# Understanding equivalent circuit of permanent magnet brushed DC motor

I started to work on a project, where I have an unknown permanent magnet brushed DC motor. To be able to control it, a model is needed. Starting with measuring armature resistance and inductance, I also got the RPMs for a range from 4-16 V. Using those measurements, I have calculated the motor speed constant via Vnoload/RPMnoload.

With these, I felt like I can start to model it in Simulink, but doing some research I got confused with the equivalent circuit. Most sources contain armature resistance, armature inductance, and the back EMF generated by the motor. Some sources say, that the armature inductance doesn't have to include in the equivalent circuit. Which one is true, and why?

Further question: I first thought the coil in the circuit is to represent the coils in the motor, which get energized and create the magnetic field. Then I thought, that this is a part of the back EMF anyways and now I am really confused. Could someone explain to me what those stand for and their differences?

Thank you in advance!

• Could be of interest (in french) stephane.genouel.free.fr/FT/Dossier_Multimedia/Moteurelectrique/… or this control.isy.liu.se/student/tsrt21/file/pm_dcmotor.pdf Feb 10, 2023 at 13:38
• You should not mix -- back EMF from Lm -- and "back EMF" from the motor (em). Feb 10, 2023 at 13:48
• Thanks for the links, looking into them now! You are right, I mixed them. Reading more into Inductance, I am still not able to clearly differentiate them from each other. Could you explain what differs back EMF and Lm? I understand, that back EMF is generated by a current flowing through a conductor in a magnetic field that is changing. In this case, it is changing because the orientation of the coil to the magnetic field changes, and, therefore, the effective magnetic field changes. For Lm, I have no understanding at all... Feb 10, 2023 at 13:59
• Back EMF for an inductor is only the voltage calculated with L*di/dt -> if varying current in the inductor (generally in terms of millisecond). em is the voltage resulting from a "rotating" inductor (with brushes !!!) in a continuous constant armature flux (does not change, quasi-continuous). Feb 10, 2023 at 14:34
• OK, thanks for the explanation. So to double check: If there would be no magnetic field around the rotor, we would still have Inductance caused by di/dt, right? So that property is reflected by the coil L in the equivalent circuit. And back EMF is the voltage induced due to the rotation of the coils inside the magnetic field? Feb 10, 2023 at 17:03

It depends as always on exactly what aspects of the system you are trying to model.

For example, you can start with the second one of your models, and note that torque is proportional to current, and back EMF is proportional to speed.

so I = (V - kR)/Ra where V is terminal voltage, R is RPM, k is a constant, and Ra is the resistance with a locked rotor.

Torque is then simply It, where I is the current and t is a torque constant.

You can then include the load friction and mass moment of inertia to get an integral that gives speed as a function of applied voltage and time, as well as computing current as a function of time during transient changes (Startup, stop), and the resulting copper losses.

This makes various assumptions about the magnetics and windings, but is often good enough for modelling gross behaviour.

Your first model is the same but assumes that winding inductance is not trivial over the time scales that matter to the system and includes the variables and constants that will matter to writing that integral. This would be the case if for example you are operating at a speed where the rise time of the current in the winding due to commutation becomes significant (Stepper motors often require far more volts then the winding resistance would imply for this reason when operated at high speed), it is not usually a major issue in small PMDC machines.

There are lots of things you can choose to model at lots of levels of detail, thermal behaviour for example, would be another slightly different model, especially if iron losses are not trivial or the motor has a fan fitted onto it's shaft, where air flow varies with motor speed.

You pick your equivalent circuit based on the things that matter to you and the level of detail that is appropriate to your problem.

• Thanks for that input, it cleared some questions already! Could you further explain how you concluded that torque is dependent on current and back EMF dependent on rotational speed? If I look at the second model, I would use KVL and could conclude, that back EMF is dependent only on speed. Which law/method is needed to conclude that Torque is current dependent only? Feb 10, 2023 at 13:03
• For torque, the Lorentz force law, the force between the wire and the magnet is proportional to the current flowing in the wire (Making some assumptions about the magnetic circuit). For back EMF, Faradays law is perhaps the most obvious expression, but you can get there a few different ways. Feb 11, 2023 at 2:39