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DC motors: dynamic model and control techniques
Luca Zaccarian
- 1 Magnetic considerations on rotating coils Contents
- 1.1 Magnetic field and conductors
- 1.2 The magneto-motive force in a rotating coil
- 1.3 The back EMF effect
- 2 The basic equations of the DC motor
- 2.1 The electric equations
- 2.2 The mechanical equations
- 2.3 Geared motors and direct drive motors
- 2.4 Block diagram of the DC motor
- 3 Stator voltage control with constant armature current
- 3.1 Steady state behavior
- 3.2 Transfer function
- 4 Stator voltage control with constant armature voltage
- 4.1 Linearized equations
- 4.2 Steady state behavior
- 4.3 Transfer function
- 5 Armature-Current control
- 5.1 Steady state behavior
- 5.2 Transfer function
- 5.3 Speed control system
- 5.4 Position control system
- 5.5 Torque control system
1 Magnetic considerations on rotating coils
1.1 Magnetic field and conductors
A magnetic field is generated by a flowing current: it is in particular experienced in the neighborhood of a moving charge. The effect of a magnetic field may be experimented, for instance, by positioning a magnet close to a wire where a current is flowing; what can be observed is that the magnet experiences a force, which is due to the magnetic field generated by the current flowing in the wire. If the magnet is moved around the wire, the force changes depending on the positions assumed by the magnet. In particular, it can be observed that the magnetic field decreases as the distance from the wire increases and increases as the current increases. The above experiment aims to verify that a flowing current indeed generates a vector field B~. This field may be thought of as the sum of infinite contributions d B~ due to all the infinitesimal segments of wire d~l, where the current i flows. Each segment induces a magnetic field at any point in the surrounding space. In particular, the Biot and Savart law states that, if ~r is the vector connecting the wire segment d~l to a generic point p in the space, the contribution d B~ of the magnetic field in the point p due to the segment d~l is given by
d B~ = i
d~l × ~r |r|^3
where the symbol ~ denotes that the considered quantity is a vector of
3 and the symbol × denotes the vector product operation. Equation (1) provides a tool for computing the magnetic field associated to any conductor where a current is flowing. In particular, by suitably integrating it for the case of a solenoid, it turns out that, if the corner effects are neglected (namely, the solenoid is long enough), the field inside the windings is constant and parallel to the axis of the solenoid, and its magnitude is given by [6, p. 136]
| B~| =
μ l
N i, (2)
where l is the length of the solenoid, μ is the magnetic permeability of the dielectric inside the solenoid, N is the number of turns of the wire (see Figure 1). Consider now a surface S located in a magnetic field B~; the magnetic flux, or flux Φ flowing through the surface is defined as the integral along the surface of the normal component of the magnetic field:
S
B^ ~ · ~n dS,
where ~n is the perpendicular to the surface. In particular, if a uniform magnetic field B~ approaches a flat surface with an angle of incidence β and the area of the surface is A, then
Φ = | B~| A cos β. (3)
l
i
i
B
α
F
Figure 2: Force experienced by a current-carrying conductor located in a uniform magnetic field.
It turns out that, if α denotes the angle of incidence between the magnetic field and a straight wire of length l, then the magnitude of the force F~ is given by
| F~ | = i l | B~| sin α, (6)
and F~ is oriented on the perpendicular to the plane spanned by the magnetic field and the wire following the right-hand-screw rule (see Figure 2). Now, on the basis of equation (6), consider a rigid rectangular coil constituted by a single wire where a current i flows, suitably located in an uniform exogenous magnetic field. As it can be seen from Figure 3; 1 if l is the length of the wire perpendicular to the magnetic field,
then two forces are applied to the coil. Since the angle α of equation (6) is ±
π 2
(depending
on which side of the coil is considered), then it turns out that the magnitudes of the two forces are the same:
| F~ | = | B~| i l. (7)
Since the coil is square, the current i flows in opposite directions on the two sides of the coil; thus, the two forces F generate a torque T exerted at the center of the coil that is dependent on the angular position θ of the coil with respect to the magnetic field. In particular, if the length of an edge of the square is d, then
T = 2 | F~ |
d 2
sin θ = | F~ | d sin θ; (8)
whence, taking into account equation (7), equation (8) yields
T = | F~ | d sin θ = | B~| i l d sin θ. (9)
(^1) As usual, a dot · means that the flowing current is exiting the page, while a cross × means that the
current is entering the page.
F
F
T
θ
B d/ 2
Figure 3: Torque experienced by a coil in a uniform magnetic field.
Tmax
π 2
π 2 π θ
T
Figure 4: Profile of the torque T experienced by the coil in a magnetic field, as the rotation angle θ varies.
Consider now equation (9) and notice that the torque highly depends on the rotor coil angular position θ. Imagine that the coil is in the rotor of a motor; then the resulting torque is highly dependent on the motor position; moreover, if no load torque is present, the motor keeps turning clockwise and counter-clockwise; as a matter of fact, if the coil turns of an angle π, the torque exerted has the same amplitude but opposite sign (see Figure 4). Since the goal is to have the motor to exert a constant torque for any position θ, the solution adopted is to insert on the rotor shaft a commutator constituted by two segments connected to the rotor windings and brushes that slide between the segments as the rotor turns(see Figure 5). In such a configuration, the sign of the current flowing in the coil changes at each half revolution of the motor; thus, if the segments are properly positioned with respect to the coil position, the torque profile of Figure 4 will be suitably inverted during half revolution (see Figure 5). Now, once this solution is adopted, although the torque has always the same sign, still it is highly dependent on the rotor position; the obvious solution to this problem is to increase the number of coils in the rotor and the segments of the commutator, connecting each pair of opposite segments to a coil in such a way that when the brushes activate that coil, the rotor
angle is θ = ±
π 2
(i.e., the maximum torque position). It turns out that, if N independent
Ti
θ
θ
T
θc θc θc θc θc θc θc θc θc
Figure 6: Torque exerted by the motor, when a multiple segments commutator is used.
In this particular case, notice that the flux flowing in the coil is given by equation (3) substituting β with the angle θ(t) of the motor and A with the area of the internal surface of the coil. Whence, computing the derivative in equation (12), the back electro-motive force or, more easily, the back EMF may be computed as
e = −
d dt
(| B~| A cos θ(t))
= | B~| A θ˙(t) sin θ(t) = | B~| A ω(t) sin θ(t),
where ω :=
dθ(t) dt
denotes the angular speed of the motor. Noticing that, due to the presence of the commutator, the coil always operates in a neighborhood the position θ = π/2, then sin θ(t) ≈ 1 and the back EMF may be written as e = | B~| A ω(t). Finally, similarly to the case studied in equations (10) and (11), the back EMF may be expressed as a function of the flux Φ, and it can be verified easily that
e = KΦ Φ ω, (13)
where KΦ is the same constant as the one in equation (11).
2 The basic equations of the DC motor
The set of equations here reported, constitutes a model of the DC motor, which may
be represented as a nonlinear dynamic system. The main restrictions of this model, with respect to a real motor are
- the assumption that the magnetic circuit is linear (such an assumption is approximate, since the metal parts are not perfectly smooth and there is some flux dispersion inside the motor, moreover, due to saturation of the metal, equation (4) does not hold for high values of i);
- the assumption that the mechanical friction is only linear in the motor speed; namely, only viscous friction is assumed to be present in the motor (such an assumption is approximate since Coulomb friction is usually experienced in motors).
2.1 The electric equations
Following the process described in Section 1.1, in a DC motor, the magnetic flux is generated by windings located on the stator. Although the physical reason why electrical power is transformed in mechanical power is the one explained in Section 1.2, the actual implementations of this result are various, as a matter of fact, since the magnetic field B arises from the stator coils, not only the rotor coils may rotate with respect to the stator, but also the stator supply may rotate (in an electrical sense) by increasing the number of coils and by a more sophisticated supply. In this handout, a simple model, which applies to the above cases (provided proper transformations are performed on the system variables) will be introduced. The stator of the motor will be assumed to have a single coil characterized by an induc- tance Le due to the windings and a resistance Re due to dispersions in the conductor (see Figure 7). The equation associated with such an electric circuit is given by
ve(t) = Le
d ie dt
Since relation (14) is linear, by transforming in the Laplace domain the signals, it can be written
ie(s) ve(s)
Ke 1 + τe s
where Ke :=
Re
is the stator gain and τe :=
Le Re
is the stator time constant. The rotor is assumed to be a single coil characterized by inductance La and resistance Ra (see Figure 7), but it has to be taken into account the back EMF of the motor in equation (13). The equation associated with such an electric circuit is given by
va(t) = La
d ia dt
Again, since relation (16) is linear, by transforming in the Laplace domain the signals, it can be written
ia(s) va(s) − e(s)
Ka 1 + τa s
2.3 Geared motors and direct drive motors
Often (e.g., in robotic applications), the speed required by the load is too low as compared to the nominal speed of the motor. 2 In this cases, gears are introduced between the motor and the load, thus reducing by a factor n the angular velocity of the load itself. Besides the increase of damping and inertia due to the presence of the additional rotating cogwheels of the gear, the mechanical coupling between the load and the motor is altered by the gear itself. To correctly understand the effects of the gear, the fist thing to remark is that damping and inertia are not the same if measured at the input or at the output of the gear. Since we are interested in the complete characterization of the motor block, let’s refer to the output quantities, and denote by FG the internal damping of the gear and by JG the internal inertia of the gear. Then, notice that, since the power exerted by the motor is the same at the input and at the output of the gear, denoting by T (^) M′ and ω′^ the torque and the speed at the output of the gear, it can be written TM ω = T (^) M′ ω′,
and, since ω′^ = ω/n, then T (^) M′ = n TM. Substituting the above equations in equation (22), and taking into account the increase of damping and inertia due to the cogwheels of the gear, it turns out 3
T (^) M′ − TL = (JG + n^2 J)
d ω′ dt
By a comparison between equations (22) and (24), it is stressed that the presence of the gear highly increases the inertia and the damping of the motor from the point of view of the load. In addition, when a gear is inserted in an actuator system, backlash is experienced on its output due to the coupling between the cogwheels of the gear. This gives rise to nonlinearities that may lead to instability effects. For this reason, especially in high precision systems, direct-drive motors are used. Such motors may exert reasonable torques at low speeds, whence they do not need gears to drive the load. However, these motors may not be adopted for high power tasks, since the maximum torque exerted has physical limitations.
2.4 Block diagram of the DC motor
By implementing equations (14), (16), (20), (21), and (22) in a nonlinear block diagram, the result shown in Figure 8 is obtained. In the block diagram, the variable θ represents the rotor angular position (whence, ω = θ˙). The nonlinear model results in a two-input, one-output map, having a disturbance input TL and with four state variables, related to
- the energy stored in the inductance Le;
- the energy stored in the inductance La; (^2) The nominal speed of the motor is the speed corresponding to the efficiency maximum. (^3) Note that TL is exerted by the load, whence it should not be scaled.
ve = Le
die dt
K
va = La K
dia dt
ω
TL
TM
va
ve ie
ia ∫ θ T = J
dω dt
Figure 8: Nonlinear block diagram of a DC motor.
- the kinetic energy of the rotor (related to J);
- the position θ of the rotor.
Remark 2.1 Note that the nonlinear model of the motor is indeed constituted by three linear relationships between physical quantities, constituted by the transfer functions (15), (17) and (23), and two multipliers, which represent the system nonlinearities. Various control techniques performed on the motor, aim to linearize such a block diagram, by suitably controlling the system by means of the two inputs va and ve. ◦
Remark 2.2 When a geared motor is considered, a constant gain block equal to 1/n should be added in the block diagram of Figure 8 right before the integrator of ω (so that ω′^ will be integrated instead) and divides by a factor n the disturbance input TL. Note that the feedback branch is related to ω and not to ω′, since the gear does not change the electrical properties of the motor. ◦
To simplify the nonlinear block diagram in Figure 8, three main control techniques are introduced in the following sections, showing the performance of the system in each case and giving the simplified block diagram related to each technique.
3 Stator voltage control with constant armature cur-
rent
Assume that a constant current supply is available, regardless of the voltage absorbed by the load; then, supplying the armature circuit with such a device, a stator voltage control configuration is obtained. It should be noticed that, since the armature current is constant, the nonlinear block diagram in Figure 8 becomes linear; as a matter of fact, the whole feedback branch is erased because the rotor current is imposed by the current supply. The main problem associated with this control is that a current generator is quite expen- sive, as far as it works for high power applications. The drawbacks of this control technique, on the other hand, are several.
TM
increasing
ω
va va 0
ω 0
TL 0
Figure 9: Steady state relationship between torque and speed in the stator voltage control with constant armature current.
K ia
Km 1 + τm s
Ke 1 + τe s
s
TM
TL
ve ie ω θ
Figure 10: Block diagram of the DC motor with constant armature current.
The transfer matrix of the whole system is easily computed from the block diagram as:
θ(s) = [W 1 (s) W 2 (s)]
[
ve(s) TL(s)
]
where
W 1 (s) :=
KAC
s (1 + τe s) (1 + τm s)
, KAC := Ke K Km ia.
W 2 (s) :=
Km s (1 + τm s)
As remarked above, the motor damping may, in many cases, be negligible. So that the mechanical time constant approaches infinity. In such a case, it has been noticed that there is no steady state value for the speed ω. This can be here motivated by the fact that the
transfer function related to the speed
ω(s) ve(s)
KAC
s (1 + τe s)
has a pole at the origin.
4 Stator voltage control with constant armature volt-
age
In this second control technique, the control is based on a constant voltage supply for the armature. It can be immediately seen from the non-linear block diagram in Figure 8, that the system thus controlled is still non-linear; however, there are several drawbacks also with this control configuration:
- the constant voltage generator is cheap, also for high power purposes;
- similarly to the control case of Section 3, the control signal is a low power signal, while the power is supplied to the system by means of the constant voltage generator;
- since this control scheme usually operates in the neighborhood of an operating point, the armature voltage is always balanced by a non null back EMF, thus limiting the armature current peaks.
4.1 Linearized equations
With reference to the general non-linear model, a linearization is here computed with respect to all the internal variables (except for va which is assumed to be constant) around an operating point. First, from equations (14), (16), (20), (21) and (22), the relations between the values ve 0 , ie 0 , va 0 , ia 0 , ω 0 , TM 0 , TL 0 of the quantities ve, ie, va, ia, ω, TM and TL, respectively, at the operating point, are written:
ve 0 = Re ie 0 va 0 = Ra ia 0 + K ie 0 ω 0 TM 0 = K ie 0 ia 0 TM 0 − TL 0 = F ω 0 ;
whence, the operating point must satisfy the above conditions. The same equations may be written as the variations of the above quantities with respect to their nominal values, neglecting the second order terms:
δve = Re δie + Le δi˙e (26a) 0 = δva = Ra δia + La δi˙a + K (ie 0 δω + ω 0 δie) (26b) δTM = K (ie 0 δia + ia 0 δie) (26c) δTM − δTL = F δω + J δω,˙ (26d)
where the subscript 0 denotes the value of a quantity at the operating point and the δ symbol denotes its variation around that value.
y 0
x 0
y
x
z ⇒
δx
δy
˜z ≈ z
Figure 12: Transformation of a multiplier after linearization.
Ke 1 + τe s
K ie 0
K ie 0
K ia 0
K ω 0
δie
δTL
Km 1 + τm s
δve δTM - δω δθ
Figure 13: Block diagram of the linearization of the DC motor under armature voltage control.
equation is
δTM = K Ke
(1 + τa s) ia 0 − K Ka ie 0 ω 0 (1 + τa s)(1 + τe s)
δve − Ka
(K ie 0 )^2 1 + τa s
δω. (28)
Considering the relationship between the torque and the speed established by equation (23), the two equations may be rearranged to compute the transfer function between the inputs δve, δTL and the output δω (or, equivalently δθ if a pole in the origin is added):
δω(s) δve(s)
Km Ka K ((1 + τa s) ia 0 − K Ka ie 0 ω 0 ) (1 + τe s) ((1 + τm s)(1 + τa s) + Km Ka (K ie 0 )^2 ) δω(s) δTL(s)
= −Km
1 + τa s ((1 + τm s)(1 + τa s) + Km Ka (K ie 0 )^2 )
5 Armature-Current control
The most common control technique for DC motors is the armature-current control. Such control is performed by keeping the flux constant inside the motor. To this aim, either the stator voltage is constant or the stator coils are replaced by a permanent magnet. In the latter case, the motor is said to be a permanent magnet DC motor and is driven by means of the only armature coils. Permanent magnet DC motors are totally equivalent to armature-controlled DC motors; as a matter of fact, in both of them the flux Φ is constant. Besides this, what makes the difference between them is dependent on equation (11). It can be seen that, with standard DC motors, the desired torque may be obtained by increasing ie (namely, the flux), ia, or both of them. Conversely, permanent magnet motors may be controlled by means of the only available current ia. This is bad from the heating point of view; as a matter of fact, the rotor coils are hard to be cooled (since they are inside the motor) and, indeed, in the permanent magnet motors, are the only power source of the motor itself. Referring to Figure 8, note that keeping the flux constant, the block diagram becomes linear. In addition, the motor has an intrinsic negative feedback structure, whence at the steady state, the speed ω is proportional to the reference input va. This two facts, in addition to the cheaper price of a permanent magnet motor with respect to a standard DC motor (as a matter of fact only the rotor coils need to be winded), are the main reasons why armature controlled motors are widely used. However, several disadvantages arise from this control technique.
- Although the flux is constant (hence the back EMF never goes to zero when the motor is running), the rotor current could take, in several cases, high values, thus bringing the motor into dangerous operating conditions. In particular, the speed of the motor could decrease to zero due to an equilibrium between the load and the motor torque, or, equivalently, the current could take high values during the transient, after a step has been applied at the input. In this latter case, the mechanical time constant will delay the increase of speed ∆ω corresponding to the increase of armature voltage ∆va and during this delay the rotor current will raise at high values.
- The reference input and the power input of the motor are the same. This often leads to a trade off, between accuracy with respect to a desired reference value and maximum power exerted. As a matter of fact, the fidelity of a linear power amplifier is inversely proportional to the maximum power exerted by such power amplifier.
5.1 Steady state behavior
Consider equations (16), (18), (19), and (22); the steady state behavior of the armature controlled motor may be easily determined by putting to zero the time derivatives of the variables. The resulting equations are:
TM =
KΦ Φ
Ra
va −
(KΦ Φ)^2
Ra
ω (29a)
TM − TL = F ω. (29b)
Km 1 + τm s
s
Ka 1 + τa s
K ie
K ie
va TM
TL
ie ω θ
Figure 15: Block diagram of the linearization of the DC motor under armature-current control.
The transfer matrix of the system may be written as
ω(s) = [W 1 (s) W 2 (s)]
[
va(s) TL(s)
]
where
W 1 (s) :=
Ka KΦ Φ Km (1 + τa s) (1 + τm s) + Ka Km (KΦ Φ)^2
W 2 (s) :=
Km (1 + τa s) (1 + τa s) (1 + τm s) + Ka Km (KΦ Φ)^2
5.3 Speed control system
As remarked in Section 5.1, the armature current current controlled DC motor is intrin- sically a velocity control scheme. In particular, consider the steady-state equations (29) and the dependence of the speed ω upon the values of the inputs va and TL. Considering, for the sake of simplicity, the case F = 0 (the general case is similar), it turns out:
ω ≈
KΦ Φ
va −
Ra (KΦ Φ)^2
TL. (30)
It can be seen that if the load torque TL is different from zero, a steady state error will be experienced on the system. Now, it can be seen that, the more KΦ Φ is high, the less the steady state error is large. Whence, it seems obvious to perform a control technique in which an additional speed feedback is performed. By means of a tachometer (i.e., a sensor for the measurement of the speed of the motor), an additional feedback loop is closed over the block diagram in Figure 15. Suppose the tachometer has a constant transfer function with a gain KT , then placing a gain KA in the direct branch, the control loop in Figure 16 is obtained. Now, with reference to equation (30), noticing that for the control scheme in Figure 16, it holds va = KA (vr − KT ω), then the following equation is obtained:
ω ≈
KA
KΦ Φ + KT KA
vr −
Ra (KΦ Φ) (KΦ Φ + KT KA)
TL.
Km 1 + τm s
Ka 1 + τa s
KΦ Φ
KA
va TM
TL
ie ω
KΦ Φ
KT
vr -
Figure 16: Block diagram of the speed control system for an armature controlled DC motor.
From the above equation, for sufficiently high values of KA, it turns out:
ω ≈
KT
vr ,
as desired. It should be noticed that, with this control scheme, a steady state error is always found, although this error may be decreased as much as necessary by the choice of the gain KA. Since the two feedback loops in Figure 16 may be represented as a unique feedback loop with suitable feedback and direct branch gains, the root locus of the system having the two poles related to τm and τa, may be traced to analyze the behavior of the closed loop system for different values of KA.
Re
Im
s-plane
− (^) τ^1 a − (^) τ^1 m
Figure 17: Root locus of the system in Figure 16, as KA varies.
The resulting diagram, reported in Figure 17 shows that, as KA increases, the system step response gets faster and overshooted.
