Model of a DC Generator Driving a DC Motor (which propels a car) John Hung 5 July 2011 The dc is connected to the dc as illustrated in Fig. 1. Both machines are of permanent magnet type, so their respective magnetic fields are constant and not adjustable. Torque T is applied to the shaft of the. The dc may have an external load torque T L. An electromechanical model is given in Fig. 2, and explained in the following sections. If the is being used to propel a car, then the mass and friction of the vehicle can be accounted by changing the moment of inertia J m and the friction coefficient B m. All other parameters will remain unchanged. The speed of the car can be modeled as being proportional to the speed of the. 1 Generator Dynamics The rotor has a moment of inertia J g, and linear friction coefficient B g. Generator speed is denoted by ω g, and voltage developed in the winngs is proportional to the speed. At the same time, the electrical load T T L Figure 1: Connection of a dc to a dc. 1
R g L g R m L m T g i B g J g ω g g ω g i V t m ω m m i T L B m J m ω m mechanical electrical electrical mechanical Figure 2: Electromechanical model of dc connected to a dc. produces a load torque on the. That load is proportional to current i in the winng. 1.1 Generator Mechanical Dynamics The dynamics of the speed are described by: T B g ω g g i = J g ω g. (1) The three terms on the left-hand side of (1) collectively describe the net torque that acts upon the shaft. The first term represents the externally applied torque from a prime mover such as a turbine or engine. The second term represents linear friction, which is proportional to speed. The third term models the torque induced upon the by the electrical load; the constant g is called a torque constant. The net torque causes the speed to change (accelerate or decelerate) accorng to Newton s Law. The right-hand side of (1) states that the angular acceleration ω g is proportional to the moment of inertia J g. The mechanical dynamics are first-order with respect to the shaft speed ω g, and are coupled to its electrical load through the variable i. 1.2 Generator Electrical Dynamics The voltage induced in the winng is given by g ω g. The proportionality constant g is called the voltage constant. Note that the same symbol g is used to denote the torque constant described earlier. The 2
torque constant and the voltage constant have the same numerical value under the SI system of units (i.e. the metric system). Therefore, a single symbol g is used here to represent both constants. If there is no electrical load on the, then the induced voltage appears at the terminals. If there is non-zero load current i, then some voltage is lost internally in the winng. The winng effects include resistance R g and inductance L g. The electrical dynamics of the are described by irchoff s voltage law: g ω g R g i L g dt = V t. (2) On the left-hand side, the first term represents the voltage induced within the. The second and third terms model the resistive and inductive effects, respectively. The net voltage equals V t, which represent the voltage appearing at the terminals. Generator electrical dynamics are first-order with respect to the winng current i, and are coupled to the mechanical dynamics through the speed ω g. 2 Motor Dynamics Voltage V t applied to the terminals produces a winng current. The develops torque proportional to the winng current. The rotor winng has resistance R m and inductance L m. Denote the mechanical moment of inertia by J m, and linear friction coefficient by B m, respectively. Motor speed is denoted by ω m. 2.1 Motor Electrical Dynamics The electrical dynamics of the are described by irchoff s voltage law: V t = R m i + L m dt + mω m. (3) Here, V t is the voltage applied to the terminals. On the right-hand side, the first two terms represent the resistive and inductive effects of the winng. The last term models the induced voltage of the winngs; the constant m is the voltage constant. The electrical dynamics are first-order with respect to the current i, and are coupled to the mechanical dynamics through the speed ω m. 3
2.2 Motor Mechanical Dynamics The dynamics of speed are described by: m i B m ω g T L = J m ω m. (4) The three terms on the left-hand side of (4) collectively describe the net torque that acts upon the shaft. The first term represents the electrically developed torque, which is proportional to winng current i. The torque constant is m (numerically the same value as the voltage constant, under SI units). The second term represents linear friction, which is proportional to speed. The third term models the external load torque. The net torque causes the speed to accelerate or decelerate accorng to Newton s Law. The right-hand side of (4) states that the angular acceleration ω m is proportional to the moment of inertia J m. The mechanical dynamics are first-order with respect to the shaft speed ω m, and are coupled to its electrical dynamics through the variable i. 3 Overall Model At first glance, it would appear that the and system are described by four coupled, first-order equations (1), (2), (3), and (4). The and are electrically connected, however, so two equations can be combined. 3.1 Coupling of Motor and Generator Electrical Dynamics Consider the connection of the and (Fig. 1), as well as the electromechanical model (Fig. 2.) The net voltage developed by the is the voltage applied to the terminals, denoted by V t. Adtionally, the and are connected in such a way that both winngs have the same current i. Therefore, equations (2) and (3) must be combined to yield a single electrical equation: g ω g R g i L g dt = R mi + L m dt + mω m, which can be rearranged in the form: g ω g m ω m = (R m + R g )i + (L m + L g ) dt. (5) 4
If the and are identical type machines, then R m = R g = R, L m = L g = L, and m = g =. Then the combined electrical equation reduces to: ( (ω g ω m ) = 2 Ri + L ). (6) dt In other words, the connected and results in electrical dynamics modeled by a single first-order equation in the variable i. The speeds of both the and affect their winng current. 3.2 The Final Model The overall model consists of three first-order equations: the speed dynamics (1), the and current dynamics (6), and the speed dynamics (4). Note that both speed ω g and speed ω m are coupled to their identical winng current i. 4 Simulink Model The and dynamics can be modeled by three blocks in SIMULIN, as illustrated in Fig. 3. Here, Transfer Fcn (transfer function) blocks are employed. The left-hand rectangular block represents the speed dynamics (1); the block input represents net torque and the output represents speed. The triangular block above it represents the torque introduced by the electrical load. Current dynamics (6) of the and are modeled by the center rectangular block; the input is the fference between speed and speed (ω g ω m ), and the block output is the winng current. Motor speed dynamics (4) are described by the right-hand rectangular block; the block input is current, and the output is speed. Transfer function block parameters can be specified in the MATLAB Workspace, or by constructing an m-file to define the parameters. Parameters that must be defined are listed in Table 1. 4.1 Model inputs Signal source blocks can be used to model the externally applied torque (at the ) T and the external load (at the ) T L. Recall that the speed block input is a current. Therefore, the s external load should be scaled by the torque constant (vide by ) so that the torque effect is expressed in units of current. 5
Table 1: SIMULIN model parameters symbol J g B g J m B m R L description moment of inertia friction coefficient moment of inertia friction coefficient winng resistance winng inductance torque or voltage constant 4.2 Model outputs The signals of interest can be stored in the MATLAB Workspace by using the To Workspace sink block. Values stored in the Workspace can be plotted and manipulated using MATLAB commands. Alternatively, the signals can be monitored in a live manner by using the Scope block. 4.3 An Electric Car If the is being used to move a car, then the mass and friction of the vehicle can be accounted by changing the moment of inertia J m and the friction coefficient B m. All other parameters will remain unchanged. The speed of the car can be modeled as being proportional to the speed of the. 6
Step Torque, T torque due to electrical load torque constant 1 Jg.s+Bg speed 2*Ls+2*R current Step1 winng current effect of external load on T_L / Jm.s+Bm speed Scope w_m To Workspace w_g To Workspace1 Figure 3: SIMULIN model of the and system. 7