DC Motors

15.2.4 DC Motors

15.2.4.1 Voltages and currents

Variable speed DC motors are mainly used in the oil industry for driving drilling equipment such as the drill string, draw-works, mud pumps, cement pumps, winches and the propulsion systems in semi-submersible rigs and barges. They are typically rated at approximately 800 kW, 750 volts, and several motors may be operated mechanically in parallel e.g. the draw-works motors. Each bridge that supplies a motor has a typical current rating of 2250 amps. Within its control system is a manually adjustable current limiting potentiometer to safeguard the bridge and to limit the torque produced by the motor. The bridges are fed from a three-phase 600 volt power source which is usually earthed by a high resistance fault detection device, that gives an alarm but does not trip the source.

Assume that the secondary phase-to-neutral emf of the supply transformer is E and the fun- damental reactance of each phase winding is X l , and the DC load current is I d , then for Mode 1

operation the DC output voltage V d is,

3 √ 6E 3X c I d

( 15.1) Where R is the DC circuit resistance.

cos α

− π =I d R +E m

E m is the emf in the motor armature.

X c = 2X l is the commutating reactance.

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An alternative expression for V d in terms of the commutation angle u, is,

3 √ 6E

( cos α

− cos(α + u))

Hence it can be seen that u is a function of I d , as will be shown below.

√ The factor 3 6/π applies to a three-phase bridge and is derived from,

Where n is the ripple number, in the above case n =3 and V do is the average ripple voltage at no-load.

The current I d can also be given as a function of α and u,

√ 6E

( 15.3) It can be seen from (15.1) that for a given delay angle α the output voltage has declining or

( cos α

2X c − cos(α + u))

‘drooping’ value as the DC current rises, for example as the load on a motor is increased causing it to slow down and to reduce its emf. Figure 15.3 shows a family of curves of output voltage against current, as a function of angle α.

Figure 15.3 Voltage versus current regulation of DC thyristor bridge used for a drilling system DC motor.

HARMONIC VOLTAGES AND CURRENTS

The power factor of the fundamental phase current in the reference phase of the secondary winding can be found from the in-phase and quadrature Fourier coefficients of the current. Let these

be a 1 and b 1 respectively. Hence the fundamental instantaneous current is,

i 1 = ˆI 1 (a 1 sin ωt +b 1 cos ωt) = ˆI 1 c 1 sin(ωt +Ø 1 )

Where the power factor is cos Ø 1 , and the suffix 1 refers to the fundament component. Reference 4 gives an expression for a 1 and b 1 in terms of the angles α and u that is suitable

for Mode 1 operation,

( 15.4) and

a 1 = cos α + cos(u + α)

2[cos α − cos(u + α)]

where α and u are in radians. From which,

c 1 = a 1 2 +b 1 2 ( 15.6) and

cos Ø 1 =

and

= cos c radians

The fundamental components of the rms current I in the phases of the secondary winding are, Real part,

a 1 ( 15.8)

and Imaginary part,

b 1 ( 15.9)

and the rms magnitude is,

c 1 ( 15.10)

The coefficient c 1 has a maximum value of 2 when α is zero and the commutation angle u is assumed to be negligibly small.

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In this case the maximum rms value of I is,

I max =

I d ( 15.11)

15.2.4.2 Active and reactive power

The rectifying elements of the bridge are assumed to be free of ohmic power losses. Therefore the power input to the DC motor must be equal to the AC power input to the bridge. Hence the sum of the active power in each phase of the supply transformer must equal the motor input power.

The input power P d to the motor is, P d =V d I d ( 15.12)

The output volt-amperes of the transformer is,

The active and reactive powers at the output of the transformer are,

The power factor of the fundamental current I is,

15.2.4.2.1 Worked example Certain operations that take place when drilling oil wells require the DC motors to operate at reduced

speed and to produce a moderate or high torque, e.g. reaming holes, running casing, stuck pipe removal, working over a well. Consider an example where a draw-works is running casing and several series-wound motors operate in parallel to drive the line drum.

The motor design details are Rated output power

750 kW

Rated efficiency

Rated voltage 750 volts Rated current

1075 amps Rated speed

975 rev/min Rated torque

7350 nm Armature and field circuit resistance (hot)

0.0488 ohms Armature and field circuit inductance

0.006 henry

HARMONIC VOLTAGES AND CURRENTS

The motor running details are,

Running output power

217.8 kW

Running voltage

323.3 volts

Running current

761.1 amps

Running speed

400 rev/min

Running torque

5200 nm

Running input power

246.1 kW

The transformer that feeds the bridge has the following ratings, Rated kVA

Voltage ratio, volts/volts

Leakage reactance in per-unit

Leakage reactance at 346 volts/phase 0.0024 ohms Commutating reactance = 2 × 0.0024 = 0.0048 ohms.

For (15.1) the variables and parameters are,

V d = 323.1 volts

E = 346.0 volts

I d = 761.1 amps

X = 0.0048 ohms

+ cos(66.215 + u))

0.7975 = 0.4033 + cos(66.215 + u) u = 66.215 − 65.677 = 0.538 degrees

= 0.00939 radians

From (15.4)

a 1 = cos 66.215 + cos 66.753 = 0.4033 + 0.3947 = 0.7980

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b 1 × 66.753) − sin(2 × 66.215) − 2 × 0.00939 = 2(0.4033 − 0.3947)

= −1.834 indicating a lagging power factor From (15.8), (15.9) and (15.10)

2 = −544.2 amps and

= 593.46 amps per phase

From (15.13), (15.14) and (15.15) the volt-amperes at the bridge AC terminals are, S sec =P sec + jQ sec

3.1415926 = 565.52 kVA r

and S sec = 616.75 kVA

The power factor of the fundamental current is,

616.75 = 0.3990 lagging or

cos Ø 1 =

a 1 0.7980

c = 1 √ 0.7980 2 2 + 1.834 = 0.3990 lagging

HARMONIC VOLTAGES AND CURRENTS

Note, a ‘rule-of-thumb’ expression for the power factor is,

ω o cos Ø 1 ≃ 0.7 + 0.2 ω n

Where ω o is the running speed of the motor and ω n is the rated speed of the motor. Hence,

975 = 0.4872 which is a little optimistic but a satisfactory estimate.