# 2 Modeling of DC machine

```2 Modeling of DC machine
• 2.1 Theorem of operation
• Maximum torque is produced when two
•
2.2 Induced EMF
• From Faraday’s law, the induced emf is
dφf
φf
e=Z
=Z
dt
t
where t is the time taken by the conductors
to cut φf 1flux lines.1 Therefore,
t=
2 &times; frequency
=
p n
2( )( r )
2 60
• Thus,(P: poles; Z: armature conductors; φf:
a flux per pole; nr: rotation speed)
e=
Zφf Pn r
60
• If the armature conductors are divided into ‘a’
parallel paths, then
e=
Zφf Pn r
60
wave winding: a = 2; lap winding: a = P.
• The usual expression
e = Kφfωm
• If the field flux is constant, then emf is
e = Kbωm
2.3 Equivalent circuit and electromagnetic torque
• The terminal relationship is
v = e + R a ia + La
dia
dt
• In steady state, the armature current is
constant and hence
v = e + Raia
• The power balance
via = eia + Raia2
•
• The power of mechanical form, Pa called
the air gap power, is expressed in term of
the electromagnetic torque and speed as
Pa = ωmTe = eia
• Hence,
Te =
eia K bωmia
=
= K bi a
ωm
ωm
2.4 Electromechanical modeling
• The acceleration torque, Ta, drives the load and
is given by
J
dωm
+ B1ωm = Te − T1 = Ta
dt
where J: a moment of inertia (kg-m2/sec2)
B1: a viscous friction coefficient N&middot;m/(rad/sec)
2.5 State-space modeling
• The dynamic equations in state-space form
⎡ Ra
−
⎢
⎡ pia ⎤
La
=
⎢ pω ⎥ ⎢ K
⎣ m⎦ ⎢ b
⎢⎣ J
Kb ⎤
⎡ 1
−
La ⎥ ⎡ i a ⎤ ⎢ La
⎥⎢ ⎥ + ⎢
B1 ⎥ ⎣ωm ⎦
⎢ 0
−
⎢⎣
j ⎥⎦
⎤
0 ⎥ V
⎡ ⎤
⎥⎢ ⎥
1 ⎥ ⎣T1 ⎦
−
J ⎥⎦
• The roots of the system are
R a B1
R a B1 2
R a B1 K 2b
)
−(
+ )&plusmn; (
+ ) − 4(
+
La
J
La
J
JLa
JLa
λ1, λ 2 =
2
2.6 Block diagrams and transfer functions
• From (2.13) and (2.19), we get
I a (s ) =
V(s) − K bωm (s)
R a + sLa
K b Ia (s) − T1(s)
ωm =
( B1 + sJ )
• The transfer functions
Kb
ω (s)
G ωV (s) = m = 2
V(s) s ( JLa ) + a ( B1La + JR a ) + ( B1R a + K 2b )
ω (s )
− ( R a + sLa )
G ωl (s) = m = 2
Ts (s) s ( JLa ) + s( B1La + JR a ) + ( B1R a + K 2b )
• The speed response due to the simultaneous
voltage input and load torque disturbance is
ωm(s) = GωV(s)V(s) + Gωl(s)T1(s)
2.7 Field excitation
• Separately excited dc machine
• Shunt-excited dc machine
• Series-excited dc machine
• DC compound machine
• Permanent-magnet dc machine
2.8 Measurement of motor constants
• Armature resistance:
It is measured between the armature terminals by
applying a dc voltage. (need to subtract the brush
and contact resistance)
• Armature inductance:
By the test schematic shown in Figure 2.11, the
inductance is
Va2
− R a2
fs: the frequency
Ia
La =
2 πf s
Ra: the armature ac resistance
• EMF constant
Specified field voltage is applied and kept
constant, and the shaft is rotated by another
dc motor, and then the armature is
connected a voltmeter.
•
```