Simulation of DC Machines in Microsim Design Lab 8.0 - An Article from Suresh Kumar K.S

[ This article attempts to offer qualitative explanation for various terms found in the voltage and torque equations of a "commutator primitive machine".The development of PSpice models for DC Motor and Generator using the equations of Commutator Primitive Machine Equations and simulation of such machines using the models etc. are also covered.]

1. The Commutator Primitive Machine

1.1 The Magnetic Axis of a Commutator-Brush Winding

Consider a simple two pole DC Machine shown in Fig.1.Two coils are identified by their respective coil planes a-a' and b-b'. The axis of the field winding which is the same as the North-South direction of the magnetic field created by the field winding is called the D-axis and these two coils are located at equal angles on either side of this axis. The magnetic field generated by the coil whose plane is a-a' will be perpendicular to the line a-a' and similarly the field created by the coil whose plane is b-b' will be perpendicular to the line b-b'. Both fields can be resolved into components along the direct axis and an axis perpendicular to direct axis (can be called quadrature axis). Clearly the direct axis components of the two coil fields cancel each other and the quadrature components strengthen each other. Similarly the entire periphery of the armature can be covered in terms of this kind of coil pairs and it can be seen that the total field produced by the entire winding will have zero direct axis component and net quadrature axis component. Notice that this conclusion is independent of the speed of rotation of the armature because the both the number of coil sides to the left and right of brush axis always remain the same and the current distribution in these two halves also remain the same. It is true that the identity of the individual conductors making up the left and right half go on changing with rotation. But as far as the total magnetic field by the entire coil system is concerned it does not matter that the individual participants are contributing different amounts to the total at different instants of time. It is the current distribution pattern that matters.

Hence the magnetic field generated by a Commutator winding will be along only the brush axis at all speeds. This same flux could have been created by a field winding structure in the quadrature structure. Thus, the Commutator winding can , equivalently, be thought of in terms of a fictitious stationary winding located in the quadrature axis.

1.2 Mutual Flux Between Direct axis Stator Coil and Quadrature axis Commutator Winding

Assume that one side of the coil plane a-a' is colored and is indicated by the arrow in Fig.2.Then by the time this coil takes up the position b-b' the colored surface would have gone under as indicated by the arrow on b-b'. This implies that the flux linkage in coil b-b' due to current in coil D is equal and opposite of the flux linkage of coil a-a' and they cancel out. The entire winding can be covered in terms of such pairs and hence the total flux linkage in the Commutator winding due to direct axis flux is going to be zero. The source of this direct axis flux does not affect this conclusion. For example this direct axis flux in the air gap could have been created by a commutator winding in that direction i.e. another pair of brushes located in the direct axis on the same armature winding. Even then the flux linkage in the quadrature axis. Also the time dimension did not come in the above reasoning. Neither did the armature speed.

Hence the mutual flux between a direct axis stationary coil and quadrature axis commutator winding will be zero for any kind of currents in them at any armature speed. This conclusion is true for quadrature axis stationary winding and direct axis commutator winding. Similarly it is true for a pair of commutator windings along direct axis and quadrature axis on the same armature. The mutual flux, mutual inductance and hence transformer e.m.f (mutually induced e.m.f) will be zero in all such cases irrespective of whether the currents and voltages involved are d.c or a.c.

1.3 Mutual Flux Between Direct axis Stator Coil and Direct axis Commutator Winding

In this case the quantity of flux linking the two coil planes a-a' and b-b' are the same and the 'sense' also is the same. Thus the two coil flux linkages add. Similarly the entire periphery can be covered in terms of such pairs of coils. The net flux linkage will not be zero. Moreover the net flux linkage will be maximum compared to any other placement of brushes. When the armature rotates the contribution of individual coils to the total flux linkage will change; but the total will remain the same. This is true for any kind of current flow since time did not figure in the reasoning. Also the axis of flux is immaterial to this conclusion and hence the conclusion holds well even if the stationary coil is in the quadrature axis and the brush is in quadrature axis.

Hence the mutual flux, mutual inductance and mutual e.m.f between a stationary winding and a commutator winding is at a maximum when the stationary coil axis and the brush axis coincide either on direct axis or on quadrature axis.

1.4 Speed e.m.f in a Quadrature Axis Commutator Winding due to Direct Axis Flux

Clockwise rotation is assumed. The surface velocity at the a-a' coil location and b-b' coil location are marked. When a conductor moves in a field the voltage induced in the conductor due to motion in the field is given by the vector cross product of the velocity vector and field vector. By the rule of cross product it can be seen that the two coils under consideration will get equal e.m.f s in the same direction in them. Since these two coils come in series under the brushes their voltages add. Taking two coils at a time like this can cover the entire periphery of the armature. Hence it can be seen that all the coil voltages add in series. And this is the brush position for maximum speed e.m.f. Obviously the speed e.m.f is proportional to speed and the current which produces the field. The conclusion that there will be maximum speed e.m.f is independent of time and independent the reason for the existence of direct axis fluxes. Thus there will be speed e.m.f in the quadrature axis brushes if another commutator winding produces direct axis flux in the air gap. Placing another pair of brushes in the direct axis on the same armature can form this commutator winding.

Hence direct axis fluxes produce maximum speed e.m.f in quadrature axis brushes and quadrature axis fluxes produce maximum speed e.m.f in the direct axis brushes.

1.5 Speed e.m.f in a Direct Axis Commutator Winding due to Direct Axis Flux

The velocity vectors and field vectors at a-a' and b-b' coil locations are drawn in Fig.5.By visualizing the directions of the cross products of velocity and field vectors at the two locations it is clear that the speed e.m.f s induced in these two coils will be equal but in opposite directions. The coils come in series and hence the speed e.m.f will cancel out. The entire periphery of the armature covered this way leads to the conclusion that the net speed e.m.f across the brushes will be zero. This conclusion is independent of the time variation of the direct axis flux nor was it dependent on the identity of the axis. Thus it is equally true for a stationary coil on quadrature axis and a commutator winding in that axis.

Hence the speed e.m.f in a commutator winding is zero if the brush axis of the winding is coincident with the flux axis of the flux being traversed.

1.6 The Mutual Inductance Coefficients and Speed e.m.f Coefficients

The machine shown has windings in both axes in the armature. The coil d on direct axis will have mutual flux with the coil D on direct axis and the coil q will have speed e.m.f in it due to the coil D on direct axis. Also there will be speed e.m.f in coil q due to flux in coil d and vice versa. But what is the relation between the various mutual inductances and self-inductances and the speed e.m.f coefficients?

Speed e.m.f is proportional to the speed and the flux being traversed. The flux being traversed is proportional to some current through some inductance or self-inductance. Thus speed e.m.f will be written as a product of speed, current and some quantity which has dimensions of inductance. There is no reason in general for any direct relationships to exist between these inductance like coefficients in speed e.m.f equations and the self/mutual inductances of the coil system. However it can be shown that if the flux distribution under a pole is sinusoidal these speed e.m.f coefficients are equal numerically to appropriate self/mutual inductances. This will be true in the case of a.c machines. This will not be exactly true in the case of d.c machines. But there is experimental evidence to indicate that this is more or less true even in the d.c machines and not much error will be incurred if this assumption is made in the case of d.c machines too.

Hence the speed e.m.f coefficient for 'q' due to 'D' will be same as mutual inductance between 'd' and 'D'. Speed e.m.f coefficient for 'q' due to 'd' will be self inductance of 'd' winding and speed e.m.f coefficient for 'd' due to 'q' will be self inductance of 'q'. The polarity of speed e.m.f s in the quadrature axis coils will be negative and in the direct axis coils will be positive. This may be worked out by visualizing the direction of cross product of velocity and flux in the conductors for the relevant cases.

1.7 The Commutator Primitive Machine and its Voltage Equation

The primitive commutator machine with the relevant sign convention is given in Fig.7.The armature is assumed to rotate at w_r mechanical rad/sec in the clockwise direction. The number of poles in the machine is 'n'.

The voltage equation of the machine will contain resistive drops, self and mutually induced e.m.f s and speed e.m.f s. The equation can be written in matrix form as given below.

This can be written as V = R i + L p i + G w_r i , where bold faced quantities are either vectors or square matrices and 'p' is the differential operator (d/dt). By power balance equation it is possible to show that the torque developed in the machine is given by T = i^*_tG i .

Newtom's Second Law of Motion applied to the motor shaft gives the mechanical system equation;

J (dw_r/dt) + Bw_r = i^*_tG i - T_m(w_r) , where J is the moment of inertia, B is the friction coefficient and T_m(w_r) is the mechanical torque applied to the shaft which could be a function of speed.

The equations describing the primitive commutator machine (which can be specialized to yield various kinds of dc motors and a.c series motors) are non-linear in a general context due the speed-current products in the equations. They are non-linear differential equations and have to be solved by numerical integration. For particular conditions, however, considerable simplifications can often be made. If speed is constant and the variables are steady d.c the equations become real algebraic ones. And if speed is constant and electrical variables are transients the equations become linear differential equations with constant coefficients. In that case Laplace Transforms, Eigen Value analysis etc can be used. If speed is a known function of time and electrical variables are transients the equations are linear with time varying coefficients and numerical methods will be needed for solution. But in all these cases only voltage equations needs to be solved. If speed is unknown, then both voltage and torque equations will have to be solved simultaneously. However if the aim is to solve the system for 'small changes' around the steady operating point linearisation of the non-linear equations is possible. The resulting small signal system will be linear differential equations with constant coefficients. And for small oscillation analysis the equations will become complex algebraic and solution is in complex numbers.

2. The Connection Matrix

The primitive commutator machine in the section 1 had two windings in each axis. It is possible to add one more winding on each and make it a six winding commutator primitive and the voltage equation can be written down by inspection using the rules developed in the earlier section.

Practical commutator machines differ from the commutator primitive machines in two aspects. Firstly, there may not be all the windings present in the practical machine. For example a simple d.c shunt motor has only D and q windings. Its voltage equation can be obtained from that of the four winding commutator primitive by deleting the columns and rows corresponding to the Q and d windings.

The windings in a practical machine are either connected to sources or interconnected among them. For example, in a d.c compound motor there are two stationary windings on direct axis and one commutator winding in the quadrature axis. The second direct axis winding is connected in series with quadrature axis winding. It is necessary to deduce the voltage equation for the connected machine from the commutator primitive equation. This is where the connection matrix becomes important.

For example, consider the case of a separately excited d.c generator driving a d.c motor as shown in Fig.8.

There are four current variables; but only three are independent by virtue of connection. The three currents are identified as i_fg,i_fm and i_qm.The following relations connect the primitive machine currents to the currents in the connected system.

I_fg = I_fg

I_qg = -I_fm - I_qm

I_fm = I_fm

I_qm = I_qm

These relations are put in matrix form I_old = C I_new , where I_old is a column vector containing the current variables in the primitive windings , I_new is the column vector of the independent current variables identified in the connected machine system and C is the connection matrix between them.

From Power Invariance condition it is possible to show that when the currents are transformed as per the above equation the machine impedance matrix transformed according to the relationship

Z_new= C_tZ_oldC where subscript t is used to indicate conjugate transpose operation. Similarly the following relation gives the interconnected machine applied voltage vector in terms of primitive machine voltage vector V_new = C_t V_old .

The electromagnetic torque developed in a primitive machine is given by T = i^*_tG i and using this the equations for the torque developed by the machines can be derived.

3. Dynamical Analysis of Electrical Machines-Simulation of DC Machines Using Design Lab8.0

The equations of the primitive commutator machine can be set up for simulation in Design Lab by using inductors, Voltage controlled voltage sources(E devices), Current controlled current sources(F device),Current Controlled Voltage Sources(H devices),Voltage controlled current Sources(G devices),multiplying blocks, gain blocks and resistors etc. The simulation diagram suitable for simulating a 4-winding primitive commutator machine is given below.

Design Lab8.0 does not come with machine models. It would have been convenient if the above simulation model could be converted into a symbol with a model associated with it. PSpice allows us to do this. The steps involved in converting a schematic into a symbol that can be used in any other schematic are listed below.

Simulate the schematic and verify its correctness. Then decide which are the inputs and outputs which should appear in the symbol to be constructed and place IF_IN and IF_OUT ports suitably. Name the ports properly and save the file.
Select 'Tools/Create Subcircuit' menu in Schematics program. This results in the creation of file DCMachine.sub in the working directory. Open this file in any text editor. The file is shown below.

* Schematics Subcircuit *

.SUBCKT DCMachine GND Tm Speed Torque Vqr Vds Vdr Vqs

G_G1 0 $N_0003 $N_0001 $N_0002 1

R_R1 $N_0003 0 {Mds/Lds}

E_Edr/ds c 0 $N_0003 0 1

L_Ldr $N_0004 $N_0005 {Ldr} IC=0

G_G4 0 $N_0006 $N_0004 $N_0005 1

R_R4 $N_0006 0 {Mds/Ldr}

E_Eds/dr a 0 $N_0006 0 1

F_F6 0 m VF_F6 1

VF_F6 $N_0007 d 0V

R_R21 m 0 1

R_R18 k 0 1

G_G7 0 k VALUE {V(o,0)+V(e,0)}

E_E23 f g VALUE {V(k,0)*V(h,0)}

G_G8 0 n VALUE {V(j,0)+V(i,0)}

R_R19 n 0 1

E_E25 d c VALUE {V(h,0)*V(n,0)}

G_G3 0 $N_0010 $N_0008 $N_0009 1

R_R3 $N_0010 0 {Mqs/Lqr}

E_Eqs/qr b 0 $N_0010 0 1

F_F5 0 l VF_F5 1

VF_F5 $N_0011 f 0V

R_R20 l 0 1

G_G2 0 $N_0014 $N_0012 $N_0013 1

R_R2 $N_0014 0 {Mqs/Lqs}

L_Lqs $N_0012 $N_0013 {Lqs} IC=0

E_Eqr/qs g 0 $N_0014 0 1

E_MULT3 $N_0015 0 VALUE {V(m)*V(n)}

E_MULT2 $N_0016 0 VALUE {V(k)*V(l)}

E_SUM1 Torque 0 VALUE {V($N_0015)+V($N_0016)}

E_GAIN9 $N_0017 0 VALUE {{B} * V(h)}

G_INTEG1 0 $$U_INTEG1 VALUE {V($N_0018)}

C_INTEG1 $$U_INTEG1 0 {1/{1/J}}

R_INTEG1 $$U_INTEG1 0 1G

E_INTEG1 h 0 VALUE {V($$U_INTEG1)}

.IC V($$U_INTEG1) = 0v

R_R23 $N_0019 Speed 1

E_GAIN11 $N_0019 0 VALUE {9.5493 * V(h)}

E_DIFF2 $N_0018 0 VALUE {V($N_0020,$N_0017)}

R_R24 Speed 0 1000k

L_Lqr $N_0008 $N_0009 {Lqr} IC=0

L_Lds $N_0001 $N_0002 {Lds} IC=0

R_Rad Vdr $N_0004 {Ra}

R_RQ Vqs $N_0012 {Rqs}

R_R33 Vqs 0 100k

R_RD Vds $N_0001 {Rds}

R_R30 Vds 0 100k

R_Raq Vqr $N_0008 {Ra}

R_R31 Vqr 0 100k

R_R32 Vdr 0 100k

R_R6 i 0 {Mqs*P}

R_R5 o 0 {Mds*P}

R_R8 e 0 {Ldr*P}

R_R7 j 0 {Lqr*P}

F_F1 0 o VF_F1 1

VF_F1 $N_0002 a 0V

F_F3 0 j VF_F3 -1

VF_F3 $N_0009 $N_0011 0V

F_F4 0 e VF_F4 1

VF_F4 $N_0005 $N_0007 0V

F_F2 0 i VF_F2 -1

VF_F2 $N_0013 b 0V

E_SUM2 $N_0020 0 VALUE {V(Tm)+V(Torque)}

.ENDS DCMachine

Edit the file and add the following immediately after the subcircuit definition and save the file.

+ PARAMS: Lds=30H, Lqs=30H, Ldr=0.02H, Lqr=0.02H, Mds=0.73H, Mqs=0.73H, Rds=230, Rqs=230, Ra=1.3, P=2, B=0.0127, J=0.5

These are the parameters to be passed on to the subcircuit whenever it is being invoked and later these will become the attributes of the constructed symbol. The values are the default attribute values that will appear in the attribute dialog box of the symbol.

Open a new empty file in Schematics program, give a suitable name and save it. Draw a block in it by using the 'Draw' menu. Double click on the block and associate it with the file DCMachine.sch. It then becomes a hierarchical block.
Select the block. Select 'Convert Block' from edit menu. The dialog boxes asking for the name of the new symbol and the library to which the symbol must be saved will appear. Give the name as DCMachine and save it to the library called dcmachines.slb. This empty symbol library must have been created earlier by using the Symbol Editor program.
Click on the symbol and choose to edit it from the edit menu in Schematic. The Symbol Editor program opens for editing. Choose 'Part\Attribute' menu and create the attributes such that the attribute box finally looks like the following.

REFDES=DCMachine
PART=DCMachine
MODEL=DCMachine
TEMPLATE=X^@REFDES %GND %Tm %Speed %Torque %Vqr %Vds %Vdr %Vqs @MODEL /n+ PARAMS: Lds=@Lds Lqs=@Lqs Ldr=@Ldr Lqr=@Lqr Mds=@Mds Mqs=@Mqs Rds=@Rds Rqs=@Rqs Ra=@Ra P=@P B=@B J=@J
Lds=30
Lqs=30
Ldr=0.02
Lqr=0.02
Mds=0.73
Mqs=0.73
Rds=230
Rqs=230
Ra=1.3
P=2
B=0.0127
J=0.5

Note that the order in which the nodes are listed in the TEMPLATE attribute must be same as the order in which they appear in the file DCMachine.sub. Also the MODEL name and PART name must be same as the subcircuit definition file name.

Choose 'Edit/Model' menu in Symbol Editor. Dialog box will report that there is no model and will indicate that a new model will have to be created. Choose to do so and choose 'text' mode in the ensuing dialog box.The program displays the Model Editor window. Simply copy the file DCMachine.sub which must be open in some text editor and paste it into the Model Editor window. Save the model and save the symbol. Choose 'Edit/Set Schematic' menu. Choose 'Delete' in the resulting dialog box. Close it and save the symbol. The resulting symbol and its attribute dialog box are shown below.

This Symbol can now be used the same way any other component is used in Design Lab.A Dc Shunt motor circuit set up using this symbol is shown below. The mechanical torque input i.e the the load torque has to follow the impressed torque convention and is hence negative.1 volt of torque is 1Nm and 1 volt of speed is 1 r.p.m. Speed and the torque developed by the motor are the outputs. The starting transient of the motor is simulated and the results are shown along with the schematic.

It will be necessary to include the library file dcmachines.lib in Schematics program by 'Analysis/Library and include Files' menu before the new symbol can be used for simulation.

Four Symbols were generated this way and they are shown below. Using these, any d.c commutator machine with upto 4 windings can be set up for simulation.The windings in "DC_motor" model are referred to a common ground and it will not be possible to connect two windings in series. This is solved in "DC_Motor1" model in which the windings are floating ones.Similarly for the Generator models too.

Inertia and Friction Coefficient values can be set for DC Motor models ; but they can not be set for DC Generator models. This is so because the input for a generator is taken as speed whereas speed is an output from DC Motor model. If the inertia and friction of DC Generator is to be taken into account it can be done by suitably changing the values of inertia and friction of the driving motor. The PSpice schematic of DC Motor driving a DC Generator is shown below for illustration.

An Osnos Generator was simulated using these symbols. The simulation set up illustrates the use of all the three kinds of analysis-AC sweep to study the frequency response on open loop, DC Sweep to obtain the steady state characteristics and Transient Analysis to obtain the time domain dynamics.

Osnos Generator Simulation Diagram in Design Lab using DC Machine Symbols.

The above Probe outputs show the result of transient analysis, nested d.c sweep and a.c sweep respectively. The transient analysis was using 1500 r.p.m and 25 ohms load across the generator. Upper trace shows the output. In the DC Sweep the main sweep was on speed and nested sweep was on the load resistance. Hence the traces show output voltage Vs speed for various values of load resistance and display the typical behaviour of an Osnos generator. The output voltage of Osnos generator is independent of speed and load resistance except for low speeds and low values of load resistance. The control voltage to output voltage frequency response is shown in the third set of waves and reveal the highly underdamped nature of response for the chosen set of parameters.

The Msim Library "dcmachines.slb" and "dcmachines.lib" can be downloaded from here. Also available for download is the set of example schematic files which can be run in Microsim Design Lab 8.0. Copy the "dcmachines.slb and dcmachines.lib" into the UserLib directory of Microsim Design Lab installation and configure Schematics to use this Library in the "Global" mode.It must be possible to load this library and run these example schematic in Orcad PSpice 9.1 (and up). But I haven't verified.

Download the Symbol and Model Library for DC Machines
Download the Schematic Files behind the models and some example schematics.