Static and Dynamic Characteristics of DC Servo Motors

Static Characteristics of DC Motors

In the general case for all types of DC motor control, the following equations express the steady-state behavior of the motor:
V_t= I_aR_a + KI_fω(1)

T= KI_fI_a(2)

Where:
V_t= armature terminal voltage, in volts
I= armature current, in amperes
I_f= field current, in amperes
R= armature resistance, in ohms
K= a constant
T= torque produced by motor, in newton-meters
ω= speed of armature-shaft rotation, in radians/sec (ω= 2π X rpm/60)

The above general equations may be modified for the following particular types of servo motor control:
(a) Armature control with constant field current
(b) Field control with constant armature current
(c) Field control with constant armature voltage
(d) Series motor control

Armature Control with Constant Field Current

Since the field current is constant in this type of servo control, the term KI_f, may be combined with a new term K₁, and substituted into Eqs. (1) and (2). Then Eq. (2) is substituted into Eq. (1) which is solved for w to obtain
ω= V_t/K₁ - TR_a/K₁²

Solving Eq. (1-3) for T
T= K₁V_t/R_a - K₁²ω/R_a(4)

The stall torque T_M which occurs at the maximum armature terminal voltage V_Tm is found from Eq. (4) when T= T_M, V_t= V_Tm, and ω= 0.
T_M= K₁V_Tm/R_a(5)

Solving Eq. (1-5) for K₁/R_a
K₁/R_a= T_M/V_Tm(6)

Substituting Eq. (6) into Eq. (4)
T= T_MV_T/V_Tm - T_MK₁ω/V_Tm(7)

The no-load speed ω_M at the maximum armature terminal voltage V_Tm is found from Eq. (3) when ω= ω_M, V_T= V_Tm, and T= 0.
ω_M= V_Tm/K₁(8)
K₁/V_Tm= 1/ω_M(9)

Substituting Eq. (1-9) into Eq. (7)
T= T_MV_T/V_Tm - T_Mω/ω_M(10)

Dividing Eq. (10) by T_M results in
T/T_M= V_T/V_Tm - ω/ω_M(11)

A plot of T/T_M versus ω/ω_M characteristics with V_T/V_Tm as a parameter is shown in the following picture. The curves are non-dimensionalized and based on Eq. (11).
The curves can be dimensionalized if the manufacturer's data for a particular motor are known. For example, a DC servo motor has an armature resistance of 5 ohms and a no-load speed of 3000 rpm at a maximum armature terminal voltage of 90 volts.

To dimensionalize a plot of T/T_M versus ω/ω_M characteristics, find the value of no-load speed
ω_M in radians/sec and the value of stall torque T_M.
From Eq. (1), ω is defined as 2ω X rpm/60. At no-load speed, ω= ω_M which is
ω_M= 2π X 3000/60
= 314 radians/sec

At the point ω/ω_M= 1, the no-load speed ω_M is 314 radians/sec. All other points on the ω/ω_M axis are dimensioned proportionally.
Substituting 314 for ω_M and 90 for V_Tm in Eq. (9) and solving for K₁ gives
K₁= V_Tm/ω_M
= 90/314
= 0.286

From Eq. (5), the stall torque is
T_m= K₁V_tm/R_a
= 0.286/5 X 90
= 5.15 newton-meters

At the point T/T_m= 1, the stall torque T_m=T=5.15 newton-meters. All other points on the T/T_m axis are dimensioned proportionally. A value of 90 volts is assigned to curve V_t/V_Tm= 1 (maximum armature voltage), and the other curves are dimensioned proportionally.