Fundamentals of Electrical Drivess - Chapter 11 pdf

Chapter 11 ANALYSIS OF A SIMPLE DRIVE SYSTEM 11.1 Introduction In this chapter we will look at some basic drive implementations with a DC machine as discussed in section 10.5 starting at page 272. Our aim is to arrive at generic models of all major drive components, (excluding the DC machine which has already been discussed) which we can then transpose to a Simulink and/or Caspoc type environment in the tutorials at the end of this chapter. In the sequel of the chapter a so-called ‘predictive dead-beat’ current control algorithm will be presented [Svensson, 1988] which will allow precise torque control of the DC machine used. The techniques described here are fundamental not only to the DC machine but to all the machines discussed in this book. 11.2 Basic single phase uni-polar drive circuit An elementary drive model as shown in figure 11.1 has almost the same structure as the general drive model given in figure 1.2 on page 4. In this exam- Figure 11.1. Basic electrical drive 296 FUNDAMENTALS OF ELECTRICAL DRIVES ple the mechanical ‘load’ module has been removed. Furthermore, the power source is now shown in a two-wire configuration (+, −) which is helpful here because a symbolic implementation of the ‘converter’ module is shown. The DC motor is represented as an R-L-e network as was discussed in section 10.5.2 on page 276. The purpose of the modulator is to control the converter switch shown in figure 11.1 on the basis of a set-point given by the controller. A simple controller structure is also given in figure 11.1 which shows a micro-processor (µP or DSP), which is a digital computational element that implements the con- trol algorithm of the drive. The input to the control module is the load current i (t) which is obtained via a current sensor which measures the load current, i.e. the armature current of the DC machine in this case. A user input value i ∗ is also shown which represents the reference current level. The aim of this drive circuit is to control the current in the motor in such a way that the reference current value matches the actual load current under all circumstances, i.e. transient as well as in steady-state. A typical situation to be discussed is to apply a step change to the reference current and our aim is to ensure that the load current will match this step change, within the limits of the system. To achieve this aim we will need to initially discuss in some detail the functioning of the modules shown in figure 11.1. Afterwards we will develop a control structure which can be implemented in the micro-processor (µPor DSP) as to realize our task. 11.2.1 Power source A DC voltage source is assumed here which has a value of u DC . The bottom side is set to 0V which means that the upper wire (red) shown in figure 11.1 has a potential of u DC . The voltage source is ‘uni-polar’ which means that there is only one voltage level other than zero. At a later stage in this chapter we will replace the power source by a bipolar power source which gives us a positive and a negative supply voltage level with respect to 0V. The term ‘uni-polar drive’ reflects the ability to operate with a variable but single positive supply value. 11.2.2 Converter module The converter module shown in figure 11.1 consists of a single two-way switch. Inrealitysuchaswitch isformedby twoswitchesasshowninfigure 11.2 which also gives the power source module. The switches are controlled by two logic signals Sw t , Sw b where logic 1 corresponds to a ‘closed’ switch state and logic 0 to an ‘open’ switch state. In this case there are four possible switch combinations of switch states: Sw t =1, Sw b =1(both switches closed; ‘shoot-through’ mode, this state should always be avoided) and Sw t =0, Sw b =0(both switches open; ’idle’ mode, normally used to disable the inverter Analysis of a simple Drive System 297 Figure 11.2. Two switch converter with power source output), both are not considered in the following part. The remaining two states are: Sw t =1, Sw b =0(top switch closed/bottom switch open), Sw t =0, Sw b =1(top switch open/bottom switch closed). In the first case (Sw t =1, Sw b =0), the converter output is connected to the positive supply line, i.e. u = u DC , while in the second case Sw t =0, Sw b =1, the output line is connected to the lower supply line, i.e. u =0. The two switches can therefore in symbolic form be replaced by a single two-way switch as shown in figure 11.1, where the logic signal Sw is used to control its state. The state Sw =1 corresponds to the switch in the ‘up’ state, i.e. the output voltage is given as u = u DC . As expected, the switch state Sw =0corresponds to the switch in the ‘down’ state, i.e. the output voltage is given as u =0. 11.2.3 Controller module Today, the controller is in most cases digital. This means that the analog input variables, here in the form of the measured current i (t) and user defined reference currenti ∗ (t), need to be convertedtoadigital form. We havetherefore introduced in figure 11.1 a new building block in the form of an ‘analog-digital’ (A/D) converter. The function of the unit is readily shown with the aid of figure 11.3: an input function x (t) to the A/D converter. The diagram shows an example waveform together with a set of discrete time points t k−1 , t k , t k+1 where k can be any integer value. The difference in time between any two time points is constant and equal to the ‘sampling interval period’ T s . For drive systems the sampling time is in the order of 100µs, 1ms. The output of the converter module is such that at these time points the input is ‘sampled’ i.e. the output is then set to equal the input value. Hence, at these ‘sampling points’ the output changes to match the instantaneous value found at the input of the converter. The output is therefore held constant during the sampling time. For example, the output x (t k )) represents the value of the input variable as sampled at the time mark t k . 298 FUNDAMENTALS OF ELECTRICAL DRIVES Figure 11.3. A/D converter unit with example input/output waveforms The A/D units are used to sample the measured current and reference current values. These inputs, at for example t k , are then used by the micro-processor or DSP to calculate an output variable known as the ‘reference incremental flux’ ∆Ψ ∗ (t k ), which acts as an input to the modulator. We will define the variable ∆Ψ ∗ (t k ) in the next section. 11.2.4 Modulator module The basic task of the modulator module is to control the switch or switches of the converter module in such a way that the condition according to equa- tion (11.1) is met (within the constraint of this unit) for each sampling interval. ∆Ψ ∗ (t k )=∆Ψ(t k ) (11.1) where ∆Ψ (t k ) is known as the incremental flux level which is defined as ∆Ψ (t k )=  t k+1 t k u (τ ) dτ (11.2) The term u (t) shown in equation (11.2) represents the instantaneous voltage across the load (output from the converter) within a sample period, in this case between sample points t k ,t k+1 . We have in the past (see equation (2.7)) commented on the fact that it is the incremental flux which controls the current in an inductance. The inductance forms a key element for our machine models and it is therefore appropriate to work with the ‘incremental flux’ as a control variable [Svensson, 1988]. Condition (11.1) in fact states that the modulator should set the converter switches during each sampling interval in such a way as to ensure that the reference incremental flux value at, for example, time t k (as provided by the controller) matches the converter incremental flux value (as defined by equation (11.2)). We will now consider two basic ‘single edged’ modulation strategies by examining the converter incremental flux as a function of the switch on/off Analysis of a simple Drive System 299 Figure 11.4. Incremental flux and output wave forms: rising edge modulation time within a sample interval t k t k+1 . We will in the first instance make use of the converter configuration as shown in figure 11.1. The so-called ‘rising edge’ type modulation strategy calls for the switch Sw to be placed in the ‘up’ (switch logical control level 1) position after a time t r measured from the start of the sampling interval. The switch is placed in the ‘down’ (switch logical control level 0) position at the end of each sampling interval. An example of the output voltage waveform which appears as a re- sult of this modulation strategy is given in figure 11.4 for the sampling interval t k t k+1 . Also shown in figure 11.4 is the incremental flux value as a func- tion of the rise time t r . This variable can change between zero and T s .For a particular value t r we can evaluate the incremental flux by making use of equation 11.2 which in this case gives the function ∆Ψ (t r )=u DC (T s − t r ), which is illustrated in figure 11.4. It is noted that this function repeats each sample interval and this gives us the possibility to find the required t r value for a given incremental flux reference value. The basic algorithm for finding the rise time t r is based on the use of equa- tion (11.1). Basically, we compare for each sample interval the required refer- ence value with the incremental flux function ∆Ψ (t r ) and move the converter switch to the ‘up’ position when the condition ∆Ψ ∗ ≥ ∆Ψ (t r ) is met. An example as given in figure 11.5, shows two consecutive sampling intervals where the reference incremental flux levels (as provided by the controller) are takentobe∆Ψ ∗ (t k−1 ) and ∆Ψ ∗ (t k ) respectively. The switching point for 300 FUNDAMENTALS OF ELECTRICAL DRIVES Figure 11.5. Switch algo- rithm for rising edge modula- tion the converter (which corresponds to the required t r value) is identified by com- parison with the ∆Ψ (t r ) function for each sample interval. We note that the converter will provide the correct incremental flux value which means that the modulator will achieve its aim. A further two observations of figure 11.5 are of interest. Firstly, this modula- tor-converter combination will provide an incremental flux value between zero and u DC T s , which is why this converter topology is ‘uni-polar’. Secondly, the required incremental flux value produced by the converter is realized by adjusting the width of the output voltage pulse for each sample interval. This is why this modulation strategy is known as ‘pulse width modulation’ (PWM). So far, we have discussed a ‘single rising edge’ PWM scheme which operates with a uni-polar converter. The so-called ‘falling edge’ type PWM modulation strategy calls for the switch Sw to be placed in the ‘up’ (switch logical control level 1) position at the start of each sampling interval and moved to its ‘down’ (switch logical control level0) positionaftera time t f (measured fromthestart ofthesampling interval). An example of the output voltage waveform which appears as a result of this modulation strategy is given in figure 11.6 for the sampling interval t k t k+1 . Also shown in figure 11.6 is the incremental flux value as a function of the fall time t f . This variable can change between zero and T s . For a particular value t f we can evaluate the incremental flux by making use of equation (11.2), which in this case gives the function ∆Ψ (t r )=u DC t f which is also illustrated in figure 11.6. Analysis of a simple Drive System 301 Figure 11.6. Incremental flux and output wave forms: falling edge modulation The algorithm for finding the time t f is again based on the use of equa- tion (11.1). Basically, we compare for each sample interval the required refer- ence value with the incremental flux function ∆Ψ(t f ) and move the converter switch to the ‘down’ position when the condition ∆Ψ ∗ < ∆Ψ(t f ) is met. How this is achieved is illustrated in figure 11.7 on page 302, which shows two consecutive sampling intervals where the reference incremental flux levels (as provided by the controller) are taken to be ∆Ψ ∗ (t k−1 ) and ∆Ψ ∗ (t k ) respec- tively. The switching point for the converter (which corresponds to the required t f value) is identified by comparison with the ∆Ψ(t f ) function for each sample interval. We note that the converter will provide the correct incremental flux value which means that the modulator will achieve its aim. Note that in this case we have discussed a ‘single falling edge’ PWM scheme which operates with a uni-polar converter. A generic implementation of a ‘falling edge’ PWM strategy is given in fig- ure 11.8 on page 302. Shown in figure 11.8 are two A/D modules which take the incremental reference flux value (from the controller) and the measured u DC value from the converter module. We in fact use this value as to allow us to ad- just the converter switch or switches as to accommodate voltage changes as will be discussed shortly. The sampled DC voltage is multiplied by a gain T s which gives us the maximum sampled incremental flux value u DC T s and this value is multiplied by a ‘saw tooth’ function which is in fact the ∆Ψ(t f ) function with a maximum value set to 1. We could have implemented this function directly 302 FUNDAMENTALS OF ELECTRICAL DRIVES Figure 11.7. Switch algo- rithm for falling edge modu- lation Figure 11.8. Generic model of falling edge PWM in the ‘function generator’ module shown. However, the chosen implementa- tion allows us to take into consideration changes to the supply voltage u DC .A summation module compares the two incremental flux values and its output ε is used by a so-called comparator module. This is a new addition to our building block library and its function is given by equation (11.3). if ε>0 comparator output = 1 (11.3a) if ε ≤ 0 comparator output = 0 (11.3b) In our case the output of the comparator is known as Sw and drives the converter switch. A logical level Sw =1moves the converter switch to the ‘up’ position and Sw =0sets it to the ‘down’ position. It is left to the reader to consider Analysis of a simple Drive System 303 Figure 11.9. Falling edge PWM, with change of supply voltage how figure 11.8 should be changed when a rising edge PWM strategy is to be implemented. The modulator generic structure as given by figure 11.8 was specifically cho- sen to allow changes to the supply voltage which (within limits) will not affect the ability of the modulator/converter combination to meet condition (11.1). With the aid of figure 11.9, we will show how the modulator/converter combi- nation is able to cope with a change in supply voltage. In this example, the incremental flux reference ∆Ψ ∗ is held constant. The supply voltage has been changed with time and this is reflected by the sampled DC bus voltage, which in the second sample is arbitrarily taken to be lower than in the first sample. The immediate effect is that the incremental flux function ∆Ψ(t f ) in the second sample will have a lower gradient than in the first. The consequence of this is that the fall time for the second sample is increased. Furthermore, the output waveform voltage in the second sample has reduced (because the converter sup- 304 FUNDAMENTALS OF ELECTRICAL DRIVES Figure 11.10. Double edged PWM strategy ply was lowered). What in fact has occurred is that the modulator has increased the fall time in the second sample to offset the reduced output voltage level of the converter. This means that the incremental flux value delivered to the load remains unaffected by the supply voltage change as long as the supply voltage is of sufficient magnitude. At the conclusion of the discussion on single edged PWM we will look at an important modulation strategy known as ‘double edged’ PWM. This modu- lation strategy combines the rising and falling edged PWM strategies into one. Basically this new modulation strategy alternates between the two single edged PWM options every sample. For example, in the first sample we use rising edge PWM and in the next we use falling edge PWM. How this operates in practice is shown in figure 11.10. Two sample intervals are shown and in the first sample interval a rising edged PWM strategy (as given in figure 11.5) is shown. In the second sample a falling edge strategy is shown (as given ear- lier in figure 11.7). The incremental flux reference for the second sample was arbitrarily set to a lower value than in the first sample. The generic module according to figure 11.8 is readily modified given that the ‘saw tooth’ generator is now replaced by a triangular function (of unity amplitude). Several important observations are to be made with respect to this modulation strategy. Firstly, the fundamental frequencyf mod of thenewmodulatoris nowchanged from 1/T s to 1/(2T s ), i.e. it is halved. Secondly, each so-called modulator period or carrier-wave period 1/f mod consists of two samples. This type of [...]... (see figure 11. 25) in order to find the required control output which is the reference incremental reference flux value When building this module define the gains ‘kp’ and ‘ki’ by way of the parameters L, R and Ts as shown in equation (11. 15) The m-file given at the end of this tutorial is used to generate the results given in figure 11. 26 320 FUNDAMENTALS OF ELECTRICAL DRIVES Sub-plot(a) of figure 11. 26 shows... for T this ‘discrete’ controller are set to L + R2 s and R Ts respectively 310 FUNDAMENTALS OF ELECTRICAL DRIVES Figure 11. 14 Predictive dead-beat controller 11. 5 Tutorials for Chapter 11 11.5.1 Tutorial 1 This tutorial considers a simple example which consists of a modulator, unipolar converter and a load in the form of an ideal inductance A single ‘rising’ edge type modulator will be considered... equation (11. 13) is basically a so-called proportional type controller In practice, a so-called proportional-integral type structure is preferable and this may be achieved by rewriting the term R Ts i (tk ) as R Ts i (tk ) ∼ R Ts = j=k−1 (i∗ (tj ) − i (tj )) (11. 14) j=0 which means that the current i (tk ) is composed of a series of difference terms as can be observed from figure 11. 13 Use of equation (11. 14)... value at the end of each interval How this is achieved is shown in figure 11. 13, where the reference and actual current are shown for a sequence of sampling intervals from t = t0 to t = t4 with i (t0 ) = 0 The modulator/converter module will insure that the following condition is satisfied ∆Ψ∗ (tk ) = tk+1 u (τ ) dτ tk (11. 5) 308 FUNDAMENTALS OF ELECTRICAL DRIVES Figure 11. 13 Predictive dead-beat control... (τ ) dτ (11. 7) tk Equation (11. 7) forms the basis for determining a generic control structure that is able to calculate the required incremental inverter flux quantity capable of satisfying condition (11. 4) It is noted that this set-point value can only be determined on the basis of a detailed knowledge of the load parameters R, L For a discrete (use of sampled data for processing via a micro-processor)... modulator flux waveform amplitude is also reduced, as can be observed by comparing figures 11. 17(b) and 11. 18(b) This in turn implies that the width of the output pulses will be increased given the same sampled incremental flux reference m-file Tutorial 1, chapter 11 %Tutorial 1, chapter 11 %inductive load 100 mH close all Ts=1e-3; subplot(3,1,1) plot(datout(:,6), datout(:,1)); hold on plot(datout(:,6), datout(:,5),’r’);... which upon use of (11. 4) gives R Ts ∗ (i (tk ) + i (tk )) + L (i∗ (tk ) − i (tk )) + e (tk ) Ts (11. 11) ∆Ψ∗ (tk ) ∼ = 2 A further simplification is possible by rewriting the resistive term in equation (11. 11) as R Ts ∗ R Ts ∗ (i (tk ) + i (tk )) = (i (tk ) − i (tk )) + R Ts i (tk ) 2 2 (11. 12) in which case the set-point incremental flux value is given as R Ts (i∗ (tk ) − i (tk )) + e (tk ) Ts (11. 13) ∆Ψ∗... Figure 11. 26 Simulink results: Modulator/converter with R/L load m-file Tutorial 4, chapter 11 %%Tutorial 4, chapter 11 close all Ts=1e-3; R=10; L=200e-3; subplot(3,1,1) plot(datout(:,6), datout(:,1),’r’); hold on plot(datout(:,6), datout(:,3),’b’); grid ylabel(’ current (A)’); xlabel(’ (a) time (s)’) % sample time % load resistance % load inductance % reference current % load current 321 Analysis of a... Workspace Clock Figure 11. 15 Simulink model of converter with single edged PWM 311 Analysis of a simple Drive System 2 Udc Sum3 u_DC Udc 1 1 In1 u 0 Phase a u Udc1 In1 u_DC Uni−polar converter Figure 11. 16 Simulink model of uni-polar converter this module, set the sampling time to Ts Define Ts = 10−3 s in the MATLAB Workspace prior to running your simulation Simulink then ‘knows’ this value of Ts for any block... (found in the non-linear library) is used Open the dialog box for this unit and enter the following settings: ‘switch on point:’ eps, ‘switch off point:’ eps, ‘output when on:’ 1, ‘output when off:’ 0 The output of the comparator module is led to the ‘converter’ sub-module which can, for example, be of the form given in figure 11. 16.‘ The ‘converter’ module has a ‘switch’ (found in the non-linear library) . RT s respectively. 310 FUNDAMENTALS OF ELECTRICAL DRIVES Figure 11. 14. Predictive dead-beat controller 11. 5 Tutorials for Chapter 11 11.5.1 Tutorial 1 This tutorial considers a simple example which consists of a. (τ ) dτ (11. 5) 308 FUNDAMENTALS OF ELECTRICAL DRIVES Figure 11. 13. Predictive dead-beat control where u represents the voltage across the load (see figure 11. 1). An observation of figure 11. 1 learns. implemented this function directly 302 FUNDAMENTALS OF ELECTRICAL DRIVES Figure 11. 7. Switch algo- rithm for falling edge modu- lation Figure 11. 8. Generic model of falling edge PWM in the ‘function