DC motors - System Analysis and Control

Direct current (DC) motors are ubiquitous in control systems applications, from small hobby motors to industrial servo motors to electric vehicle traction motors. In this section, we will derive a mathematical model for a basic brushed DC motor from first principles, starting with the underlying electromagnetic laws. We will then make some idealizing assumptions to simplify the model for control design and analysis.

But first, let’s talk about the basics. A DC motor is a device that converts electrical power (voltage and current applied to input terminals) into mechanical power (torque and angular velocity of a spinning shaft). For this reason, we call DC motors electromechanical devices.

DC motor with input voltage v, current t, output torque T, and angular velocity \omega. — Figure 1:DC motor with input voltage $v$ , current $t$ , output torque $T$ , and angular velocity $\omega$ .

Electromagnetic principles¶

The key principle behind DC motors is how electric currents interact with magnetic fields to produce forces. This interaction is described by Lorentz’s force law^[1], which states that a charged particle with charge $q$ moving with velocity $\v$ in the presence of an electric field $\E$ and magnetic field $\B$ experiences a force $\F$ (the Lorentz force) given by:

\F = q \left(\E \;+\; \v \times \B \right)

(1)

Let’s see how the Lorentz force leads to the operation of a DC motor. Consider what happens when we place a copper wire in a magnetic field $\B$ and apply a voltage at either end of the wire to drive a current $i$ . The sequence of events is as follows:

The applied voltage causes an electric field $\E$ to form that is aligned with the conductor, which causes a Lorentz force $q \E$ on the charged particles (electrons) in the wire. This force accelerates the electrons, causing them to flow; we get a current $i$ .
The moving particles now experience a Lorentz force $q\v \times \B$ due to the magnetic field. This force is perpendicular to both the direction of current flow and the magnetic field.
Since the charges are confined to the wire, the perpendicular Lorentz force is transferred to the wire itself, causing it to accelerate in a direction perpendicular to the current flow (Newton’s second law).

A wire carrying current i in a magnetic field \B experiences a Lorentz force \F perpendicular to both the current and field. This causes the wire to accelerate upwards. — Figure 2:A wire carrying current $i$ in a magnetic field $\B$ experiences a Lorentz force $\F$ perpendicular to both the current and field. This causes the wire to accelerate upwards.

Slow electrons, fast current?

In metals, current is carried by electrons. In a vacuum, electrons can be accelerated to a substantial fraction of the speed of light. But in a wire, they move much more slowly due to collisions with atoms in the material. The average drift velocity of electrons in a typical copper wire is on the order of millimeters per second, even when the current is quite large! So why does current seem to flow instantaneously when we flip a switch? This is because the electric field that causes the electrons to move propagates along the circuit at a speed set by the geometry and surrounding materials, often a significant fraction of the speed of light. As this disturbance travels, electrons throughout the conductor begin to drift in response. Just like sound propagating through air, the individual air molecules move slowly, but the pressure wave moves quickly.

The wire is now accelerating, so the charged particles in the wire develop some upward velocity $\v$ . This leads to an additional sequence of events:

By Lorentz’s force law, the charged particles now experience a Lorentz force $q\v \times \B$ , which points in the opposite direction to the current $i$ .
This force on the charges causes them to redistribute along the wire, creating a voltage $V_{\textsf{emf}}$ that opposes the original voltage that caused the current $i$ . This induced voltage is also known as back EMF^[2].
The back EMF reduces the net voltage across the wire, which reduces the current $i$ flowing through it, thereby reducing the force on the wire. Eventually, the wire reaches a steady velocity where everything is in balance.

A wire moving with velocity \v in a magnetic field \B experiences an induced voltage (back EMF) v_b that opposes the applied voltage, reducing the current i flowing through the wire. — Figure 3:A wire moving with velocity $\v$ in a magnetic field $\B$ experiences an induced voltage (back EMF) $v_b$ that opposes the applied voltage, reducing the current $i$ flowing through the wire.

DC motor construction¶

The basic principles described above can be exploited to produce rotational motion. The idea is to use a loop of current, which will produce torque that causes a shaft to spin. Here is a diagram showing the basic idea:

Figure 4:A current-carrying loop in a magnetic field experiences forces that produce torque, causing the loop to rotate.

From this basic principle, we can build a DC motor by arranging multiple current-carrying loops (windings) around a central shaft. Some useful terminology:

Stator: The stationary part of the motor that produces the magnetic field.
Rotor: The rotating part of the motor.
Commutator: A switching mechanism that changes the direction of current or the orientation of magnetic fields ensuring continuous torque in one direction.

All DC motors follow a similar causal chain of events:

The most common DC motor (small and cheap) is the permanent-magnet brushed DC motor (PMDC). It is most similar to the diagram of Figure 4:

the stator is a pair of permanent magnets that provide the constant magnetic field.
the rotor is a set of windings arranged around a central shaft.
the commutator is a mechanical switch that reverses the current direction every half-turn.
brushes (usually made of carbon) are pressed against the rotating commutator to transfer current from the external power source to the rotor windings and prevent the wires from getting tangled.

Here are two excellent videos that illustrate how a PMDC works. The first video gives a very basic introduction, while the second video goes into more detail about a particular design called a three-pole motor.

Video explaining how permanent-magnet brushed DC motors (PMDC) work.

Video going into more detail, specifically for three-pole PMDCs.

Types of DC motors¶

DC motors can be broadly classified into two categories based on how the commutation is achieved: brushed and brushless DC motors. The PMDC motor is an example of a brushed DC motor. Here are the defining characteristics of each type:

Brushed DC motors: These motors use mechanical commutation with brushes and a commutator to switch the current direction in the windings. They are simple, inexpensive, and easy to control, but require regular maintenance due to brush wear. Since large permanent magnets can be expensive, larger brushed DC motors often use wound-field stators, where the magnetic field is generated by electromagnets. Different types exist depending on whether the same power source is used for both the field and rotor windings (series, shunt, compound) or if they are powered separately.
Brushless DC motors: These motors do not use windings in the rotor, so brushes are not needed. Instead, the rotor has permanent magnets and the stator uses coils to produce a magnetic field. Sensors detect the rotor angle and a controller orchestrates rapid electronic switching of the stator magnetic field to produce torque (electronic commutation). Different topologies can be used for the rotor/stator depending on the application. Brushless motors are more efficient, have a longer lifespan, and require less maintenance than brushed motors, but are also more expensive.

DC motors in practice

Here are examples of different types of commercially available DC motors.

PMDC

Wound-field

Inrunner

Outrunner

Pancake

Permanent-magnet brushed DC motors (PMDC) are the most common type of small DC motor. They are inexpensive and simple to control, making them ideal for toys, small actuators, and automotive accessories. The example below is a small hobby motor. You can see the two permanent magnets on the stator. The rotor has seven poles, each with windings. You can also see the commutator with its seven segments and the brushes making contact on either side.

Permanent-magnet brushed DC motor (PMDC). Example: small hobby motor. Image credit: control.com.

Here is a flowchart summarizing different types of DC motors with examples.

Figure 12:Flowchart of different types of DC motors.

Modeling a DC motor¶

We will develop a mathematical model for a basic permanent-magnet brushed DC motor (PMDC). The model consists of two main parts: the electrical dynamics governing the current flow in the motor windings, and the mechanical dynamics governing the rotation of the motor shaft. Here is a schematic diagram of the motor model we will use:

Figure 13:Model circuit for a brushed DC motor, showing electrical and mechanical components.

The windings are modeled as a series resistance $R$ and inductance $L$ , together with a back-EMF voltage source $v_b$ that opposes the applied input voltage $v_{\textsf{in}}$ . On the mechanical side, the motor produces a torque $T$ that must overcome a load torque $T_L$ through a rotor with inertia $J$ and viscous friction $b$ .

The inputs signals are the applied voltage $v_\textsf{in}(t)$ and load torque $T_L(t)$ .
The outputs signals are the motor angular velocity $\omega(t)$ and current $i(t)$ .

We will model the electrical and mechanical dynamics separately, and then combine them to obtain the complete equations of motion for the DC motor.

Electrical dynamics¶

The electrical dynamics are a straightforward RL circuit with two voltage sources: the applied voltage $v_\textsf{in}$ and the back-EMF voltage $v_b$ .

Examining Figure 14, applying the constitutive equations for the resistor and inductor yields the following differential equation:

L \frac{\dd i}{\dd t} + R\, i + v_b = v_\textsf{in}

(2)

The back-EMF voltage $v_b$ is proportional to the angular velocity $\omega$ of the motor shaft, so it satisfies a relationship of the form

v_b = K_b \, \omega

(3)

where $K_b$ is the back-EMF constant of the motor, measured in volts per radian per second (V/(rad/s)), or volts per rpm (V/rmp). This constant depends on the motor construction, including the number of windings, magnetic field strength, and geometry.^[3]

Mechanical dynamics¶

The mechanical dynamics are a rotational inertia (the rotor) with viscous friction and two torques: the load torque $T_L$ and the motor-produced torque $T$ .

Applying Newton’s second law for rotational motion to Figure 15 yields the following differential equation:

J \dot \omega + b\,\omega = T - T_L

(4)

The motor-produced torque $T$ is proportional to the current $i$ flowing through the windings, so it satisfies a relationship of the form

T = K_t \, i

(5)

where $K_t$ is the torque constant of the motor, measured in newton-meters per ampere (Nm/A). This constant also depends on the motor construction

Relationship between torque and back-EMF constants

The back-EMF constant $K_b$ and torque constant $K_t$ are actually equal when expressed in consistent units. Specifically, if $K_b$ is in V/(rad/s) and $K_t$ is in Nm/A, then $K_t = K_b$ (the units are actually the same!). This equality follows because power delivered to the electromagnetic (EM) coupling equals the power delivered from the EM coupling:

\begin{aligned} P_\textsf{to} &= v_b \, i = (K_b \, \omega) \, i = K_b \, i \, \omega \\ P_\textsf{from} &= T \omega = (K_t \, i) \, \omega = K_t \, i \, \omega \end{aligned}

(6)

Since $P_\textsf{to} = P_\textsf{from}$ , it follows that $K_t = K_b$ . This relationship is true despite losses due to resistance and friction, which dissipate power as heat; those losses occur outside the EM coupling itself. The complete flow of power through the motor is as follows:

Electrical power provided by source: $v_\textsf{in} i$
Some of this power is lost to electrical resistance: $R i^2$
Remaining power is delivered to EM coupling: $v_b i = v_\textsf{in} i - R i^2$ from Eq. (2)
Equal to the power delivered from EM coupling: $T \omega$
Some of this power lost to mechanical friction: $b\omega^2$
Remaining mechanical power delivered to load: $T_L \omega = T\omega - b\omega^2$ from Eq. (4)

So the equality $K_t = K_b$ holds true as a consequence of energy conservation within the electromagnetic coupling itself; it does not depend on the losses that occur elsewhere in the motor. For this reason, DC motors are often specified using a single constant $K$ (in Nm/A or V/(rad/s)) that serves as both the torque constant and back-EMF constant. It is often just called the motor constant $K$ .

Note: There is a different “motor constant” defined as $K_m = K/\sqrt{R}$ , which measures how efficiently the motor converts power into torque, regardless of how the motor is wound. Unfortunately, many textbooks and datasheets use the term “motor constant” to loosely refer to any of $K_b$ , $K_t$ , $K_v$ , or $K_m$ . However, once you specify the units, the meaning becomes clear.

Since the entire motor design can be boiled down to a single constant $K$ , we should talk a bit about what this constant means.

Meaning of the motor constant

The motor constant $K$ captures how strongly the electrical and mechanical domains are coupled in a motor. From the definitions,

T = K i\qquad\textsf{and}\qquad \omega = K^{-1} v_b

(7)

Let’s interpret these relationships:

From the first equation, a larger value of $K$ means that the motor can produce more torque for the same amount of current. This suggests that a larger $K$ is better for applications requiring high forces or rapid acceleration when current is limited (e.g., due to thermal limits, driver limits, or supply constraints).
From the second equation, a smaller value of $K$ means that the motor spins faster for the same back-EMF voltage. This suggests that a smaller $K$ is better for when high speed is required under limited voltage (e.g., fixed supply voltage).

You can think of $K$ as an electrical gear ratio: the total power capability (torque times speed) is roughly the same, but you can trade off speed versus torque by changing $K$ .

Generally, to obtain a larger $K$ , one must either use stronger magnets, make the motor larger (larger radius means larger moment arm and therefore more torque for the same force), or increase the number of windings. Increasing the windings also increases the resistance $R$ and inductance $L$ , which are detrimental, but this can be avoided by using thicker wire (more copper). So ultimately, increasing $K$ usually means making a motor larger, heavier, and more expensive.

Equations of motion¶

The equations (2), (3), (4), (5), describe the dynamics of the DC motor. Let’s rewrite them here for convenience, and we’ll set $K_b = K_t = K$ to simplify the notation:

\begin{aligned} L \frac{\dd i}{\dd t} + R\, i + v_b &= v_\textsf{in} && \textsf{(Electrical dynamics)}\\ v_b &= K \omega && \textsf{(Back-EMF voltage)}\\ J \dot \omega + b\,\omega &= T - T_L && \textsf{(Mechanical dynamics)}\\ T &= K i && \textsf{(Torque-current relationship)} \end{aligned}

(8)

We can eliminate the variables $v_b$ and $T$ by substituting the second and fourth equations into the first and third equations, respectively. This yields the pair of ODEs:

\boxed{\begin{aligned} L \frac{\dd i}{\dd t} + R i + K \omega &= v_\textsf{in} \\ J \dot \omega + b\,\omega - K i &= - T_L \end{aligned}}

(9)

In the next section, we will make some idealizing assumptions and solve modeling problems involving motors and other electromechanical devices.

Test your knowledge¶

Solution to Exercise 1 #

The two constants are the same, but they are expressed in different units. The speed constant $K_v$ is in rpm/V, To convert this to radians per second per volt, we can use the conversion factor $1 \text{ rpm} = \frac{2\pi}{60} \text{ rad/s}$ :

K_v = 135 \, \frac{\text{rpm}}{\text{V}} \times \frac{2\pi}{60} \approx 14.1 \, \frac{\text{rad/s}}{\text{V}}

(10)

The torque constant $K_t$ is given in Nm/A, which is the same as V/(rad/s). So based on this, the two constants should be reciprocals of one another. Let’s check:

\frac{1}{K_v} = \frac{1}{14.1} \approx 0.0707 = K_t

(11)

Footnotes¶

This principle is distinct from Maxwell’s equations. Maxwell’s equations govern how electric and magnetic fields are generated by charges and currents and how the fields influence each other, while the Lorentz force law specifies how those fields exert forces on charged particles.
↩
EMF stands for “electromotive force,” which is a historical term for voltage. Despite the name, EMF is not actually a force; it is measured in volts (V), not newtons (N).
↩
Sometimes the relationship is written the other way, as $\omega = K_v v_b$ , where $K_v = 1/K_b$ is called the velocity constant, measured in (rad/s)/V or rpm/V.
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References¶

Hansen, P. (2022). On Automotive Electronics. ATZelectronics Worldwide, 17(9), 24–27. 10.1007/s38314-022-0821-1