Equations to set up an understanding of rotating machinery
¶ Lorentz equation and torque on armature loop
¶ Back EMF derived from Faraday's equation
¶ Invoking Lenz' Law for sign of back EMF
¶ Torque Speed curve from KVL
¶ Calculating power delivered to load
¶ Generator and Braking sections of curve
¶ Brushless (no commutator) DC motor design
from Kenjo & Nagamori, Permanent Magnet and Brushless DC Motors, Oxford University Press, 1985. pages 2-17 of chapter 1.
David McCauley, The (new) Way Things Work, Houghton-Mifflin (1998), pp 280-81.
Consider the Lorentz force:
where v is the the velocity of charge q, B is the magnetic field and E is the electric field.
Put the second term of the Lorentz equation in differential form:
Let a wire of length L carry a current I in a direction perpendicular to magnetic
By the right hand rule the direction of the force is into the page.
Now integrate the differential form:
If that wire is part of a rectangular loop with upper and lower sides L and height 2R ("radius") then following diagram, with the loop going into the page, and the current source in the form of a battery Vs and resistor Ra shown, we have the following forces:
therefore there is a torque on the loop, with 2*F*R the magnitude of the torque, substituting our force formula,
How to get around the 3/8 turn limit? Commutator!
CW or CCW? As given to you in the example above, with you facing the commutator, the rotation is counter-clockwise (CCW). What change would be necessary for the rotation to turn CW? Reverse the sign of the voltage source sending current through the coils!
Example: How a cone-type speaker works, with a permanent
magnet surrounding a coil. The coil has the sound wave AC current pass through
take the cross product of the wires coming out of the page and going into the page, with the B field being reversed (top and bottom) to see the movement of the cone the coil is attached to, going left and right above. The AC in the coils creates audio-range sinusoids. Audio amplifiers increase the current through the coils.
Note that KE, the EMF factor, is the same formula as KT, the torque factor. What about the minus sign in the back EMF formula? We do know from Lenz' Law that the correct sign of the back EMF must be as shown below. Therefore the notation "KΩ" refers to the back EMF correctly labelled. We will carry the constant through the succeeding calculation.
If torque is 0 then ΩMAX = V0 (the "maximum" rotational speed) = Vs/K.
If rot. speed Ω=0 then T0 = KV/Rs
Notice what happens as the external voltage Vs drops: both max torque and max speed drop. The red line above moves closer to the origin.
Where Z (the number of turns) comes in:
And with Z turns the torque exerted by the motor is now T = Z*K*I(t).
Example of a Motor Quiz
Power delivered: Power is Torque X
speed. At zero speed or zero torque power is zero, and so too at Ω-MAX
. If you multiply the expression above for torque by Ω you will have a
second order expression for power as a function of rotation speed--an upward
pointing parabola. Where will Power be maximum?
A different answer would appears when considering rotational speed for maximum power in Hill's equation for muscle. See notes for Lab VT...
When the motor is made to turn against its natural direction, it acts as a brake. The electric energy brakes the mechanical. Both mechanical and electrical energy enter the motor and it is in danger of overheating. How will electric braking work in an electric train going downhill? Reverse the voltage to the motor!
When the motor is turned in the direction to produce a back EMF, it acts as a generator (the tach in the Fan Lab). Consider the case where no battery voltage is applied: an EMF will be seen; or if there is a chemical battery (like in car engine) the battery can be charged. A generator looks like a motor with no power supply!
next: Brushless DC motors, with no commutator or brushes; wherein the surrounding magnetic fields have their polarities switched. Below: H-bridge for switch current direction through brushless DC coil.