CME – HW#1 2016

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CME – Controle de Máquinas Elétricas – 2016

Due Date: Weds., Sept. 28 Name: ____________________________________

Required Homework #1 - DC Machine

Note: Please fill in the answers on these sheets to simplify grading but attach separate sheets that include all calculations used to complete your homework so that the grader can assign partial credit when appropriate.

1. Using data from the DC machine data sheet, consider two dc machines, one rated at 10 hp,

500 rpm and the second at 50 hp, 2500 rpm. (Use "hot" resistance for all calculations =

1.2 x Ram @ 25°C):

a) Assuming operation at rated speed with rated flux φR, find the steady-state values for

the rated torque TR, rated current IR, rated armature voltage VR and the full-load

efficiency Eff (include the field circuit loss). Also find the back-emf voltage e and the

no-load speed ωNL at rated voltage. Plot the torque (y-axis in N-m) vs. normalized speed

(x-axis, dimensionless) curve for both machines on the same plot. Normalize the speed

values for each machine by its rated speed (e.g., ωn = ωr (in rpm)/500 for the 10 hp

machine, so that ωn =1 for rated speed). Plot over a speed range from 0 to 1.2 pu speed,

but limit the torque axis range to ±300 Nm.

10 hp Machine 50 hp Machine

TR = ___________________ [Nm] TR = ___________________ [Nm]

IR = ___________________ [A] IR = ___________________ [A]

VR = ___________________ [V] VR = ___________________ [V]

Eff = ___________________ [%] Eff = ___________________ [%]

e = ___________________ [V] e = ___________________ [V]

ωNL = ___________________ [rpm] ωNL = ___________________ [rpm]

b) Find the speed ωRG at which each machine will operate and the corresponding torque

TRG and armature current IRG when operating as a generator delivering rated power P =

- PR at armature voltage V = Vs = 250 V so that P = - PR = Vs * IRG. Assume that the

machines are operating at rated flux φR.

10 hp Machine 50 hp Machine

ωRG = ___________________ [rpm] ωRG = ___________________ [rpm]

TRG = ___________________ [Nm] TRG = ___________________ [Nm]

IRG = ___________________ [A] IRG = ___________________ [A]

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c) Determine the rotor speed, armature current, and efficiency when a series resistor Rs

with a value equal to ten times the machine's hot internal resistance (i.e., Rs = 10 * 1.2

* Ram) is placed between the positive terminal of the excitation voltage source Vs = 250

V and the machine’s armature terminal. Each machine is delivering its rated torque TR.

In addition, calculate the back-emf e and the no-load speed ωNL (i.e., speed at zero

output torque) for each machine with the additional series resistance in place. Include

the losses in this extra series armature resistance in the efficiency calculation as well as

the field losses (assume full field excitation). Prepare a new figure with the torque vs.

normalized speed curves for both modified machines on the same plot using the rated

machine speeds (500 rpm and 2500 rpm) for the speed axis normalizations, as done in

part a). Use the same speed and torque axis ranges given in part a).

10 hp Machine 50 hp Machine

ωr = ___________________ [rpm] ωr = ___________________ [rpm]

I = ___________________ [A] I = ___________________ [A]

Eff = ___________________ [%] Eff = ___________________ [%]

e = ___________________ [V] e = ___________________ [V]

ωNL = ___________________ [rpm] ωNL = ___________________ [rpm]

d) Find the variable values listed below for operation at the same machine torque and speed

operating points as in part c), delivering rated torque using armature voltage control (full

field excitation). In this part the only resistance is the internal hot armature resistance

(1.2*Ram). The no-load speed is the speed at zero torque for the armature voltage

calculated to give the specified speed at rated torque. Prepare a new figure with the

torque vs. normalized speed curves for both machines on the same plot using the rated

machine speeds (500 rpm and 2500 rpm) for the speed axis normalizations. Use the

same speed and torque axis ranges as in part a).

10 hp Machine 50 hp Machine

I = ___________________ [A] I = ___________________ [A]

V = ___________________ [V] V = ___________________ [V]

Eff = ___________________ [%] Eff = ___________________ [%]

e = ___________________ [V] e = ___________________ [V]

ωNL = ___________________ [rpm] ωNL = ___________________ [rpm]

e) We wish to operate each of the machines at 250% of its rated speed (i.e., 1250 and 6250

rpm) using field weakening (φ ≤ φR). Assume that the field flux in each machine is

reduced to 40% of its rated full field value and the armature voltage is 250 V. Determine

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the torque TFW that is developed when the fieldweakened machines are operated at 1250

and 6250 rpm, respectively. Calculate the output power PFW, no-load speed ωNL, and

the efficiency Eff at this operating point, assuming that the field winding losses vary as

the square of the operating flux (i.e., Wf = Wf (at full flux) * (φ / φR)2 ). Prepare a new

figure with the resulting torque vs. normalized speed curves for both machines on the

same plot for V = 250 V and 40% field flux. Use a torque axis range of ±150 Nm and

a speed axis range from 0 to 4.0 pu.

10 hp Machine 50 hp Machine

TFW = ___________________ [Nm] TFW = ___________________ [Nm]

PFW = ___________________ [kW] PFW = ___________________ [kW]

ωNL = ___________________ [rpm] ωNL = ___________________ [rpm]

Eff = ___________________ [%] Eff = ___________________ [%]

2. Calculate the following for two 20 hp dc machines, one rated at 3500 rpm and other rated

at 500 rpm. Use data from the DC machine data sheet, making use of "hot" armature

resistance values for all calculations. Note that the value of K is proportional to the field

flux, and the printed value is for rated (100%) flux. Both the load moment of inertia JL and

the viscous friction coefficient B are zero unless stated otherwise:

a) Calculate the eigenvalues (real or complex) for operation at rated (100%) flux and at

50% of rated flux.

3500 rpm Machine 500 rpm Machine

100% Flux λ1 = ______________ [sec-1] λ1 = _______________ [sec-1]

λ2 = ______________ [sec-1] λ2 = _______________ [sec-1]

50% Flux λ1 = ______________ [sec-1] λ1 = _______________ [sec-1]

λ2 = ______________ [sec-1] λ2 = _______________ [sec-1]

b) Calculate the dominant time constant τ of the 3500 machine and the natural frequency

ωN and damping factor ζ of the 500 rpm machine (assume 50% of rated flux for both

machines). Use them to determine the approximate percentage overshoot and settling

time for the rotor speed’s natural response for each machine following a step change in

the armature voltage. Assume zero load inertia. Plot the transient response of the rotor

speed ω (in rpm) for both machines for a step in the armature voltage from 75% to 100%

rated voltage, assuming no steady-state load torque (i.e., TL=0) and an initial rotor speed

corresponding to the no-load speed at 75% rated voltage. Calculate the initial and final

speed values for both machines.

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3500 rpm Machine 500 rpm Machine

τ = ____________ [sec] ωN = _____________ [rad/s]

ζ = _____________

Overshoot = ____________ [%] Overshoot = _____________ [%]

Settling time= ____________ [%] Settling time= _____________ [%]

ωinit = ____________ [rpm] ωinit = _____________ [rpm]

ωfinal = ____________ [rpm] ωfinal = _____________ [rpm]

c) Find the value of a series resistance for both machines that will limit the steady-state

stall current (i.e., speed = 0) with rated voltage to 150% of rated current. With this

resistor in the circuit, repeat the eigenvalue calculation of part a) for both machines.

Assume rated field flux (100%).

3500 rpm Machine 500 rpm Machine

Radd = ______________ [Ω] Radd = _______________ [Ω]

λ1 = ______________ [sec-1] λ1 = _______________ [sec-1]

λ2 = ______________ [sec-1] λ2 = _______________ [sec-1]

d) For each machine with rated field flux, find the time required to accelerate from zero to

90% of rated speed (T90) with a rigidly-connected load inertia equal the motor’s inertia

(JL = Jm) if the current is constant and equal to the rated current. Then find the value of

this same acceleration time (T90) to reach 90% of rated speed (for the first time) for the

same combined motor/load inertia if the rated armature voltage is suddenly applied and

held constant (armature current is no longer constant).

3500 rpm Machine 500 rpm Machine

Rated

current

(constant)

T90 = ______________ [sec] T90 = _______________ [sec]

Rated

voltage

(constant)

T90 = ______________ [sec] T90 = _______________ [sec]

e) Copy the eigenvalues for the 20 hp, 500 rpm dc machine at rated (100%) flux with JL =

0 from part a). Calculate estimates of the resulting overshoot and settling time for a step

armature voltage input.

λ1 = ____________ [sec-1] λ2 = ____________ [sec-1]

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Overshoot = ___________ [%] Settling time = ____________ [sec]

f) Now assume that a closed-loop current regulator is introduced. What are the new

eigenvalues of the motor-plus-regulator system if the current regulator gain Ki is set to

20? Ignore the mechanical damping constant (B=0). Calculate estimates of the resulting

overshoot and settling time for a step current input.

λ1 = ____________ [sec-1] λ2 = ____________ [sec-1]

Overshoot = ___________ [%] Settling time = ____________ [sec]

g) Now consider a closed-loop speed control system. Assume that the current regulator

dynamics are very fast compared to the outer speed loop and the current regulator gain

is high enough so that it is effectively “ideal”. Calculate the eigenvalue of this

closedloop system for Kp=50, τz=∞. (Assume B=D=0 again.) Calculate the resulting

output speed overshoot and approximate settling time of the closed-loop system for a

step speed command input.

λ = ____________ [sec-1]

Overshoot = ___________ [%] Settling time = ____________ [sec]

3. Solve using data from the DC machine data sheet, using the "hot" resistance value for all

calculations. Note that the value of K on the sheet is for rated (100%) field flux. Assume

that the machine can safely handle as much current as the power converter can deliver to

it. In addition, assume that all power semiconductors are ideal with zero voltage drop when

they are on. Consider a 15 hp, 1750 rpm dc machine connected to a three-phase, fully-

controlled thyristor bridge, assuming that the series inductance is large enough to justify

the assumption of infinite output inductance. The amplitude of the ac voltage source is 220

Vrms, the maximum thyristor current is 40 A and the phase control angle α can be varied

over the full range from 0 to π rad. Assume that the maximum machine speed is 4000 rpm

in either direction (i.e., ± 418.8 rad/s)

a) Plot the complete boundary envelope (i.e., capability curve) of the drive system’s

torque-speed operating capabilities on a set of T-ω axes, with torque T on the y-axis [in

N-m] and speed ω on the x-axis [in rad/s]. Assume that the field flux is constant at its

rated (100%) value. In addition, assume that the maximum armature current is

determined by the maximum thyristor current rather than the rated machine current

based on its rated power. Similarly, assume that the maximum applied armature voltage

amplitude is determined by the maximum positive and negative rectifier output voltage

(@α = 0 and α = π) rather than the rated machine voltage. What are the torque, speed,

machine shaft power, and phase control angle α at the four corners of the operating

envelope? (Hint: Do not be surprised if P does not equal the rated machine power (15

hp = 11.2 kW)).

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Upper left T =_______[Nm] ω =_______[rad/s] P =______[W] α =_____[rad]

Upper right T =_______[Nm] ω =_______[rad/s] P =______[W] α =_____[rad]

Lower left T =_______[Nm] ω =_______[rad/s] P =______[W] α =_____[rad]

Lower right T =_______[Nm] ω =_______[rad/s] P =______[W] α =_____[rad]

b) Assume that a second identical single-phase thyristor bridge is added in anti-parallel.

Plot the complete operating envelope (i.e., capability curve) of the drive system’s

torque-speed operating capabilities on a set of T-ω axes, assuming that the field flux is

still constant at its rated value. What are the torque, speed, machine shaft power, and

phase control angle α at the four corners of the operating envelope?

Upper left T =_______[Nm] ω =_______[rad/s] P =______[W] α =_____[rad]

Upper right T =_______[Nm] ω =_______[rad/s] P =______[W] α =_____[rad]

Lower left T =_______[Nm] ω =_______[rad/s] P =______[W] α =_____[rad]

Lower right T =_______[Nm] ω =_______[rad/s] P =______[W] α =_____[rad]

c) Replot the complete operating envelope in part b) with a dual anti-parallel thyristor

bridge converter assuming that the field can be weakened. What is the maximum value

of torque that can be achieved at the speed extremes of –4000 rpm (=-418.9 rad/s) and

+4000 rpm (=+418.9 rad/s), and what are the corresponding values of machine output

power P, machine torque-per-Amp constant K and control phase angle α?

ω = + 4000 rpm

Positive

torque

T =_______[Nm] K =_______[Nm/A] P =______[W] α =_____[rad]

Negative

torque

T =_______[Nm] K =_______[Nm/A] P =______[W] α =_____[rad]

ω = - 4000 rpm

Positive

torque

T =_______[Nm] K =_______[Nm/A] P =______[W] α =_____[rad]

Negative

torque

T =_______[Nm] K =_______[Nm/A] P =______[W] α =_____[rad]

d) Consider a 2 hp, 850 rpm dc machine connected to a single-phase, fully-controlled

thyristor bridge. Again assume infinite output inductance and constant field flux at its

rated value. If the amplitude of the input ac voltage is 210 Vrms line-to-line and the

maximum thyristor current is 10 A, determine the outer boundary of the drive’s torque-

speed operating capabilities, assuming that the phase control angle α can vary from 0 to

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π radians. What are the torque, speed, and phase control angle α at the four corners of

the capability curve?

Upper left T =_______[Nm] ω =_______[rad/s] α =_____[rad]

Upper right T =_______[Nm] ω =_______[rad/s] α =_____[rad]

Lower left T =_______[Nm] ω =_______[rad/s] α =_____[rad]

Lower right T =_______[Nm] ω =_______[rad/s] α =_____[rad]

HP Rating and Frame Sizes Table

DC Machine Parameters and Performance Data

rpm Machine base speed in revolutions per minute = Machine speed at rated output powerR Machine armature resistance in ohms at 25o (x 1.2 for hot value at 85 degC)L Machine armature inductance in henrys (unsaturated)

K Machine torque constant in Newton-meters per Ampere @ Rated field flux= Machine voltage constant, CEMF, in volts/radian/second @ Rated field flux

Frame rpm R L K

283 3500 0.153 0.0013 0.605 2500 0.301 0.0023 0.85 1750 0.615 0.0045 1.21 1150 1.426 0.0104 1.84

850 2.608 0.0192 2.5 500 7.56 0.055 4.23 300 19.5 0.153 7.07

284 3500 0.142 0.0011 0.59 2500 0.279 0.0021 0.82

1750 0.570 0.0043 1.171150 1.36 0.0100 1.78 850 2.42 0.0185 2.42 500 6.71 0.0532 3.98 300 19.34 0.147 6.85

286 3500 0.070 0.00070 0.655 2500 0.137 0.00140 0.917

1750 0.280 0.00281 1.311150 0.657 0.00650 1.98 850 1.19 0.0120 2.69 500 3.32 0.0344 3.99 300 9.5 0.095 7.60

288 3500 0.045 0.00073 0.610 2500 0.089 0.00144 0.850

1750 0.180 0.00293 1.221150 0.415 0.00677 1.85 850 0.762 0.0125 2.50 500 2.21 0.0360 4.27 300 6.1 1.00 7.10

365 3500 0.022 0.00055 0.63 2500 0.041 0.0011 0.88

1750 0.086 0.0022 1.26 1150 0.199 0.0051 1.91 850 0.368 0.0094 2.6 500 1.06 0.027 4.42 300 2.91 0.075 7.3

Frame rpm R L K

366 3500 0.0168 0.00026 0.64 2500 0.0328 0.00050 0.896

1750 0.067 0.0010 1.281150 0.155 0.0024 1.95 850 0.284 0.0044 2.56 500 0.772 0.013 4.37 300 2.27 0.035 7.25

367 2500 0.0203 0.00052 0.88 1750 0.0415 0.0011 1.26

1150 0.0963 0.0025 1.92 850 0.176 0.0046 2.58 500 0.478 0.013 4.3 300 1.41 0.036 7.35

368 1750 0.0363 0.00085 1.26 1150 0.0964 0.0020 1.92

850 0.153 0.0036 2.60 500 0.417 0.011 4.41 300 1.24 0.29 7.33

503 1750 0.0144 0.0011 1.271150 0.089 0.0025 1.95 850 0.066 0.0045 2.77 500 0.168 0.013 4.35 300 0.500 0.036 7.4

504 1750 0.0100 0.00085 1.271150 0.0237 0.0020 1.95 850 0.0420 0.0036 2.6 500 0.115 0.011 4.45 300 0.342 0.029 7.4

505 1750 0.0099 0.00073 1.271150 0.0206 0.0017 1.95 850 0.0380 0.0031 2.6 500 0.109 0.0090 4.45 300 0.350 0.025 7.4

506 1150 0.0125 0.0012 1.95 850 0.0230 0.0023 2.63

500 0.0660 0.0065 4.47 300 0.188 0.019 7.5

Nominal Performance Constants

Tpu 1.0 per unit torque in pound-feet (continuous torque only of DC 1150 rpm or above or blower-ventilated)

Wf Power for full field in watts

Jm Motor inertia in pound-feet second^2 Note: Multiply by 1.355 for value in kg-m^2Tm Motor inertia time constant in seconds (JR, /Kt, Kv)Cfm Forced air in cubic feet per minuteP Static pressure drop in inches of water1/T Bandwidth in radians per second (w)

VentilationFrame Wf Jm Cfm P

283 150 0.050 150 1.00284 160 0.065 150 1.00286 180 0.087 150 1.00288 200 0.115 150 1.00365 210 0.218 350 1.25366 220 0.292 350 1.25367 230 0.340 350 1.25368 242 0.412 350 1.25503 325 1.34 800 1.9504 410 1.43 800 1.9505 430 1.63 800 1.9506 500 2.08 800 1.9

Note: For an application requirement, the horsepower rating and frame size can be chosen from thetable. Considerations are ventilation, enclosure, continuous rms torque (or horsepower) and peak torque.Ventilation and enclosure affect the continuous rms torque capacity of a given frame size.

The rms torque or the peak momentary overload torque may be the limiting requirement. Using

the rated or 1.0 per unit torque (Tpu ) for the frame size chosen for thermal rating, use the maximummomentary load curves of Figure A to identify the overload capability. peak torque = T x (per centoverload/100). If the peak torque capability is not sufficient, then a new frame size must be chosen basedon peak torque.

The curves of Figure A are defined as follows:

1. Instantaneous loads are defined as 0.5 seconds duration or less repeated not oftener thanonce every minute.

2. Occasionally repeated loads are defined as 5 seconds duration or less repeated not oftenerthan once every 5 minutes.

3. Frequently repeated loads are defined as 1 minute duration or less repeated not oftenerthan once in a period 20 times the duration.

4. Curves apply regardless of whether speed is obtained by armature voltage or shunt fieldcontrol.

5. Curves also apply for regenerating operations.With the frame and base speed chosen, performance data can be taken from the table.

900

800

700

600

500

400

300

200

100

50 100 150 200

% Base Speed

Figure A Maximum momentary loads

Horsepower Rating and Frame SizesDrip-proof 60oC Rise

Frame SizeSpeed in rpm

hp 3500 2500 1750 1150 850 5001 - - - - - 2832 - - - - 283 2843 - - - 283 284 2865 - - 283 284 286 288

7 1/2 - 283 284 286 288 366

10 - 283 284 286 288 36715 283 284 286 365 366 36820 284 286 288 366 367 50325 286 288 365 366 368 50430 286 288 366 367 368 505

40 288 366 366 368 503 50650 - 366 367 503 504 -60 - 367 368 503 505 -75 - - 503 504 505 -100 - - 503 505 - -

125 - - 504 506 - -150 - - 505 - - -

Implementation

Occasionally Repeated

Frequently Repeated

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