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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