MC-B3 · DC motor torque & speed

Torque and Speed of the DC Motor

The turning armature generates even in a motor, and that back emf pushes against the supply voltage and sets the speed. Change the field flux, see which way the speed goes, and check for yourself a result that runs against intuition.

Weaken the field flux — where does the speed go?

As the bar diagram shows, the back emf Eb = kΦN nearly matches the terminal voltage V and balances it, so the speed is set by N = (V - IaRa)/(kΦ). Before turning the flux slider down, pick below what you expect the speed to do; noticing that Φ sits in the denominator lets you guess the answer in advance. That result, where intuition and the formula disagree, is the question of this lesson.

Field flux ΦΦ = 1.00

Slide for continuous flux. The curve is the speed N.

Speed set by the back-EMF balance

N = (V - IaRa)/(kΦ) = 1500 rpm

It seems that reducing the flux should slow the motor, but it is the opposite. The back emf Eb = kΦN has to stand against a nearly constant terminal voltage, so when Φ gets smaller, N must grow for the balance to hold.

Lower Φ — N goes?

Misaligned

Back-EMF is the latch that limits current

A motor's armature also turns in the flux and generates. This back emf Eb = kΦN pushes against the supply voltage, so the armature current is set by Ia = (V - Eb)/Ra, and the larger the back emf, the smaller the current. At the moment of starting, N = 0 so Eb = 0, and the current V/Ra becomes very large, which is why large motors hold down the starting current with a starting resistor or a reduced voltage. The back emf acts as a bolt that holds the current back.

ObserveEb = kΦN

The back emf is proportional to the product of flux and speed.

ChooseIa = (V - Eb) / ?

The armature current is the leftover voltage divided by the armature resistance.

Torque is flux times current

A conductor carrying current in a flux feels a force F = BIl, and adding up the torque on all the armature conductors gives T = kΦIa. When the load grows, the speed drops a little, the back emf falls by that much, Ia = (V - Eb)/Ra rises, and the torque grows with it until a new balance is reached. This property of the motor following its load without any separate control is thanks to the back emf.

On your ownT = ?

Torque is proportional to the product of flux and armature current.

Three handles to set the speed

N = (V - IaRa)/(kΦ) has three handles. Field control reduces Φ to raise the speed above rated (constant power, high speed); voltage control lowers V to bring the speed down over a wide range below rated (constant torque); and resistance control inserts series resistance in the armature to lower the speed, but the loss in that resistor is large and the efficiency poor. All three differ only in which term of the same formula they touch. If the field circuit of a shunt motor breaks, though, Φ nearly vanishes and the speed shoots up dangerously, so no separate fuse or breaker is put in the field circuit.

Fill inN = (V - IaRa) / ?

Speed has the form of back emf divided by flux.

Back to the first screen

Reducing the flux made the motor faster because the back emf Eb = kΦN has to balance against a nearly constant terminal voltage. When Φ in the denominator shrinks, the point where the balance is restored is a larger N. In the same formula N = (V - IaRa)/(kΦ), lowering V lowers the speed, and so does raising IaRa with series resistance. Torque is T = kΦIa, the product of flux and current, and when the load changes, the back emf shifts and the current follows, so the torque is matched. The picture keeps IaRa constant, but it is worth remembering that in reality, to produce the same torque, the armature current must grow by as much as Φ shrinks.

The DC motor: the spinning armature’s back-EMF Eb = kΦN opposes the terminal voltage, so the armature current is Ia = (V - Eb)/Ra, the speed N = (V - IaRa)/(kΦ), and the torque T = kΦIa. With Φ in the denominator, weakening the field raises the speed. Speed control is the three handles V, Φ and R of the same equation.

Once you hold this balance

The back-emf balance is the root of every characteristic a DC motor shows. Depending on whether the field is connected in series or in parallel, the relation between Φ and Ia changes, giving the different torque-speed curves of series, shunt, and compound motors. The next lesson covers the external characteristic and voltage buildup when the same machine runs as a generator (MC-B4). There, too, the starting point is Eb = kΦN.