Fundamentals of Electrical Drivess - Chapter 10 pot

Chapter 10 DIRECT CURRENT MACHINES 10.1 Introduction The genesis of the ideas required to build an electrical machine can be traced back to the discovery of electromagnetism by the Danish scientist Oersted in 1819-20. Oersted discovered that a current in a wire could deflect a compass needle. Thus the connection between a current carrying conductor, a magnetic field and mechanical movement was established. A German chemist named Schweigger, who studied Oersted’s experiment, found that if the wire carrying the current was wound into a coil then the deflection of the magnet was greatly increased. The Professor of Chemistry at Cambridge, Cumming, coined the term ‘Galvanometer’ for this configuration and used it as a current detector. At around the same time Amp ` ere developed a theory to support the observations made about current carrying coils of wire. In 1825 Sturgeon found that putting an iron core in the coil increased the magnetic field strength considerably for the same current. Meanwhile, a laboratory assistant by the name of Faraday, working for the Royal Institution in England, developed what could be called the first electrical motor in September 1821. It was a crude device that caused a wire suspended in a basin of mercury, with a vertical magnet, to rotate around the magnet when current flowed through the wire and mercury. It produced continuous motion. In 1837 Davenport filed a patent for an electromagnetic engine. It incorpo- rated a crude commutator and a multi-pole stator with electromagnets rather than permanent magnets. He could use this machine to drill 6mm holes in wood and steel. Gramme introduced a ring type rotor configuration in 1870, which in turn led to a rotor concept introduced by Von Hefner in 1873, which is still in use today. 266 FUNDAMENTALS OF ELECTRICAL DRIVES The direct current (DC) machine is still in use given that it has certain ad- vantages in terms of controllability and low manufacturing costs. In household appliances and automotive applications, series DC machines or universal ma- chines (these run on both DC or AC) are often used for the reasons mentioned above. However, the development of adjustable speed drives has demonstrated that the same type of performance can also be achieved with for example an asynchronous machine in combination with a power electronic converter. Asyn- chronous machines are considerably cheaper to manufacture at most power levels and very much easier to maintain than a DC machine. In this chapter we will consider the use of the generalized IRTF based model for representing the DC type of machine. It will be shown that the IRTF concept is particularly useful for modelling this type of machine, despite the fact that the DC machine configuration is decidedly different (in terms of actual physical structure) than the generalized IRTF based machine concept (see figure 7.13 on page 183). Particular care should therefore be taken when it comes to interpreting the variables which appear in the IRTF model, against those which exist in the actual machine. Two dynamic IRTF based models will be presented, which make use of a new simulation component that is able to represent the brush/commutator assembly, which is a key component of the DC machine. In the sequel to this chapter the steady-state behaviour of this type of machine is discussed. 10.2 Machine configuration A simple two-pole stator of a DC machine is shown in figure 10.1. The field winding consists of an N turn concentrated winding of which each half is wrapped around a pole. The two parts of the winding are connected in series and attached to a DC power supply. The frame is in this case also the Figure 10.1. DC machine stator Direct Current Machines 267 yoke-part of the magnetic flux path. The use of a field winding gives us the ability to control the magnetic flux in the circuit in terms of amplitude and polarity. Permanent magnets are often used to replace the field winding which leads to a more compact and efficient machine. However, we loose in practical terms one degree of freedom, as we are now unable to control (in electrical terms) the flux magnitude during operation. We also loose the potential of operating the machine on an AC source, therefore universal machines (AC/DC) always have a field winding and no permanent magnets. A very simple example of a DC rotor is given in figure 10.2, which shows the same single turn winding introduced for the synchronous machine. In this case, the slipring/brush combination is replaced by a so-called commutator. This commutator consists, for this simple rotor, of two brushes and two commutator segments. Segment 1 and 2 are connected to coil side A and B respectively. The Figure 10.2. Highly simpli- fied DC machine rotor brushes are connected to a direct current (DC) power supply. The purpose of the commutator is to reverse the current polarity in the rotor winding every half revolution in this case. For example, in the case shown, coil side A is connected via segment 1 and a brush to terminal 1. Likewise side B is connected via segment 2 and a brush to terminal 2. When we rotate the rotor by 180 degrees, side A will be electrically connected to terminal 2 and side B to terminal 1. This means that a current reversal in the winding will take place twice during one period. In reality the number of segments is greater than 2 and this has a marked effect on the torque ripple as will become apparent from the tutorial at the end of this chapter. 10.3 Operating principles The operating principles are discussed on the basis of the IRTF model pre- sented in section 7.3 on page 178. A non-rotating magnetizing flux vector  ψ m = ˆ ψ m is in this case present, hence ω s =0. The current source shown in figure 7.9 on page 179 must (according to equation (7.21)) generate a current 268 FUNDAMENTALS OF ELECTRICAL DRIVES vector  i xy = ˆ ie j(ω r t+ρ r ) which is stationary and orthogonal to the flux vector. This implies that the angular frequency ω r must be set to ω r = −ω m , while the angle ρ r can be set to π/2, if we assume that the rotor angle is given as θ = ω m t. The machine is fed from the rotor side with a constant current ˆ i, which means that a brush/commutator assembly is required to realize the counter rotation (with respect to the rotational direction of the shaft) of the current vector  i xy . The current vector in stator coordinates is reduced to  i = j ˆ i. The torque ob- tained with this machine is found by use of equation (7.22), with ρ m =0, ρ r = π 2 rad which leads to T e = ˆ ψ m ˆ i (10.1) Equation (10.1) shows that the torque produced by the ideal DC machine can be independently controlled by the flux amplitude on the stator side and the DC current amplitude ˆ i on the rotor side. The disadvantage of this type of machine is the need for a brush/commutator assembly. The space vector type presentation as discussed in section 7.3 on page 178, is for this type of machine given by figure 10.3. Figure 10.3. Space vector diagram for DC machine 10.4 Symbolic model, simplified form The generalized model as discussed in section 7.4 on page 183 is used as a basis for the development of a simplified DC machine model. The leakage inductance, magnetizing inductance, rotor and stator resistance are initially ignored, which reduces the symbolic model according to figure 7.15 on page 185 to the configuration shown in figure 10.4. A non-rotating magnetizing vector  ψ M is assumed, the magnitude of which is taken to be quasi-stationary. This means that variations of the flux value are taken to be slow in comparison with Direct Current Machines 269 Figure 10.4. Symbolicrepresentation ofa DCmachine connectedtoa currentsource, simplified version the electrical time constants of the machine, which is a realistic assumption. A DC voltage source is connected to the stator to provide the flux  ψ M . In this case we need a constant flux level, hence the voltage source only needs to deliver a pulse (as shown in figure 10.4). A DC current source with magnitude i a is connected to the rotor side of the IRTF via a brush/commutator assembly shown in figure 10.4. A new ‘armature’ current vector  i xy A is introduced, which according to figure 10.4 is defined as  i xy A = −  i xy R (10.2) The subscript A for ‘armature’ is introduced, given that the rotor of this type of machine is referred to as the armature. Note that figure 10.4 also shows the stator based rotor current vector  i R , with components i Rα , i Rβ . The compo- nent i Rα will be present in case a finite magnetizing inductance is introduced. Furthermore, a component i Rβ is present in the IRTF based model which is the current that would appear in the so-called ‘compensation winding’ of the machine. The basic machine model discussed in section 10.2 does not carry a compensation winding, hence the current component i Rβ is not found in that machine configuration. The term armature refers to that section of the machine which is primarily responsible for the energy conversion process. For the DC machine the armature is the rotor. The purpose of the brush/commutator shown in figure 10.4 ensures, as was mentioned earlier, that the current vector  i xy A ro- tates in a direction opposite to the rotor. The speed of rotation (with respect to the rotor based complex plane) is the same as the rotor speed, hence the current vector will be stationary when viewed from the stator. Furthermore, we need to orientate this vector perpendicular with respect to the flux vector to maximize the torque output of the machine. Given the above the ‘ideal’ commutator needs to perform the transformation as shown by  i xy A = ji a e −jθ (10.3) 270 FUNDAMENTALS OF ELECTRICAL DRIVES The net result of this transformation is that the current vector as viewed from a stationary reference frame, i.e. the stator, is indeed stationary and of the form  i A = ji a . Further observation of figure 10.4 shows that the commutator has a second variable namely the motional voltage e a , which for this model is an ‘output’ caused by the presence of a rotating (when viewed from the rotor) flux vector  ψ xy M . This rotating flux vector corresponds to a voltage vector e xy M = d  ψ xy M dt . The imaginary component of this vector when transformed by the commutator represents the motional voltage e a =   e xy M e jθ  . The equation set which corresponds with the symbolic model given by figure 10.4 is as follows  ψ xy M =  ψ M e −jθ (10.4a)  i xy A = ji a e −jθ (10.4b) e xy M = d  ψ xy M dt (10.4c) e a =   e xy M e jθ  (10.4d) T e = −   ψ ∗ M  i A  (10.4e) The magnetizing vector  ψ M = −ψ f is introduced at this stage, which with the aid of equation (10.4e) allows the torque to be written as T e = i a ψ f .The reason for choosing a negative real flux vector  ψ M is to ensure that a positive armature current i a gives a positive torque value. 10.4.1 Generic model The generic model which corresponds to the symbolic model of figure 10.4 and equation set (10.4) (given without the mechanical equations) is shown in figure 10.5. The IRTF module normally calculates the torque on the basis of the flux vector  ψ M and current vector  i R . In this case the armature current vector is used, which implies that the torque is calculated using equation (10.4e). Hence, a gain module with gain −1 must be inserted in the torque output. Alternatively, a −1 gain module could have been placed in line with the current input, as to allow this module to use the vector  i R instead of  i A . It is again noted that the IRTF has a stator based armature current vector  i A of which the real component will be zero (given the chosen orientation of this vector, i.e. perpendicular to the flux vector) and the imaginary component will be equal to i a . The generic implementation of the commutator function as given in fig- ure 10.6, shows the two conversion modules required to realize the conversion from rotor to stator coordinates (to provide the variable e a ) and vice versa (to implement equation (10.3)). Direct Current Machines 271 Figure 10.5. Generic representation of a DC machine connected to a current source, simplified version Figure 10.6. Ideal Commu- tator/brush assembly It may be argued that a simpler generic implementation of the DC machine can be realized without the IRTF module, i.e. use of equation (10.1). This is indeed the case with the ‘ideal’ commutator. However, the use of an IRTF mod- ule is advantageous given the possibilities of extending the machine model to include for example a compensation winding to counter the so-called ‘armature reaction’ of the machine. Furthermore, the torque ripple which will occur in the machine due to the presence of a finite set of commutator segments in the brush/commutator unit can be modelled. A detailed analysis of such effects 272 FUNDAMENTALS OF ELECTRICAL DRIVES is not within the context of this book. However, in the tutorial at the end of this chapter an example is given, which shows how the instantaneous torque is effected by the use of an IRTF based model with an ideal and non-ideal (4 commutator segments) commutator. The generic model in its present form contains a differentiator module which should be avoided given that the inclusion of modules of this type can cause simulation problems. In the tutorial at the end of this chapter an alternative implementation of the differentiator module is used (see figure 6.8 on page 155) which avoids simulation problems. 10.5 General symbolic DC machine model The DC model in its simple form as discussed above, assumes the presence of a current source on the input of the commutor/brush assembly, hence a rotor current vector  i xy A is imposed, which together with a constant stator flux leads to a constant torque level. This concept is broadened in this section, given the fact that the armature of the machine is usually connected to a voltage source via the brush/commutator assembly. The term ‘general model’ refers to a model which can be used in a wider context, i.e. with voltage excitation. For modelling this type of machine it is helpful to use the ‘two inductance’ model as discussed in section 3.7.2 on page 59, where the leakage inductance is positioned on the rotor side and the ITF module is replaced by an IRTF module. This is in fact the only time when we consider this alternative model form. The resultant symbolic model of the machine with brush/commutator assembly is given in figure 10.7. The magnetizing inductance L 1 is in this type of machine Figure 10.7. Symbolic representation of general DC machine finite, with value L f , in the the real (α axis) and zero in the imaginary (β axis) direction, given the nature of the magnetic circuit in this type of machine. Hence the inductance L 1 is represented in matrix format namely L 1 =  L f 0  .A current source i F is now connected to the stator side, which in combination with the magnetizing inductance component L f provides the magnetizing flux of this machine. Direct Current Machines 273 A DC voltage source u a connected to the rotor via the brush/commutator assembly must convert this variable to a vector in a similar manner as shown for the simplified model (see figure 10.6) for the current i a . The conversion which an ideal commutator must realize is of the form given by u xy R = ju a e −jθ (10.5) Note that equation (10.5) is almost identical to equation (10.3), the difference being that we have replaced the current with a voltage variable as required. The inclusion of the ‘j’ operator in the right hand side of equation (10.5) is to ensure that the resultant current vector will be in a direction which is perpendicular to the flux vector provided by the stator side, i.e. to maximize the torque output of the machine. From the symbolic model we can deduce the equation set (10.6) for the machine in its present form.  ψ M = L f i f  ψ f +L f i Aα (10.6a)  ψ xy R =  ψ xy M + L  σ  i xy A (10.6b) e xy R = d  ψ xy R dt (10.6c) u xy R =  i xy A R R + e xy R (10.6d) i a =    i xy A e jθ  (10.6e) T e = −   ψ M  ∗  i A  (10.6f) The scalar variable ψ f is introduced in equation (10.6a), which represents, as was mentioned earlier, the excitation flux that can be provided by a field (stator based) winding or permanent magnet. Also shown in this equation is the term L f i Aα which will be set to zero by the IRTF based DC machine model, as will be discussed at a later stage. Under these conditions the magnetizing flux vector will be equal to  ψ M = ψ f . The leakage inductance L  σ represents the so-called armature inductance of the machine and is henceforth referred to as L a . The rotor resistance R R is for DC machines also renamed as the armature resistance R a . The armature current is in this case an output variable and is given by equa- tion (10.6e). The conversion process used to find this variable is similar to that described for the simplified model and the conversion from e xy M → e a as shown by figure 10.6. 274 FUNDAMENTALS OF ELECTRICAL DRIVES 10.5.1 Generic model The generic model, which corresponds with the symbolic model given in figure 10.7, can be developed using equation set (10.6). The generic model as given in figure 10.8 shows the IRTF module, which now uses the field flux ψ f provided by the stator side and the flux vector generated by the rotor circuit. These vectors are used together with the armature inductance L a to generate the current vector  i xy A . Not shown in this generic diagram is the sub-module which is required to maintain the condition i Aα =0in the model during dynamic and steady-state operation. No differentiator module is required. The modules linked to the torque and load side of the machine are unchanged with respect to the simple model given in figure 10.4. Figure 10.8. Generic representation of a DC machine connected to a voltage source 10.5.2 Alternative model representation The DC machine in its current form, i.e. with an ideal commutator, quasi stationary field flux excitation ψ f = − ˆ ψ f and operating at speed ω m , can also be represented in a more simplified form. This alternative symbolic model will be developed here. In addition the measures which need to be taken to realize the condition i Aα =0will also be discussed in this section. [...]... voltages 290 FUNDAMENTALS OF ELECTRICAL DRIVES Table 10. 3 Steady-state results Simulink model with variable armature voltage T ref Te (Nm) (Nm) 8 6 4 2 0 8.00 6.00 4.00 2.00 0.00 Ua =110V nm (rpm) Ua = 110* 1.2V Ua = 110* 0.8V 286.57 477.46 668.44 859.42 105 0.41 496.55 687.52 878.52 106 9.51 1260.49 76.39 267.38 458.36 649.34 840.33 namely: ua = 110 ∗ 1.2V and ua = 110 ∗ 0.8V An example of the steady-state results... + ψ f ωm dt (10. 12) ea Expression (10. 12) corresponds to the symbolic representation of the DC machine as given by figure 10. 9 Observation of figure 10. 9 learns that the rotor circuit of the machine can be represented by a series network in the form of an armature resistance Ra , inductance La and speed dependent voltage source ˆ ea = ωm ψf 276 FUNDAMENTALS OF ELECTRICAL DRIVES Figure 10. 9 Alternative... equation (10. 6d), which can with the aid of equations (10. 6c), (10. 6b) be written as xy xy uR = iA Ra + La xy xy dψM diA + dt dt (10. 7) Expression (10. 7) may also be written in its stator coordinate based form with the aid of the transformation x xy = xe−jθ , where θ = ωm t The resultant equation is of the form uR = iA Ra + La dψM diA − jωm La iA + − jωm ψM dt dt (10. 8) Substitution of equation (10. 6a)... −2 −4 −6 0 Figure 10. 22 200 400 600 800 speed (rpm) 100 0 1200 1400 1600 MATLAB results: torque/speed curves, varying armature voltage Direct Current Machines 291 analysis as indicated by figure 10. 10 An example of an m-file which can produce the results shown in figure 10. 22 is as follows m-file Tutorial 6, part 1 chapter 10 %Tutorial 6, part 1, chapter 10 close all clear all %results from Simulink model... nm1=[286.57 477.46 668.44 859.42 105 0.41]; %Ua= 110 nm2=[496.55 687.52 878.52 106 9.51 1260.49];%Ua=1.2* 110 nm3=[76.39 267.38 458.36 649.34 840.33]; %Ua=0.8* 110 %steady-state analysis DC machine %machine parameters Ra =10; % armature resistance La=0.05; %armature inductance psif=1.0; % peak flux value %%%%%%%%%%%%%%%%%%%%%%%%%%5 ua=[ 110 110* 1.2 110* 0.8]; % armature voltages ns=[0 :100 :1600]; % speed points for... 10. 23 reinforce the quantitative analysis given in section 10. 6 and figure 10. 11 in particular The m-file which corresponds to this part of the tutorial is as follows 292 FUNDAMENTALS OF ELECTRICAL DRIVES Table 10. 4 T ref (Nm) 8 6 4 2 0 Steady-state results Simulink model with variable field flux Te (Nm) 8.00 6.00 4.00 2.00 0.00 |ψf |=1.0Wb nm (rpm) |ψf |=0.8Wb |ψf |=1.2Wb 286.57 477.46 668.44 859.42 105 0.41... 1) The load module settings are set to 288 FUNDAMENTALS OF ELECTRICAL DRIVES produce a load torque of 8Nm at 300rpm Electrically the machine is to be connected to a DC voltage supply source of 110V at t=0 Add a set of display modules as indicated in figure 10. 20, so that we are able to view the shaft speed (in rpm), shaft torque and armature current at the end of the simulation Under ‘simulation parameters’... 477.46 668.44 859.42 105 0.41 119.14 417.25 715.35 101 3.45 1311.56 344.84 477.46 610. 09 742.72 875.35 12 psif=1.0*0.8 psif=1.0 psif=1.0*1.2 10 8 torque (Nm) 6 4 2 0 −2 −4 −6 0 200 Figure 10. 23 400 600 800 speed (rpm) 100 0 1200 1400 1600 MATLAB results: torque/speed curves, varying field flux m-file Tutorial 6, part 2 chapter 10 %Tutorial 6, part 2, chapter 10 close all clear all %results from Simulink model... torque value) and use of the condition iAα = 0 allows equation (10. 9b) to be written as uRβ = iAβ Ra + La diAβ ˆ + ωm ψ f dt (10. 11) Expression (10. 11) can be written in its more traditional notation form by making use of equations (10. 5) and (10. 6e) which in effect state that the variables uRβ , iAβ are normally referred to as ua and ia respectively Use of the new variables in equation (10. 11) gives ua... (rpm), ia (A) and pout (W) An example of the results which should appear with the chosen armature voltage, field flux and load module settings, is given in figure 10. 21 The m-file shown below can be used to display the results from the simulation m-file Tutorial 5, chapter 10 %Tutorial 5, chapter 10 %calculates plots for DC motor simulation %%%%%%% figure(1) clf tp=0.1;%length of time for plots subplot(2,2,1) . non-zero torque value. Figure 10. 12. Steady-state torque speed curves with series connected field winding Direct Current Machines 279 10. 7 Tutorials for Chapter 10 10.7.1 Tutorial 1 The aim of. in figure 10. 16. An observation of figure 10. 16 clearly shows the impact of using a segmented commutator. Increase the number of segments (even number of course) rerun the simula- tion and m-file and. ω m ˆ ψ f . 276 FUNDAMENTALS OF ELECTRICAL DRIVES Figure 10. 9. Alternative DC machine model with ideal commutator 10. 6 Steady-state characteristics It is instructive to consider the steady-state characteristics