Why Does the Rotation Rate of a Motor Increase When the Voltage is Raised?

The question addressed in this article is “why does the motor rotation rate increase when the applied voltage is increased?” To put it more precisely, the question under consideration is why the rotation rate increases when the voltage being applied to the windings of a motor is raised. This might seem like something that should happen as a matter of course, but quite a few physics equations are needed to provide a proper answer. The physics in question is very relevant to an understanding of motors, so that’s where we will begin our explanation.

●Statement of the question

When two batteries are connected in series to a brushed motor, the rotation rate is faster than when one battery is used. Some people may have experienced something similar with light bulbs. A light bulb shines more brightly when connected to two batteries as opposed to one. Thus it may seem natural that the motor rotation rate will increase. However, among actual motors there are those for which the rotation rate changes when the voltage is increased, and those for which the rotation rate does not change. Motors for which the rotation rate changes include brushed motors as well as brushless motors; and motors that do not exhibit such a change of rotation rate include stepping motors and induction motors. We will begin by explaining how these motors are different.

The Difference Between Motors that Undergo a Change in Rotation Rate and Those that Do Not

Motors that do and do not undergo a change in rotation rate when the voltage is changed differ in the methods they use to control the rotating magnetic field.

Motors with an Unchanging Rotation Rate

In motors with a rotation rate that does not change, the speed at which the magnetic field is rotated is controlled; the motor is rotated by causing the rotor to follow the magnetic field. This action is explained using a stepping motor as an example.

First, suppose that a control circuit has created an electromagnet (winding magnetic field) at the position in figure (1) below. The rotor is pulled by the electromagnet and rotates, and at the position indicated in (2), the winding magnetic field waits for the rotor to catch up. The actual arrival of the rotor is not confirmed, however; at the time at which the rotor should have arrived, the magnetic field is rotated to the position in (3). When the rotor moves from the position in the figure (2) below to the state in (3), it is again pulled by the electromagnet and rotates. The stepping motor rotates by repeating this action. Hence the number of rotations of the motor depends on the speed at which the magnetic field is rotated.

Let us here consider what happens when the voltage being applied to the windings is increased. In such a case, the magnetic force of the electromagnet is stronger, and so the rotor catches up more quickly. However, even if the rotor movement is quicker, if the magnetic field of the electromagnet does not move, the rotor does not rotate any further. Hence even if the voltage is increased, the states in (2) and (4) just continue for a longer amount of time; the motor rotation rate is still determined by the rotation speed of the magnetic field.

The action in an induction motor differs somewhat from this, but when the magnetic field is rotated, the rotor lags behind (“slips”) but catches up, so that once again the speed of rotation of the magnetic field can be said to determine the motor rotation rate.

Motors with a Rotation Rate that Changes

On the other hand, a motor with a rotation rate that changes rotates the magnetic field of the electromagnet according to the position of the rotor. In other words, as the rotor approaches, the magnetic field recedes from it (see the figure below). In order to compare such motors with the motors described above having a rotation rate that does not change, we use as an example a two-phase full-wave brushless motor.

When the rotor is at the position in figure (1) below, the motor control circuit grasps the position of the rotor and creates an electromagnet at the position seen in (1). By creating the electromagnet at this position, the rotor is pulled and approaches. Before the rotor arrives, however, the control circuit moves the electromagnet magnetic field to the position of figure (2). The electromagnet field recedes from the rotor, and so the rotor continues to try to catch the magnetic field. And when the rotor approaches the position in (3), the magnetic field is again moved. The brushless motor repeats this action to continue rotating. Hence the rotation rate of the motor depends on the speed at which the rotor follows the field. This control method is one reason why the rotation rate of the motor changes.

Here we will consider what happens when the voltage applied to the windings is raised; the situation is a bit complicated. Put simply, as the voltage rises, the magnetic force of the electromagnet becomes stronger, and the speed at which the rotor follows also rises, so that the rotation rate increases. However, if this were all that was happening, the explanation would again be “increased power means a higher rotation rate”. Let us give a more precise explanation.

It should be noted that the brushed motors that were the subject of the question being asked have a construction in which a magnet rotates, unlike brushless motors. However, the electromagnet switching action and other aspects are the same as for brushless motors, and so like brushless motors, a change in voltage causes a change in rotation rate.

How is the Rotation Rate of a Motor Determined?

As stated above, motors in which the rotation rate changes with the voltage are motors that change the rotating magnetic field according to movement of the rotor. It is assumed that changes of the rotating magnetic field are performed automatically. Given this assumption, we here consider the rotational force (torque) that acts directly on the rotor, as well as the relationship between this torque and the rotation rate.

The Torque Acting on the Rotor

We first explain how the force (rotational force, torque) acting on the rotor is determined. In general, the torque (T) of a motor is proportional to the current (I) flowing in the windings. K _t is the torque constant.

Next is the question of how the current flowing in the windings is determined. The current flowing in the windings of either a brushless motor or a brushed motor is determined by the impedance (Z) of the windings and the voltage obtained by subtracting a voltage called the induced voltage (V_bemf) from the applied voltage (V_in) (see the equation below). In a motor that uses a permanent magnet, the induced voltage is the voltage generated by rotation of the rotor, and is calculated by multiplying the angular velocity (ω) by an induced voltage constant (K_e).
(The equations appearing here are intended to indicate the relations between physical quantities, and do not accurately represent the actual direct current or alternating current.)

From this, we see that the rotational force (torque) acting on the rotor is determined in terms of the applied voltage and the motor rotation rate. The torque is proportional to the current, and the current is calculated from the applied voltage, the induced voltage, and other factors. The induced voltage is proportional to the angular velocity (rotation rate).

Relationship Between Torque and Rotation Rate

These relationships alone are insufficient to explain what the rotation rate will be for a given applied voltage. Next we explain the relationship between torque and rotation rate for a motor.

Many readers have probably learned, in a physics course or elsewhere, the two equations below representing the relations between the force (F) acting on an object and the velocity (v) of the object. An acceleration (a) occurs according to the force and the object mass (m), and the velocity is obtained by integration of the acceleration.

The same thing can be said of a rotating body. The relationship between a rotational force (T) and an angular velocity (ω) is expressed by the following two equations. The rotational force (torque) and an inertia (J) causes an angular acceleration (α)to occur, and by integrating the angular acceleration we obtain the angular velocity.

In the case of a motor, when a load such as a propeller (fan) is attached, the load torque (T_L) necessary to rotate the load is subtracted from the rotational force. Hence the two following equations, which include this load torque, represent the relationships between the rotational force (torque) and the rotation rate of the motor.

We have obtained the equations needed to explain the relationship between the applied voltage and the rotation rate (angular velocity). Using these equations, we now explain how the rotation rate is determined.

First, to say that the rotation rate is “determined” means that the motor is in a state in which the rotation rate (angular velocity) does not change. In other words, the above equation (E) means that the angular acceleration becomes zero. From equation (D), the angular acceleration becomes zero when the torque and the load torque have the same value. The equation for the generation of the torque is equation (F), obtained by combining equations (A), (B), and (C). Upon examining equation (F), we see that the torque is adjusted by the quantity in the numerator, (V_in－K_e・ω).

From all of this, we can say that the rotation rate of a motor when a certain voltage has been applied is the rotation rate that generates the same torque as the load torque. When the applied voltage changes, the torque changes, and with this the rotation rate changes accordingly; the motor settles into a state in which the load torque and the torque are the same. This is the answer to the question of how the rotation rate of a motor is determined.

With this, let us return to our original question of why the rotation rate increases if two batteries instead of one are used with a motor. As premises we assume that the propeller load is proportional to the square of the motor rotation rate, and that constants such as K_t, K_e, and Z are equal to one.

In equation (F), if we assume that the applied voltage is 1.5 V when one battery is connected, then at a rotation rate of 1.0 the torque and the load torque are equal. If, from this state, the number of batteries is increased to two, the applied voltage becomes 3.0 V; if the rotation rate were still 1.0, then the torque would be 2.0, and would not be the same as the load torque. The torque would be greater than the load torque, so that the motor would accelerate (equation (D)). As the rotation rate rose, the torque would fall, while the load torque would rise. And at a rotation rate of 1.65, the torque and the load torque are again equal and stable.

One might think that, if the load torque constant is larger than 0.5, then we can be sure that the rotation rate will increase. In the case of a propeller load, this is true. When the applied voltage rises and the torque increases, the only way for the torque and the load torque to become equal is for the rotation rate to increase.

This is clear if we use the equations to make a graph. In the graph below, the vertical axis is the torque, and the horizontal axis is the rotation rate. As the torque equation, if we use T=-Aω+BV_in with the constants substituted, we obtain a graph of a straight line sloping downward to the right. When V_in changes, the graph undergoes parallel movement. If the load torque equation is T_L=Cω with C as the constant, we obtain a graph of a quadratic function. The point of intersection of these two graphs indicates the rotation rate and torque at which the motor operates.

When the applied voltage (Vin) changes, the point of intersection moves and the rotation rate (ω) changes. From the graphs, we see that as the applied voltage rises, the rotation rate rises (ω_a1⇒ω_a2, ω_b1⇒ω_b2).

The above concludes our explanation of the question posed here of why the rotation rate of a motor rises with rising voltage.

To summarize:

・In a motor with a rotor that creates a magnetic field that rotates in accordance with the movement of the rotor, the rotation rate changes depending on the voltage.
・The rotation rate of a motor is stable when the motor torque and the load torque have become the same.
・The torque of a motor is related to the applied voltage and the rotation rate. When the voltage rises, the rotation rate also rises.