**DC Machine Dynamics**

DC Machine Dynamics – The DC machines are quite versatile and are capable of giving a variety of V-A and speed-torque characteristics by suitable combinations of various field windings. With solid-state controls their speeds and outputs can be controlled easily over a wide range for both dynamic and steady-state operation. By addition of the feedback circuit, the machine characteristics can be further modified. The aim of this section is to study dc machines with reference to their dynamic characteristics.

For illustration, let us consider the separately excited dc machine shown schematically in Fig. 7.68. For ease of analysis, the following assumptions are made:

- The axis of armature mmf is fixed in space, along the q-axis.
- The demagnetizing effect of armature reaction is neglected.
- Magnetic circuit is assumed linear (no hysteresis and saturation). As a result all inductances (which came into play in dynamic analysis) are regarded as constant

The two inductance parameters appearing in Fig. 7.68 are defined below:

L_{a} = armature self-inductance caused by armature flux; this is quite small* and

may be neglected without causing serious error in dynamic analysis

*L _{f}= *self-inductance of field winding; it is quite large for shunt field and must be accounted for

Mutual inductance (between field and armature) = 0; because the two are in space quadrature.

Further for dynamic analysis it is convenient to use speed in rad/s rather than rpm. Applying Kirchhoff’s law to the armature circuit,

where

Similarly for the field circuit,

For motoring operation, the dynamic equation for the mechanical system is

where J = moment of inertia of motor and load in Nms2

D = viscous damping coefficient representing rotational torque loss, Nm rad/s

Energy storage is associated with the magnetic fields produced by if and is and with the kinetic energy of the rotating parts. The above equations are a set of nonlinear* (because of products if(t)com and if(t)ia(t)) state equations with state variables if is and 04n. The solution has to be obtained numerically.

**Transfer Functions and Block Diagrams**

In the simple linear case of motor response to changes in armature voltage, it is assumed that the field voltage is constant and steady-state is existing on the field circuit, i.e./f= constant. Equations (7.98), (7.100) and (7.101) now become linear as given below

Laplace transforming Eqs (7.102) and (7.103),

These equations can be reorganized as V(s)— K;(0.(s)

From Eqs (7.106)—(7.108), the block diagram of the motor can be drawn as in Fig. 7.69. It is a second-order feedback system with an oscillatory response in general. It is reduced to simple first-order system, if *L _{a}* and therefore r

_{a}is neglected.

**Shunt Generator Voltage Build-up**

The qualitative explanation for the voltage build-up process in a shunt generator has already been advanced in Sec. 7.10. Here the mathematical treatment of this problem will be given, which in fact boils down to the solution of a nonlinear differential equation.

Referring to Fig. 7.70 it is seen that for any field current the intercept *ab, *between the OCC and the R_{f}-line gives the voltage drop caused by the rate of change of O_{f} and the intercept *be *gives the drop in the field resistance. The two together balance out the generated emf *e _{a}* (neglecting

*i*the armature drop). Thus

_{f}R_{a},The field flux **dPf **is greater than the direct axis air-gap flux * _{‘d}* because of leakage. Taking this into account

Here a is known as the *coefficient of dispersion. *Recalling Eq. (7.3),

Substituting Eqs (7.110) and (7.111) in Eq. (7.109),

Multiplying numerator and denominator by *N/F _{ag}*

where *T _{ag}* is the permeance of the air-gap/pole

It is easily recognized that the numerator is the unsaturated value of field inductance, *L _{f}, *and the denominator is the slope of the air-gap line. Both are constants. Hence,

This integral can be evaluated graphically by summing up the areas on a plot of 1/(e_{a} – R_{f}* i _{f}) *against

*e*This approach is employed to plot

_{a}.*e*against time. The theoretical time needed for the generated emf to attain the no-load value, e

_{a}_{ao}would be infinite; hence in practice the time needed to reach 0.95

*e*is taken as the time needed to reach e

_{ao}_{ao}. The variation of

*e*with time is plotted in Fig. 7.71.

_{a}The response is rather sluggish since only small voltage differences (= *e _{a} – R_{f}id *contribute to the flux build-up (19.