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To: My Elek-Trak
co-conspirators
From: Steve
Naugler
Subject: An unsolicited lecture on
DC motor theory, dynamic braking, and regeneration.
For any electrical engineers out there, this
will be somewhat shallow, but it may help others on understanding speed control
and braking of our DC motors.
First lets define some
variables:
N = shaft rpm
T = shaft torque
I = armature current
V = armature voltage
R = armature resistance
f = field strength
i = field current
v = field voltage
r = field resistance
rfw = field weakening resistance
k1 = constant1, etc
Vbatt = battery pack voltage = 36 V
DC motor theory:
A perfect DC motor
without resistance will have the following relationships:
(1a) N = k1 x V / f
(1b) V = f x N / k1
Here speed follows armature voltage if the
field strength does not change, as in a permag motor. Increase armature
voltage, go faster.
(2a) T = k2 x I x f
or
(2b) I = T / (k2 x
f)
Here torque follows armature current if the
field strength does not change, also as in a permag motor. Work the motor
harder, current draw will go up.
Now we'll add armature
resistance:
(3a) N = k1 x (V - (I x R)) / f
or
(3b) I = (V / R) - ((N x f) / (k1 x
R))
Plug equation (2b) into (3a) and get
eq (4).
(4) N = k1 x (V - (T/(k2 x f)))
This looks a little nasty, but notice here
that for any given armature voltage V, when you load the motor increasing motor
torque T, speed N goes down. That is why when you drive your tractor
uphill you slow down a little bit.
Now we will control the
field as in our traction motors.
(5) f = k3 x i
Plug equation (5) into
(4) and get eq (6).
(6) N = k1 x (V - (T/(k2 x k3 x i)))
This looks worse, but let me generalize: Armature voltage
goes up, motor speed N goes up. Torque T goes up, motor speed N goes
down. Field current i goes up, motor speed N goes down, or more
importantly to owners of E15s and E20s, if field current i goes down, motor
speed N goes up. This downwardly adjustment of i is the essence of the
field weakening circuitry in the large framed E15's and E20's. In all
speeds 4 and above, the fields are weakened and the motor speeds
up.
You might ask why we just don't control speed with
field weaking. More equations will give that answer. Now we add
armature resistance effects to the torque equation, which is just an algebraic
rearrangement of equation (6).
(7) T = k2 x k3 x i x (V -(N /
k1))
You can see that as
field current i goes down, so does motor torque T. Carried to an extreme
your field weakened motor can get fairly wimpy. So why not make field
current i go up to make the motor torque T also go up. Well, you run out
of voltage. Without a DC to DC converter your maximum field voltage v
cannot exceed your battery voltage Vbatt.
(8a) i = v / r
or
(8b) i(max) = Vbatt / r
or
(8c) i = Vbatt / (r + rfw)
Now lets
discuss dynamic braking. Dynamic braking basically means that you brake
the motor using electricity to do the work vs. mechanical braking which is done
via a mechanical brake. But how does it work. Lets look again at
equation (1a).
(1a) N = k1 x V / f
By shorting the
armature leads together, V must = 0, so N must = O rpm. Simple in concept,
but obviously the motor can't stop instantly. However my math is not up to
the task demonstrating that clearly this Sunday. You are, however, giving
the motor a zero speed command.
One final note
about braking; the torque capability of a motor is still proportional to field
strength even when braking. In equation(2a)
T = k2 x I x f you can see if field strength drops to 0,
torque must also drop to 0. That is why our tractor accessories with wound
fields cannot have dynamic braking without rewiring to separately exciting the
field with added wiring.
Finally lets discuss regeneration. Regeneration means that energy is
reabsorbed by the motor and converted back to electricity usefully. Let's
look at equation (1b) and ignore resistances.
(1b) V = f x N / k1 Here you see that as rpm N goes up, so does
voltage. Increase speed N enough and armature voltage V exceeds the
battery voltage of Vbatt and you will charge the battery. How do you
increase speed; only by applying a negative torque, i.e., make the motor a
generator. You can also see that increasing the field strength increases
voltage output. You see that in the E15s and E20s when you decrease motor
speeds to speed 3 from any higher speed. (Speeds 1 and 2 in all tractors
use resistors in series with the armature and regeneration is harder to make
happen.) Well that's enough for now. I will
follow with some AC motor theory because of a request on how braking might be
applied to an inverter fed AC motor.
Steve
Naugler |