DC motors

Lorentz force

Consider a charged particle moving through a background magnetic field. The Lorentz force is the (mechanical!) force on the particle, which depends on the velocity of the particle, the background magnetic field, and the background electric field. Charge flowing through a straight, stationary² wire with current \(i\) in a uniform background magnetic field \(\bm{B}\) is subject to the cumulative effect of the Lorentz force on each charge. Let the straight wire’s length and orientation in the \(B\)-field be described by the vector \(\bm{\ell}\), which should be chosen to be in the direction of positive current flow. It can be shown that the resultant force \(\bm{f}\) on the wire is \[\begin{aligned} \bm{f} = i \bm{\ell} \times \bm{B} \end{aligned}\] as shown in figure 4.4.

With a curved wire, then, we could take infinitesimal sections \(d\bm{\ell}\) and integrate along the wire’s path: \[\begin{aligned} \bm{f} = i \int d\bm{\ell} \times \bm{B}. \end{aligned}\]

DC motors take advantage of this electromechanical phenomenon by driving current through cleverly arranged wires to generate torque on a shaft.

Permanent magnet DC motors

In order to take advantage of the Lorentz force, first a uniform background magnetic field \(\bm{B}\) is required. Some DC motors, called permanent magnet DC motors (PMDC motors) generate this field with two stationary permanent magnets arranged as shown in figure 4.5. The magnets are affixed to the “stationary” part of the motor called the stator.

Now consider a rigidly supported wire with current \(i\) passing through the field such that much of its length is perpendicular to the magnetic field. Consider the resultant forces on these perpendicular sections of wire for different wire configurations, as illustrated in figure 4.5. We have torque! But note that it changes direction for different armature orientations, which will need to be addressed in a moment. Note that we can wind this wire—which we call the armature—multiple times around the loop to increase the torque. The rotating bit of the motor that supports the armature is called the rotor, which includes the shaft.

The trouble is, if we connect our armature up to a circuit—which is usually located alongside the stator, i.e. not rotating—the wire will wrap about itself, which is not \#winning. But we’re tricky af so let’s consider just cutting that wire and rigidly connecting it to a disk—called a commutator—with two conductive regions, one for each terminal of the armature. The commutator will rotate with the armature, but it provides smooth contacts along the perimeter of the disk.

We can then connect the driving circuit to these contacts via brushes: conductive blocks pressed against the commutator on opposite sides such that they remain in contact (conducting current) yet allow the commutator to slide easily, as shown in figure 4.6. Brushes are typically made from carbon and wear out over time. This is partially mitigated by spring-loading, but eventually the brushes must be replaced, nonetheless.

So brushes solve the “wire wrapping” problem, but do they have an effect on the “torque flipping” issue? Yes! When the armature passes through its vertical orientation, current reverses direction through the armature. So whenever the perpendicular section of wire is on the right, current flows in the same direction, regardless of to which side of the armature it belongs.

Finally, is there a way to overcome the limitation of torque variation with different armature angles? Yes: if there are several different armature windings at different angles and correspondingly the commutator is split into several conductive contact pairs (one for each armature winding), a relatively continuous torque results! Real PMDC motors use this technique.

Wound stator DC motors

Wound stator DC motors operate very similarly to PMDC motors, but generate their background field with two stationary coils in place of the permanent magnets, above. These electromagnets require a current of their own, which is usually provided through the same circuitry that supplies the armature current (DC motors typically have only two terminals).

Three common configurations of the electrical connection of these are shown in figure 4.7. These define the following three DC motor types.

shunt

The shunt DC motor has its stator and rotor windings connected in parallel. These are the most common wound stator DC motors and their speeds can be easily controlled without feedback, but they have very low starting torque.

series

The series DC motor has stator and rotor windings connected in series. These have high starting torque—so high, in fact, that it is not advisable to start these motors without a load—but their speeds are not as easily controlled without feedback.

compound

The compound DC motor has stator and rotor windings connected in both series and parallel. These can exhibit characteristics that mix advantages and disadvantages of shunt and series DC motors.

Brushless DC motors

There is yet another type of DC motor: brushless (BLDC). Brushless DC motors work on principles more similar to AC motors, but require complex solid-state switching that must be precisely timed. As their name implies, these motors do not require brushes. A brushless DC motor mathematical model is not presented here, but a nice introduction is given by (Baldursson2005?).

The brushed DC motor is still widely used, despite its limitations, which include relatively frequent maintenance to replace brushes that wear out or clean/replace commutators. Other disadvantages of brushed DC motors include their relatively large size, relatively large rotor inertia, heat generated by the windings of the stator and/or rotor, and arcing that creates electronic interference for nearby electronics. Reasons they are still widely used include that they are inexpensive (about half the cost of brushless DC motors), don’t require (but often still use) complex driving circuits, are easy to model, and are easily driven at different speeds; for these reasons, an additional reason emerges: they’re relatively easy to design with!

A PMDC motor model

We have already explored a model for a PMDC motor in example, which yielded elemental equations \[\begin{aligned} T_2 &= -TF i_1 \quad\text{and}\\ \Omega_2 &= v_1/TF, \end{aligned}\] where \(TF\) is the motor constant. That model assumed neither armature resistance nor inductance were present—that is, it was an ideal transformer model. A linear graph of a much better model for a DC motor is shown in figure 4.8. This model includes a resistor \(R\) and inductor \(L\) in series with an ideal transducer. On the mechanical side, the rotor inertia \(J\) and internal bearing damping \(B\) are included. The tail ends of \(R\) and \(2\) should be connected to external electrical and mechanical subgraphs, respectively.

Motor constants

The motor torque constant \(K_t\) and back-emf voltage constant \(K_v\) are related to the transformer ratio \(TF\) derived above to characterize a brushed DC motor’s response. If expressed in a set of consistent units—say, SI units—\(K_t\) and \(K_v\) have the same numerical value and are equivalent to \(TF\). Precisely, with consistent units, \(TF = K_v = K_t\).

However, manufacturers usually use weird units like oz–in/A and V/krpm. If they are given in anything but SI units, we recommend converting to SI for analysis.

Once in SI, we will have something like (for \(x\in\mathbb{R}\)): \[\begin{aligned} K_t &= x \ \text{N--m/A and} \\ K_v &= x \ \text{V/(rad/s)}.\end{aligned}\]

So if we are given either \(K_t\) or \(K_v\), the unknown constant can be found (in SI units) by converting the known constant to SI.³

Animations

There are some great animations of DC motor operating principles and construction. I’ve included the url of my favorite, along with some bonus animations for other important types of motors we don’t have time to discuss, here.

• Brushed DC motors: youtu.be/LAtPHANEfQo

• Brushless DC motors: youtu.be/bCEiOnuODac

• AC (asynchronous) induction motors: youtu.be/AQqyGNOP_3o

• AC synchronous motors: youtu.be/Vk2jDXxZIhs

• Stepper motors: youtu.be/eyqwLiowZiU