The actual value is largely determined by the precision of manufacture, primarily in coil winding.
typically generator output signal, 5
segment commutator
a) value of the induced voltage(1 V) with
actual ripple of 90 mV(9%)
b) (scale 10 times larger than a) minimum
theoretical ripple, frequency per revolution
equals twice the number of commutator
segments.
Figure 5
c) actual ripple value, twice per
revolution
d)signal for one complete revolution of
rotor
Tachometer Output Signal
5.0 REVIEW OF PHYSICAL LAWS GOVERNING ESCAP® MOTOR APPLICATIONS
5.1 Rectilinear Motion
5.1.1 Velocity, sometimes referred to as speed, is equal to distance (displacement) traveled per unit time (second).
Thus, v = dx
dt
(v)velocity = ∆(x)distance
∆ (t) time
meters
second
5.1.2 Acceleration is defined as the change in velocity per unit time. Therefore:
(a) acceleration = dv = d²x
dt
dt²
v = Vo + at
5.1.3 Displacement is the distance traveled after time (t).
Thus:
(x) = v0t + 1 at² + x0
2
and with zero initial velocity (when v0 = 0 and x0 = 0)
x = 1 at²
2
351
5.1.4 Force is defined as a mass (m) multiplied by the acceleration (a) which it is experiencing.
Thus: F = ma
Conversely, the mass of a body is represented by the force per unit acceleration.
5.1.5 The concept of inertia comes into play when describing the amount of force that is required to accelerate a body. A small
inertia requires less force to achieve a given acceleration than does a body with a larger inertia. Thus, load inertia is a critical
consideration when making a motor application.
5.1.6 When a force is applied to a body which is free to rotate about some axis, the product of the magnitude of the force and its
force arm is called the moment of the force about the axis.
The force arm is defined as the perpendicular distance between the line of action of the force and the axis of rotation.
It is this property of a moment of a force which produces torque and thus rotation of a body (load).
5.1.7 Work and energy
Work is defined as the product of displacement and the component of force in the direction of displacement. Thus:
W = Fx
The kinetic energy of a moving body is expressed as:
Thus, the work done in accelerating a body equals the increase in its kinetic energy.
5.1.8 Mechanical power
Average power:
and instantaneous power:
and under constant force: P = Fv
5.2 Rotational Motion
5.2.1 Angular velocity
NOTE: 2π radians = 1 rotation (360º)
By definition a radian is an angle whose arc length is equal to the radius of the arc.
Also, "speed" should not be used to describe angular velocity. Rotational speed can only be expressed as RPM or RPS.
352
5.2.2 Angular acceleration
Under acceleration angular velocity changes as follows:
ω = ωo + αt
5.2.3 Angular displacement φ after time (t) is defined as:
5.2.4 Handy equations of angular rotation physics with constant angular velocity
(ω = constant and φo = 0)
φ =ωt
alternate forms: w =
with constant acceleration (α = constant):
where ω = ωo + α t
∴a =
with constant acceleration (α) and with zero initial velocity (ωo = 0):
angular acceleration of a load (J) with constant torque (M) is:
a=
5.2.5 Moment of inertia. The moment of Inertia (J) of a body referred to an axis of rotation is the product of the mass of the
body and the square of the distance between the center of mass and the axis of rotation.
J = Σmr²
353
For a flywheel:
(J) = mr²
and for a homogeneous disc:
5.2.6 Energy, work and power
Rotational kinetic energy is expressed by:
And work (W) is defined as:
dW = Mdφ
where M is the torque causing the displacement (φ).
Under constant torque the work done over a given displacement is stated as:
W = M (φ2 - φ1)
Now
but
is rate of doing work (Power)
and
Hence, power P = M ω (under constant torque)
Net work = change in kinetic energy
and
∴M = J
= J α which is the rotational analogue of Newton's second law, F = ma in linear motion.
354
5.2.7 Momentum
The law of conservation of momentum states that the total momentum of a system can only be changed by external forces acting
on the system.
In rectilinear motion the system momentum is given as the product of mass and velocity (mv).
Whereas angular momentum is:
L=Jω
5.3 Basics of DC Circuits
5.3.1 OHM'S law
R = resistance
E =voltage
I = current
(ohms)
(volts)
(amperes)
variations:
E = lR and l =
5.3.2 Power
P = power (watts)
Power input to a network is computed as:
P = El
and the power (heat) dissipated in a resistance is given by:
p = l² R
5.3.3 Kirchhoff's rules
Point rule: The algebraic sum of the currents toward any point of a network is zero.
Σi = 0
loop rule: The algebraic sum of the voltage sources and the lR products in any loop of a network equals zero.
ΣE + ΣiR = 0
5.3.4 Series and parallel resistance
The equivalent resistance of resistors in series is the sum of the individual resistances.
Thus: R = R1 + R2 + R3 + ... RN
The equivalent resistance of resistors in parallel is computed as:
A special solution for only 2 parallel resistors is;
355