# THE UNIVERSITY OF BRITISH COLUMBIA. Department of Electrical and Computer Engineering. EECE 365: Applied Electronics and Electromechanics

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1 THE UNIVERSITY OF BRITISH COLUMBIA Department of Electrical and Computer Engineering EECE 365: Applied Electronics and Electromechanics Final Exam / Sample-Practice Exam Spring 2008 April 23 Topics Covered: Magnetic Circuits, Electromechanical Devices with Motion, DC Motors, AC Power and Transformers, Induction Motors, Synchronous Motors, Brushless DC Motors, Stepper Motors Surname: First Name: Student ID: Signature: Close notes and books. You are allowed to have only a calculator, a pen/pencil, and two double-sided pages of hand-written formulas. Show you work including derivations, comments, assumptions, and units wherever appropriate. Use back side of each page or ask for additional pages if you need extra space to write your answers. Exams suspected of cheating and/or turned in late will not be marked failed exam. You have 90 minutes to answer the following questions: Problem Points Max Total 100

2 Problem 1: Consider the magnetic system shown below. The core has permeability and dimensions such that the reluctance of magnetizing path is R m = 10 5 At / Wb. The coil has 100 turns, dc resistance of 2 Ω, and is connected to a dc source = 10V. It is also known that 20% of the total flux Φ produced by the coil leaks into the air. V dc (a) Draw an equivalent magnetic circuit, show the direction of mmf and the fluxes (b) Calculate flux linkage λ, and inductance L (c) Find the rms value of current if the coil is supplied from an ac source V ac = 10V ( rms) with the frequency f e = Hz

3 Problem 2: Consider an electro-mechanical device shown in the figure. You can use common approximations as we did in class and assume magnetically-linear core. The air-gap between the core and the plunger is denoted by x (which has units of meters). Assume that total inductance of this device may be approximated as L( x) H, and the dc resistance r is 20 Ω = x (a) Sketch the equivalent magnetic circuit and label all elements (b) Sketch the equivalent electric circuit and label all elements (c) Express the electromagnetic force f e ( x, I dc ) (d) What value of dc voltage V dc and dc current I dc should be applied to the coil in order to produce a force of 25 Nm when the plunger has air-gap of 2 mm? (e) Calculate the energy stored in the system for part (d)

4 Problem 3: Consider a Permanent-Magnet DC motor with the following parameters: rated voltage Vt = 240 V; armature resistance R a = 1.2 Ω, and friction torque T fric = 0. 5Nm = const. (a) When the motor drives a mechanical load of 9.5Nm it draws a current of 10 A. Calculate the induced armature emf E a and torque constant K t (b) Assume that mechanical load has increased to T m = 19. 5Nm. Calculate the motor speed n in rpm, speed regulation SR in %, and efficiency η also in %

5 Problem 4: Consider a 115V series-connected DC motor with the following parameters: armature resistance = 1Ω ; and field winding resistance R f = 2Ω. The motor is supplied from a dc source Vt = 115 V and is operating under nominal load at speed n = 3000 rpm drawing armature current I = 5A. (a) Draw an equivalent circuit (b) Calculate the induced back emf, (c) Calculate the torque at zero speed (starting torque), T start E a a R a

6 Problem 5: A 1.5-kVA, 60-Hz, step-up transformer has two windings with N 1 = 1000 and N 2 = 2000 turns, respectively. The leakage reactances are X 1 = 2Ω, X 2 = 8Ω (each quantity is referred to its own side), and the magnetizing reactance X = 400Ω (referred to the secondary side). The core and copper losses can be ignored. Assume m2 120V is applied to the primary side: (a) Calculate the open-circuit primary current I, oc 1 and the secondary voltage 2 oc (their rms values) (b) Assume a resistive load R Load = 40Ω is connected to the secondary side. Calculate the resulting currents (rms) in each winding. Also calculate the input power-factor angle ϕ in degrees V,

7 Problem 6: Consider a 60Hz, 208V (line-to-line), Y-connected, NEMA Clsass B Squirrel-Cage Induction Motor with the following per-phase parameters: R 1 = 1Ω, R 2 = 1.5 Ω, X 1 = X 2 = 3 Ω, and X m = 40 Ω (all referred to the stator). The motor is supplied with the nominal (rated) voltage and is driving a mechanical load. The speed of the motor shaft is n = 855 rpm. You can neglect core losses and use an approximate equivalent circuit. Recall that 1 2 R T ( I ) s e = 3 2. Determine the following: ω syn s (a) Number of poles P and slip s (b) Input stator current I 1, power factor PF, and total three-phase input power P in (c) Developed electromagnetic torque T e (d) Assume the friction torque is 5% of the developed torque T e. Calculate the useful mechanical load torque T m and the motor efficiency η in %

8 Problem 7: Consider a 3-phase, 60Hz, 208V (line-to-line) 14-pole Permanent Magnet Round-Rotor Synchronous Motor with the following parameters: per-phase stator resistance and synchronous reactance are R a = 1Ω and X s = 10 Ω, respectively. Assume that the motor outputs mechanical power Pm = 1676W and power factor is one. (a) Sketch an equivalent electric circuit (per-phase) (b) Calculate the motor shaft is n in rpm, induced voltage E f, and the rotor angle δ in degrees (c) Assume mechanical rotational losses Pmech _ loss = 50W, calculate the efficiency η

9 Problem 8: (a) Consider a 2-phase PM Stepper Motor shown here. The rotor initial position is as shown corresponds to the phase bs energized. Sketch the sequence of currents i as and i bs to drive this motor at half-step in CW direction assuming phase bs is energized first to positive value. Front Back (b) Assume a standard (one phase energized at a time) full-step operation with duration of current pulses T 1 / f = 0.01sec. Calculate rotor mechanical speed n in rpm step = step (c) List all classes/types of stepper motors that we discussed in class: (d) List some of the factors that limit the stepping rate (or speed) at which a given stepper motor can operate: We discussed this in class and you have observed that in Lab-5.

10 Problem 9: Consider a single-stack stepper motor shown below. Complete the following table: 2pts 2pts 3pts 3pts 3pts 3pts Stator tooth pitch Rotor tooth pitch Step length Number of steps per revolution (Resolution) Assume sequence of pulses as B C D A, determine the direction of rotation (CW or CCW) Assume you have a 4-phase pulse generator to supply this motor. The generator produces 12 pulses per second per phase (48 pulses per second total). What is the motor speed in rpm?

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