Motor performance

In this section we’ll dive into more depth on the performance characteristics of DC motors. We’ll derive the torque-speed curve, efficiency, and power output of a DC motor, and see how these depend on the motor parameters.

We begin with the standard DC motor equations from Eq. (1) in the main text:

We consider steady-state operation, so all time derivatives are zero.

The maximum speed (and minimum current) occur when there is no load torque, i.e., $T_L = 0$. From (2), we can solve for the no-load speed and current:

The maximum torque (and maximum current) occur at stall, i.e., $\omega = 0$. From (2), we can solve for this stall torque and current:

If there was no friction ($b=0$), the the no-load speed would be

which is larger than $\omega_\textsf{nl}$ in (3). In other words, friction reduces the no-load speed of the motor. The speed $\omega_\textsf{max}$ is often quoted as the “maximum speed” of the motor, even though it is not achievable in practice when there is friction.

The stall torque and current in (4) are unaffected by friction, since at stall the motor is not moving and thus frictional losses are zero.

Power and efficiency¶

Now let’s talk about power. The formulas for input power, output power, and motor efficiency are given by

We care about output power and efficiency, and we want to know when they are maximized and at what operating points (what speed, what load torque). This is essentially an exercise in algebra with a bit of calculus.

Since we are trying to derive the entire range of possible performance for the motor, it’s important to think about what is being held fixed and what is being allowed to vary.

Power-speed curve¶

Let’s parameterize performance in terms of motor speed $\omega$. Combining (2) and (6), we view this as a system of 5 equations in unknowns $(i,T_L,P_\textsf{in},P_\textsf{out}, \eta)$, treating $v_\textsf{in}$ and $\omega$ as fixed parameters. Solving this system, we obtain

It helps to normalize the equations by defining appropriate dimensionless variables.

where $\omega_\textsf{max}$, $i_\textsf{stall}$, $T_\textsf{stall}$ are given by (5) and (4). The reason for normalizing the power by $\frac{1}{4}T_\textsf{stall} \omega_\textsf{max}$ is that this quantity is the maximum output power, as we will see shortly. The parameter $\sigma>0$ captures the effect of viscous friction and resistance on motor performance: the larger $\sigma$ is, the more losses the motor has relative to its torque constant $K$. An ideal motor has $\sigma = 0$.

Substituting (8) into (7), we obtain normalized power-speed relationships:

We can analyze these equations to find various optimized operating points.

Maximum output power¶

To maximize output power $\bar P_\textsf{out}$, we take the derivative and set it to zero:

Converting back to non-normalized variables, we see that maximum power occurs at:

In other words, maximum power is achieved when the motor is loaded to half its stall torque, which causes it to run at half its no-load speed. The maximum output power decreases as friction and resistance increase (as $\sigma$ increases), and approaches the ideal value of $\frac{1}{4} T_\textsf{stall} \omega_\textsf{max} = v_\textsf{in}^2/(4R)$ as $\sigma \to 0$.

Maximum efficiency¶

Maximizing power output does not necessarily maximize efficiency. To find the speed that maximizes efficiency, we take the derivative of $\eta$ in (9) and set it to zero. This gives us optimal efficiency at the operating point:

In other words, the closer to ideal the motor is (the smaller $\sigma$ is), the higher the maximum efficiency. Moreover, the motor is most efficient when it runs faster and with less load torque. For example, if the motor has a maximum efficiency of 80%, then it achieves this efficiency at 90% of its no-load speed and loaded at 10% of its stall torque.

Power-speed plot¶

To see what different performance curves might look like, we can plot the solutions in Eq. (9) as a function of normalized speed $\bar \omega$ for different values of the parameter $\sigma$.

To get a sense of scale, consider a brushless drone motor. Typical values are given by

This corresponds to values of $\sigma$ ranging from 0.03 to 0.3.

From the plot in Figure 1, we can see that as $\sigma$ increases (i.e., as friction and resistance increase), both the maximum output power and maximum efficiency decrease. The shape of the power-speed curve also changes, becoming more skewed towards lower speeds. The efficiency curve also shifts downwards, indicating that the motor becomes less efficient at all operating points.

Power-torque curve¶

Instead of parameterizing performance in terms of motor speed, we can instead parameterize in terms of load torque $T_L$. Combining (2) and (6), we can view this as a system of 5 equations in unknowns $(i,\,\omega,P_\textsf{in},P_\textsf{out}, \eta)$, this time treating $v_\textsf{in}$ and $T_L$ as fixed parameters. We can carry out the same analysis as before and this time plot the results as a function of normalized load torque $\bar T_L$.

As before, we can normalize the equations using (8). Substituting into (14), we obtain normalized power-torque relationships:

This leads us to the following power-torque plot, where this time we vary normalized load torque $\bar T_L$ instead of speed, for different values of the parameter $\sigma$.