EE Pro for TI-89, 92 Plus
Equations - Motors and Generators
127
31.2 DC Generator
The first equation describes the relation between electrical radian frequency
ω
me, the mechanical radian frequency
ω
m, and the number of poles in the generator p. The next equation expresses the emf generated per turn Eta with
the relative motion of the coil with respect to the magnetic field
φ
.
ω
ω
me
p
m
= ⋅
2
Eq. 31.2.1
Eta
p
m
= ⋅
⋅
π
ω φ
Eq. 31.2.2
The next two equations illustrate two ways to express the induced armature emf Ea as a function of number of
armature coils N, the number of parallel paths ap, number of poles p, the mechanical radian frequency
ω
m, a
machine constant K, and flux
φ
. The machine constant K, is seen to be dependent purely on the characteristics of
the machine.
Ea
N
ap
p
m
=
⋅ ⋅
⋅
π
ω φ
Eq. 31.2.3
Ea
K
m
= ⋅
⋅
ω φ
Eq. 31.2.4
K
N p
ap
=
⋅
⋅
π
Eq. 31.2.5
The sixth equation shows the conversion of mechanical energy available as torque T and mechanical angular
velocity
ω
m to its electrical counterpart – namely, the emf and current in the armature Ea, and Ia and the voltage
and current in the field windings Ef and If. The next equation for torque connects T with K,
φ
, and the current Ia.
T
m
Ea Ia
Ef IIf
⋅
=
⋅ +
⋅
ω
Eq. 31.2.6
T
K
Ia
= ⋅ ⋅
φ
Eq. 31.2.7
The armature resistance is given by the equation for Ra in terms of N, ap, coil length L, area A and its resistivity
ρ
.
Ra
N L
ap
A
= ⋅ ⋅
⋅
ρ
2
Eq. 31.2.8
Vf represents the voltage across the field winding carrying a current IIf and a resistance Rf. The terminal voltage Vt
represents the induced voltage minus the IR drop in the armature..
Vf
Rf IIf
=
⋅
Eq. 31.2.9
Vt
K
m
Ra Ia
= ⋅
⋅ −
⋅
ω φ
Eq. 31.2.10
The final equation represents the shaft torque Ts needed to generate the induced emf, assuming a given value for
equivalent loss of torque Tloss
Ts
K
Ia
Tloss
= ⋅ ⋅ +
φ
Eq. 31.2.11
Example 31.2 -
A six-pole DC generator rotates at a mechanical speed of 31 rad/s. The armature sweeps across
a flux of 0.65 Wb. There are eight parallel paths and 64 coils in the armature. The armature current is 12 A. The
field is supplied by a 25 V source delivering a current of 0.69 A. Find the torque and the voltages generated in the
armature.
