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# 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

_{a}R

_{a}+ KI

_{f}ω(1)

_{f}I

_{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 K

_{1}, 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}/K

_{1}- TR

_{a}/K

_{1}

^{2}

Solving Eq. (1-3) for T

T= K

_{1}V

_{t}/R

_{a}- K

_{1}

^{2}ω/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}= K

_{1}V

_{Tm}/R

_{a}(5)

Solving Eq. (1-5) for K

_{1}/R

_{a}

K

_{1}/R

_{a}= T

_{M}/V

_{Tm}(6)

Substituting Eq. (6) into Eq. (4)

T= T

_{M}V

_{T}/V

_{Tm}- T

_{M}K

_{1}ω/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}/K

_{1}(8)

K

_{1}/V

_{Tm}= 1/ω

_{M}(9)

Substituting Eq. (1-9) into Eq. (7)

T= T

_{M}V

_{T}/V

_{Tm }- T

_{M}ω/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 K

_{1}gives

K

_{1}= V

_{Tm}/ω

_{M}

= 90/314

= 0.286

From Eq. (5), the stall torque is

T

_{m}= K

_{1}V

_{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.

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