mutual inductance Ldf between the field winding and armature winding (by that winding which exists there and is brought to the terminals through the brushes in the q-axis).
We get from Eq. (96) the equation for calculation of the time varying of the mechanical angular speed:
dΩ dt ¼1
Jpψdiq�tL
¼1
JpLdfifiq�tL
, (101)
and the electrical angular speedω, which appears in the voltage equations, is valid:
dω dt ¼p
Jpψdiq�tL
¼p
JpLdfifiq�tL
: (102)
In this way, a system of three equations (Eqs. (99), (100), and (102)), describing the smallest number of windings (three), was created. The solution of these equations brings time waveforms of the unknown variables (iq= f(t), if= f(t), andω= f(t)).
2.3.1.1 Separately excited DC motor
If a DC machine is in motoring operation, the known variables are terminal voltages, moment of inertia, load torque, and parameters of the motor, i.e., resis- tances and inductances of the windings.
Unknown variables are currents, electromagnetic torque, and angular speed.
Therefore, Eqs. (99) and (100) must be adjusted for the calculation of the currents:
dif
dt ¼ 1
Lfðuf�RfifÞ, (103) diq
dt ¼ 1
Lquq�Rqiq�ωψd
: (104)
The third equation is Eq. (102). It is necessary to solve these three equations, Eqs. (102)–(104), to get time waveforms of the unknown field current, armature current, and electrical angular speed, which can be recalculated to the mechanical angular speed or revolutions per minute:Ω¼ω=p or n¼60Ω=2πmin�1, at the known terminal voltages and parameters of the motor.
As it was seen, a very important part of the transients’ simulations is determina- tion of the machine parameters, mainly resistances and inductances but also moment of inertia. The parameters can be calculated in the process of the design of electrical machine, as it was shown in Chapter 1. However, the parameters can be also measured if the machine is fabricated. A guide how to do it is given in [8]. The gained parameters are introduced in equations, and by means of simulation pro- grams, the time waveforms are received. After the decay of the transients, the variables are stabilized; it means a steady-state condition occurs. The simulated waveforms during the transients can be verified by an oscilloscope and steady-state conditions also by classical measurements in steady state.
Designers in the process of the machine design can calculate parameters on the basis of geometrical dimensions, details of construction, and material properties. If they use the above derived equations, they can predestine the properties of the designed machine in transients and steady-state conditions. This is a very good method on how to optimize machine construction in a prefabricated period. When the machine is manufactured, it is possible to verify the parameters and properties by measurements and confirm them or to make some corrections.
2.3.1.2 Simulations of the concrete separately excited DC motor
The derived equations were applied to a concrete motor, the data of which are in Table 1. The fact that the motor must be fully excited before or simultaneously with applying the voltage to the armature must be taken into account. Demonstra- tion of the simulation outputs is inFigure 13. InFigure 13a–d, time waveforms of the simulated variables if= f(t), iq= f(t), n = f(t), and te= f(t) are shown after the voltage is applied to the terminals of the field winding in the instant of t = 0.1 s.
After the field current ifis stabilized, at the instant t = 0.6 s, the voltage was applied to the armature terminals. After the starting up, the no-load condition happened, and the rated load was applied at the instant t = 1 s.
InFigure 13e–g, basic characteristics of n = f(Te) are shown for the steady-state conditions. They illustrate methods on how the steady-state speed can be controlled:
by controlling the armature terminal voltage Uq, by resistance in the armature circuit Rq, as well as by varying the field current if.
Figure 13hpoints to the fact that value of the armature current Iqkdoes not depend on the value of the field current ifand also shows the typical feature of the motor with the rigid mechanical curve that the feeding armature current Iqis very high if motor is stationary; it means such motor has a high short circuit current. This is the reason why the speed control is suitable to check value of the feeding current, which can be ensured by the current control loop.
2.3.1.3 Separately excited generator
Equations for universal machine are derived in general; therefore, they can be used also for generating operation. If the prime mover is taken as a source of stiff speed, then the time changing of the speed can be neglected, i.e., dΩ=dt¼0, andΩ= const is taken. In addition, the current in the armature will be reversed, because now the induced voltage in the armature is a source for the whole circuit (seeFigure 14). According to Eq. (96), equilibrium occurs between the driving torque of prime mover Thnand electromagnetic torque Te, which act against each other, i.e., the prime mover is loaded by the developed electromagnetic torque. If
UqN= UfN= 84 V (in motoring) Rq= 0.033Ω
IqN= 220 A Lq= 0.324 mH
IfN= 6.4 A Rf= 13.2Ω
nN= 3200 min�1 Lf= 1.5246 H
TN= 48 Nm Lqf= 0.0353 H
PN= 16 kW J = 0.04 kg m2
p = 1 Te0= 0.2 Nm
Table 1.
Nameplate and parameters of the simulated separately excited DC motor.