Torque Maximization for Permanent Magnet Synchronous Motors Marc Bodson
Alexander Verl
Department of Electrical Engineering University of Utah Salt Lake City, UT 84112, U.S.A. Tel.: 1-801-581-8590 Fax: 1-801-581-5281 E-mail:
[email protected]
Institute for Robotics and System Dynamics German Aerospace Research Establishment (DLR) D-82234, Wessling, Germany Tel.: 49-8153-282479 Fax: 49-8153-281441 E-mail:
[email protected]
Abstract: The paper discusses the problem of maxi-
speed usually varies suciently slowly that the formulas are useful [1]. At low speeds, the current limits restrict the torque available, and torque maximization can be achieved by aligning the currents with the direction perpendicular to the permanent magnet. Interestingly, however, one nds that torque optimization throughout the speed range can involve an intermediate region where both the voltage and the current limits are active and the low speed and high speed solutions are not applicable. The objective of this paper is to solve the optimization problem in all three speed ranges, and to calculate the location of the transition points between the regions. Paper [2] also partly addresses the same problem. But since the derivations are based on a geometric approach, the stator resistance had to be neglected. In addition to the non-zero resistances of the stator windings, this paper provides explicit, analytic formulas for operation in voltage and current controlled modes which are suitable for realtime implementation.
mizing the torque of permanent magnet synchronous motors in the presence of voltage and current constraints. Formulas are given that are suitable to operation with voltage and current source inverters and to real-time computation. Generally, the available torque is limited by current constraints at low speeds and by voltage constraints at high speeds, but there is also often an intermediate range of speeds where both the voltage and the current constraints are active. The paper speci es how optimal operation can be achieved in all three ranges, and at what speeds the optimal operation transitions from one mode to another. A numerical example for a typical permanent magnet stepper motor is discussed to illustrate the contributions of the paper. Keywords: Industry day, Robotics, Energy systems.
1 Introduction The maximization of the torque produced by electric motors is an important practical consideration, since optimization may yield the use of a smaller motor for a given application or a faster operation for a given motor. In this paper, we consider the problem for permanent magnet synchronous motors, which include brushless DC motors and stepper motors, and are commonly used in robotic applications. At high speeds, the available torque of electric motors decreases because of the back-emf voltage and the limits on the source voltage. Maximization of the torque requires the use of eld-weakening which, for synchronous motors, can be achieved optimally using relatively simple strategies. These strategies are derived assuming that the speed is constant, an assumption that is often not satis ed in practice. However, the
2 State-Space Model and Problem Statement For a two-phase motor, the DQ transformation is given by
id iq
=
cos (np ) sin (np ) ? sin (np ) cos (np )
ia ib
; (1)
where ia and ib are the currents in phases a and b respectively, is the rotor angular position, and np is the number of pole pairs (or the number of rotor teeths for a stepper motor). The currents id and iq are the transformed currents in the d and q (for direct 1
and quadrature) reference frame. In the same way, the voltages va and vb applied to phases a and b can be transformed into the voltages vd and vq . With ! the rotor speed, L the inductance of a stator winding, R the resistance of a stator winding, J the moment of inertia of the rotor, L the load torque, the state-space model of a permanent magnet synchronous motor is given by di L d = dt diq = L dt d! = J dt d = dt
vd ? Rid + np !Liq
(2)
vq ? Riq ? np !Lid ? K!
(3)
Kiq ? L
(4)
!:
(5)
For simplicity, the model assumes that the rotor is smooth and that the magnetics are linear. For constant speed !, equations (2) to (4) yield vd = Rid ? np !Liq vq = Riq + np !Lid + K! el = Kiq = L :
(6) (7) (8)
In (8), el is the electrical torque developed by the motor. The voltage and current constraints are incorporated in the problem statement by using the fact that, at constant speed, the phase voltages and the phase currents are sinusoidal, with peak magnitudes q q given by vd2 + vq2 and i2d + i2q respectively. Therefore, for phase voltages bounded by Vmax and phase currents bounded by Imax , the following constraints apply to the DQ variables V= I=
q
vd2 + vq2 Vmax
(9)
i2d + i2q Imax :
(10)
q
3 Torque Maximization
3.1 Speed Ranges and Transition Speeds
Because the torque may, in theory, be made arbitrarily large by suciently increasing the voltage and current levels, optimal operation under constraints is always achieved with the limits on either the currents or the voltages being reached. It is also possible for both
limits to be reached together. A typical motor is characterized by three ranges of speed: a low-speed range where operation is constrained by the current limit only, an intermediate speed range where operation is constrained by both the voltage and the current limit, and a high-speed range where operation is constrained by the voltage limit only. If the current limit is very high, it is possible for the whole speed range to be constrained by the voltage limit only. However, this is not typically the case. At a speci c speed, torque maximization will be achieved for a speci c choice of id and iq (for a currentcontrolled drive), or a speci c choice of vd and vq (for a voltage-controlled drive). We will denote the optimal values by id, iq , vd , and vq . Knowing what the solution is in each of the speed ranges, it remains to determine what the boundaries of those ranges are in order to determine which solution is to be applied. We will call the two speeds where one transitions from one type of constraint to another the rst and second transition speeds, and we will denote them !1 and !2 . We will show how these speeds can be calculated after having obtained the form of the optimal solution in each speed range.
3.2 Maximization without the Voltage Constraint We rst consider the case where only the current constraint is active. In this case, the optimum id must be zero to maximize the current iq . The overall solution is simply id iq vd vq
= = = =
0
(11) (12) (13) (14)
Imax ?np !LImax RImax + K!:
3.3 Maximization without the Current Constraint Torque maximization under the voltage constraint alone yields a solution which is referred to as optimal eld weakening and is well-known (cf. [3], p. 266.). To obtain the solution, one solves (6) and (7) for id and iq , so that
id iq
= Z1
Rvd + np !L(vq ? K!) ?np !Lvd + R(vq ? K!) ; Z = R2 + (np !L)2 :
(15)
If one de nes V and so that vd = V cos ( ); vq = V sin ( );
(16)
the current iq is given by ?n !LV cos ( ) + RV sin ( ) ? K!R : (17) iq = p Z
(17) shows that the maximum current iq , and therefore the maximum electrical torque, is obtained for V = Vmax d (?np !L cos ( ) + R sin ( )) = 0: d v
sin ( ) = pR
(19)
Z
cos ( ) = ?npp !L ; Z
and therefore vd =
?npp !L V
max
Z R vq = p Vmax : Z
(20) (21) (22)
(n !L) (K!) ; id = ? p Z
(23) which can be viewed as the optimal eld weakening current. Using (17), (19), and (20), with V = Vmax , gives iq
p
(24)
3.4 Maximization under Voltage and Current Constraints
In the case where both the voltage and the current constraints are active, the optimization reduces to the solution of a system of algebraic equations, because there are as many constraints as degrees of freedom. The steady-state equations (6), (7), with vd2 + vq2 = 2 Vmax yield Z (i2d + i2q ) + K 2 !2 + 2RK!iq + 2np LK!2id 2 : = Vmax
(26)
= 4R2 !2 K 2 + 4n2p!4 L2 K 2 (27) 2 2 2 2 = ?4RK!(Vmax ? K ! ? ZImax ) (28) 2 2 2 2 2 2 2 4 2 2
= (Vmax ? K ! ? ZImax ) ? 4Imax np ! L K :
The other variables can be determined from iq using q
2 id = Imax ? iq vd = Rid ? np !Liq vq = Riq + np !Lid + K!:
(30) (31) (32)
3.5 First Transition Speed The rst transition speed !1 can be found by calculating the value of the speed for which the voltage limit is reached for the optimal solution with current con2 , (13) and (14) yield straint only. With vd2 + vq2 = Vmax a quadratic equation for the rst transition speed 2 (K 2 + (np LImax)2 )!12 + 2RKImax!1 = Vmax : (33)
3.6 Second Transition Speed
For id, one nds, using (16) and (18), that
Z ? K!R V : iq = max Z
with
iq 2 + iq + = 0;
(29)
The optimal d and q are therefore found to satisfy vd ?np !L = R = tan1( ) : (18) vq This equation gives the value of the optimal lead angle for voltage-controlled drives. Equation (18) yields v
2 leads to a quadratic Eliminating id using i2d +i2q = Imax equation for iq of the form
(25)
The second transition speed can be calculated by nding the value of the speed such that the optimum solution obtained with the voltage constraint only, reaches the current limit. By proceeding carefully, we will show that a cubic equation can be obtained, so that the solution can be calculated exactly using standard formulas. 2 , one rst nds Using (23), (24), and i2d + i2q = Imax that ?
p
2 2 = 2K!RVmax Z; (34) Z Vmax + (K!)2 ? ZImax
and, cancelling the square-root term and squaring both sides,
?
2 2 2 Z (Vmax ? R2 Imax ) ? (n2p L2 Imax ? K 2 )!2 2 = (2KRVmax )2 !2 : (35)
A cubic equation is obtained for !2, of the form (a ? x)2 (b + x) = cx;
(36)
where x = !2 2 V 2 ? R2 Imax a = 2max2 2 np L Imax ? K 2 R2 b = (np L)2 (2KRVmax)2 c = 2 (np L)2 (n2p L2Imax ? K 2 )2 :
(37) (38) (39) (40)
The most common situation is the one in which the constants a, b, and c are all positive. In that case, it can be deduced from the form of the cubic equation that there will be one negative root x < ?b, a positive root x < a, and a positive root x > a. One of the positive roots is an invalid solution introduced by the derivation and must be eliminated. Inspection of (34) and (35) reveals that the valid root is the one such that x < a. The case where the numerator of a is negative is unrealistic, and the case where its denominator is negative is possible, although requiring a large current limit. In that case, there may be a positive real root, but it would be invalid. This case occurs when the voltage constraint alone de nes the optimum for the whole speed range.
4 Implementation Issues The optimal control solution may be used, for example, to calculate the open-loop current or voltage references minimizing the time required for a pointto-point move. The results may also be used in a closed-loop con guration, in order to guarantee that the maximum torque is available at all times. However, in that case, the torque reference may not always be equal to the maximum. Two modes of operation are then typical for a current-controlled drive: given a current iq corresponding to the desired torque, the current id may be set to id = id (in order to maintain the optimal eld weakening), or to a value such that id=iq = id =iq (in order to maintain the optimal lead angle). A question to be answered is whether these two strategies will still yield to the satisfaction of the constraints. Fortunately, we will see it is the case. If one sets id = id and iq = iq , for some 0 1, the current constraint will clearly be satis ed, given that it is satis ed for the optimum strategy. As for the voltage constraint, note that the peak voltage is given by V 2 = Z (i2d + i2q )+ K 2 !2 +2RK!iq +2np LK!2id ; (41)
so that V 2 V 2 , where V 2 is the voltage required by the optimal solution. Therefore, the voltage constraint must also be satis ed. If one sets id = id and iq = iq , for some 0 1, the current constraint will again be satis ed. (41) implies that V 2 V 2 and, therefore, that the voltage constraint is also satis ed. For voltage-controlled drives, similar strategies are available, but do not guarantee that the current constraint will be met. A solution to the problem is to set vd = vd vq = vq + (1 ? )K!
(42) (43)
for some 0 1. In that case, (16) shows that id = id and iq = iq , so that, by the argument of the preceding paragraph, the satisfaction of the voltage and current constraints is guaranteed. With this strategy, is an adjustable parameter that regulates the torque, i.e., it is the main control variable. Note that if a reduced torque is required, not only will the voltage be reduced, but the lead angle for the voltages will be reduced as well, even though the lead angle for the currents will be maintained.
5 Example We consider the example of a commercial permanent magnet stepper motor, manufactured by Aerotech. The parameters were obtained using a least-squares identi cation procedure, and were determined to be R = 0:6 , L = 0:00254 H, np = 50, and K = 0:4 Nm/A. The operating limits are given by Vmax = 80 V and Imax = 6 A. In Fig. 1, the variables of interest are plotted for the useful speed range. Plots a) and c) give the peak voltage and the peak current. The voltage and current constraints are satis ed for all speeds, and at least one of the limits is reached at any speed. The intermediate speed range where both limits are reached is visible, with the two transition speeds at 90 and 120 rad/s. The maximum torque is shown in plot e) and is de ned by three smooth branches. The torque falls o as soon as the voltage limit is reached. Plot b) gives the optimum direct current id. The intermediate speed range and the high speed range constitute the eld weakening region, where a negative current id is applied to reduce the e ect of the back-emf produced by the permanent magnet. In plot d), the results are expressed in terms of the lead
angle, de ned to be lead = arctan
id : iq
References (44)
An interesting observation is that the current id as determined by the voltage constraint alone reaches its peak negative value very rapidly as the speed increases. In contrast, for a whole range of speeds, the optimal current id remains equal to zero. Figure 2 shows the results that would be obtained if the current id as speci ed by the eld weakening formula was applied for the whole speed range. While such a choice is simple, and apparently reasonable, plot e) for the torque shows that it results in a signi cant loss of torque (about 14%) in the low speed range. The results of this paper provide an answer to the problem of connecting the low speed and high speed solutions, by formally solving the combined voltage/current constrained problem.
6 Conclusions The paper has considered the problem of torque optimization for permanent magnet stepper motors. A useful feature of the solution is that it consists in explicit, analytic formulas. The most complicated calculations are square roots and cubic roots, so that the algorithm can be completely implemented in real-time, if the motor parameters are determined on-line using an identi cation procedure such as described in [4]. It was found that there is, in general, an intermediate speed range where both the voltage and the current limits are reached. Consideration of this speed range not only provides the complete optimal solution to the problem, but also allows to connect the solutions for the low and high speed regimes in a smooth manner. Compared to a solution implementing the optimal eld weakening formula for the whole speed range, the exact optimal solution gives substantial increases in torque in the low speed range. The presence of an intermediate speed range was also found for induction motors (see [5]). However, no analytic expression could be found in that case, because of higher-order polynomial equations. Further, a substantial di erence existed in induction motors between the motoring and generating mode. For synchronous motors, the generating case was not discussed because it is trivial: one simply switches the sign of the current iq , leaving the id identical. The optimal lead angle switches by 180 degrees under a torque reversal.
[1] Bodson, M., Chiasson, J.N., Novotnak, R.T., & Rekowski R.B., "High-Performance Nonlinear Feedback Control of a Permanent Magnet Stepper Motor," IEEE Trans. on Control Systems Technology, vol. 1, no. 1., 1993, pp. 5 - 14. [2] Shi, J. & Lu, Y.-S., "Field-Weakening Operation of Cylindrical Permanent Magnet Motors," to appear in the Proc. of the IEEE Conference on Control Applications, Dearborn, MI, 1996. [3] Leonhard, W.: Control of Electrical Drives, Springer Verlag, Berlin, 1990. [4] Blauch, A., Bodson, M., & Chiasson, J., "HighSpeed Parameter Estimation of Stepper Motors," IEEE Trans. on Control Systems Technology, vol. 1, no. 4, 1993, pp. 270-279. [5] Bodson, M. & Chiasson J., "A Systematic Approach to Selecting Optimal Flux References in Induction Motors," IEEE Trans. Control Systems Technology, vol. 3, no. 4, 1995, pp. 388-397.
Peak Voltage
id
100
[A]
[V]
0 50
-1 -2 -3
0 0
50 a)
100 150 [rad/sec]
-4 0
200
50
Peak Current 0
[deg]
[A]
200
Lead Angle
10
5
0 0
100 150 [rad/sec]
b)
50 c)
100 150 [rad/sec]
-50
-100 0
200
50
100 150 [rad/sec]
d)
200
[Nm]
Torque 3
V=Vmax
2
I=Imax, V=Vmax
1 0 0
I=Imax 50 e)
100 150 [rad/sec]
200
Figure 1: Motor Characteristics with Torque Optimization
Peak Voltage
id
100
[A]
[V]
0 50
-1 -2 -3
0 0
50 a)
100 150 [rad/sec]
-4 0
200
50
Peak Current
[deg]
[A]
0
5
50 c)
100 150 [rad/sec]
200
-50
-100 0
50 d)
100 150 [rad/sec]
[Nm]
Torque 3
V = Vmax
2
I = Imax, V = Vmax
1 0 0
200
Lead Angle
10
0 0
100 150 [rad/sec]
b)
I = Imax 50 e)
100 150 [rad/sec]
200
Figure 2: Motor Characteristics with Field Weakening
200
