A differential equation for the equivalent circuit can be derived by using Kirchoff's voltage law around the electrical loop. Kirchoff's voltage law states that the sum of all voltages around a loop must equal zero, or [1] v a −v R−v L −v b=0
(2.1) According to Ohm's law, the voltage across the resistor can be represented as
v R=i a Ra
(2.2) where I a is the armature current. The voltage across the inductor is proportional to the change of current through the coil with respect to time and can be written as v L =La
(2.3) where
d ia dt
(2.3) vb
The motor back back electromotive-force voltage,
which is also known as speed voltage, is expressed as v b =K v ω i f
(2.4)
Where ia=I a is a constant armature current, and K m is defined as the motor constant. Substituting eq’s. (2.1),(2.2), and (2.3) into eq. (2.4) the instantaneous armature current can be found from the following differential equation: v a =Ra i a+ L a
(2.5)
di a −K b ω=0 dt
The armature current is related to the input voltage applied to the armature by R a + La s ¿I ¿ V a ( s )=¿
(2.7)
Then armature current equation is I a ( s) =
V a ( s )−K b ω(s) R a + La s
(2.8)
Performing an energy balance on the system, the sum of the torques of the motor must equal zero. Therefore, T e−T ω −T ω−T L =0 (2.9) '
where
Te
is the electromagnetic torque,
T ω'
is the torque due
to rotational acceleration of the rotor, T ω is the torque produced from the velocity of the rotor, and TL is the torque of the mechanical load. The electromagnetic torque is proportional to the current through the armature winding and can be written as T e=K m i a (2.10) where K t is the torque constant and like the velocity constant is dependent on the flux density of the fixed magnets, the reluctance of the iron core, and the number of turns in the armature winding. T ω can be written as T ω' =
Jdω dt
(2.11)
where J is the inertia of the rotor and the equivalent mechanical load. The torque associated with the velocity is written as T ω=Bω (2.12) where B is the damping coefficient associated with the mechanical rotational system of the machine. Substituting eq’s. (2.10), (2.11), and (2.12) into eq. (2.9) gives the following differential equation: K m i a−J
dω −Bω−T L =0 dt
(2.13)
With ω ( t )=d θ(t) /dt then ω ( s )=sθ (s) which is the transform of the angular speed so the Laplace transform of eq.(2.13) [2] K m I a ( s )−J sω ( s ) +b ω ( s ) −T L ( s )=0
(2.14)
The relations for the armature-controlled DC motor are shown schematically in Figure 2.2. Using Equations (2.7), (2.14), and (2.15) or the block diagram, and Jetting T;,(s) = 0, we solve to obtain the transfer function G ( s )=
Km θ (s) = V a ( s ) ( Ra + La s ) ( Js +b ) + K b K m
The differential equations given in eqns. (6F.5) and (6F.10) for the armature current and the angular velocity can be written as d I a Ra K V = I a− b ω− a dt La La La T dω K t B = I a− ω− L dt J J J
When a separately excited motor is excited by a field current of if and an armature current of ia, flows in the armature circuit, the motor develops a back Electromotive force (emf) and a torque to balance the load torque at a particular speed. The field current, if, of a separately excited motor is independent of the armature current, ia, and any change in the armature current has no effect in the field current. The field current is normally much less than the armature current. The equations describing the characteristics of a separately excited motor can be determined from Fig. 2. The instantaneous field current, if, is described as [3] v f =R f i f + Lf
di f dt
The field current is related to the field voltage as
R f + Lf s ¿I ¿ V f ( s )=¿
The torque developed by the motor is T d=K t i f i a
The developed torque must be equal to the load torque: T d=J
The motor torque
T m ( s)
dω +Bω+T L dt
is equal to the torque delivered to the
load. This relation may be expressed as T m ( s ) =T L ( s ) +T d ( s ) T L (s )
where
is the load torque and
T d (s )
is the disturbance
torque, which is often negligible. However, the disturbance torque often must be considered in systems subjected to external forces such as antenna wind-gust forces. The load torque for rotating inertia, as shown in Figure 2.18, is written as 2
T L ( s )=J s θ ( s )+bsθ ( s )
Rearranging Equations (2.55)-(2.57), we have T L ( s )=T m ( s )−T d ( s ) T m ( s ) =K m I a ( s )
I f ( s )=
V f ( s) Rf + Lf s
Therefore, the transfer function of the motor-load combination, with T d ( s )=0 , is G ( s )=
Km K m /(J Lf ) θ( s) = = V f ( s ) s ( Js+b ) ( L f s+ R f ) s ( s +b /J ) ( s+ R f ¿ L f )
The block diagram model of the field-controlled DC motor is shown in Figure 2.19
References 1. Prof. Dr. İsmail H. ALTAŞ , Dynamic Model of a Permanent Magnet DC Motor, Karadeniz Technical University ,
Turkey
2-C dorf 3- Power-Electronics-Circuit-Devices-and-Applications-by-Muhammad-H-Rashid
