전기공학부 전기에너지변환 기말고사         2007.12.15(土)     H.K.Jung

1. Quasi static 전자계에서 에너지 관계식을 유도하고, 이때에 자계 및 전계 에너지 밀도는 각각

[eqn: w_m = 1 over 2 {vec{B} BULLET vec {B} } over mu_0], [eqn: w_e = 1 over 2 epsilon_0 {vec{E} BULLET vec {E} }]

임을 보여라.

2. 전계에 저장된 코에너지 [eqn: w_e prime]는 다음과 같은 식으로 표현될 수 있음을 보여라.

[eqn: w_e prime= int _{ 0} ^{V } {q(v prime ,x)dv prime}]

3. 균일한 공극의 균형 2상 여자 회전기에서 발생하는 출력은 [eqn: P_m = - omega_m M I_s I_r sin gamma] 임을 보여라.

(단, [eqn: L_sr (theta)=Mcos(theta)], [eqn: I_as =I_s cos (omega_s t )], [eqn: I_bs =I_s sin (omega_s t )], [eqn: I_ar =I_r cos (omega_r t )], [eqn: I_br =I_r sin (omega_r t )], [eqn: theta = omega_m t + gamma], [eqn: omega_m = omega_s - omega_r] 이다.)

4. Kirchhoff 법칙을 전자장 이론으로부터 설명하라.

5. 60[Hz] 주파수의 고정자 전원으로 구동되는 15극 동기 전동기의 동기속도(rpm)는 얼마인가?

1

6. A magnetic circuit, including a movable plunger, is shown on figure. The circuit is excited by an N-turn coil and consists of a perfectly permeable yoke and plunger with a variable air gap [eqn: x(t)] and a fixed nonmagnetic gap d. The system, with the cross section shown, has a width w into paper. The following parts lead to a mathematical formulation of the equations of motion for the mass M, given the excitation [eqn: I(t)].

(a) Find the terminal relation for the flux [eqn: lambda(i,x)] linked by the electrical terminal pair. Ignore fringing in the nonmagnetic gaps. Note that the coil links the flux through the magnetic material N times.

(b) Find the energy [eqn: W_m (lambda , x)] stored in the electromechanical coupling. This should be done by making use of part (a).

(c) Use the energy function [eqn: W_m (lambda , x)] to compute the force of electrical origin [eqn: f^e] acting on the plunger.

(d) Write an electrical (circuit) equation of motion involving [eqn: lambda] and x as the only dependent variables and [eqn: I(t)] as a driving function.

(e) Write the mechanical equation of motion for the mass. This differential equation should have [eqn: lambda] and x as the only dependent variables, hence taken with the result of (d) should constitute a mathematical formulation appropriate for analyzing the system dynamics.

2

7. A smooth-air-gap machine has a two-phase set of stator windings, each with a total of N turns. The windings are distributed sinusoidally and current in them produce surface current densities as indicated in figure. When [eqn: g<< R], the radial flux density produced in the air gap by each winding, is

[eqn: B_ra = {mu_0 N i_a } over 2g cos psi], [eqn: B_rb = {mu_0 N i_b } over 2g sin psi].

(a) For the two-phase excitation [eqn: i_a =I_a cos (omega t )], [eqn: i_b =I_b sin (omega t )], which is unbalanced in amplitude, find the total radial flux density.

(b) Express the answer to part (a) as a sum of two traveling waves. Identify the forward and backward components and show that their respective angular velocities are [eqn: omega_ f = omega] and [eqn: omega_ b = -omega].

(c) Evaluate the ratio of the amplitudes of backward and forward waves. Show that the ratio -> 0 for a balanced excitation (i.e., consider the limit for [eqn: I_b -> I_a]).

(d) Discuss how to achieve a constant amplitude backward wave only. This is the method used to reverse the direction of rotation of an ac machine.

3

8. The salient pole, synchronous machine of figure is electrically linear and lossless and has a terminal inductance expressed as

[eqn: L = L_0 over {(1-0.25 cos 4theta - 0.25 cos 8theta)}],

where [eqn: L_0] is a positive constant. This is an alternative mathematical representation to the form given by

[eqn: L_s (theta) = L_0 +L_2 cos 2theta+L_4 cos 4theta+...].

(a) Describe briefly why the dependence of this inductance on [eqn: theta] is physically reasonable.

(b) Fine the torque of electric origin [eqn: T^e] as a function of flux linkage [eqn: lambda], angle [eqn: theta], and the constants of the system.

(c) As shown on figure, the terminals are excited by a sinusoidal voltage source such that the flux [eqn: lambda] is given by [eqn: lambda (t) = Lambda_0 cos omegat], where [eqn: Lambda_0] and [eqn: omega] are positive constants. The rotor is driven by a constant-angular-velocity source such that [eqn: theta(t) = Omegat+delta], where [eqn: Omega] and [eqn: delta] are constants. Find the value of [eqn: Omega], in terms of the electrical frequency [eqn: omega], at which time-average power can be converted by the machine between the electrical and mechanical systems.

4

9. This is a problem that involves the use of a synchronous condenser to correct power factor in a power system. The correction is actually achieved by using the synchronous condenser to regulate voltage. We consider one phase of a balanced two-phase system. In the figure (a) a power system feeds a steady state load which has admittance [eqn: Ye^{-j phi}] as shown. The Thevenin equivalent circuit of the system, as viewed from the load, is the source [eqn: hatV_s] in series with the inductive reactance [eqn: jX_s]. To fix ideas assume the following parameters and excitation: [eqn: V_s = 100000 sqrt2 V_peak], [eqn: X_s = 10 Omega], [eqn: Y=0.01 mho].

(a) Find the ratio of the magnitudes of the load voltage [eqn: V] and the source voltage [eqn: V_s] for [eqn: phi=0 DEG] and [eqn: phi=45 DEG].

(b) Now a synchronous condenser is connected across the load as shown in figure (b) and draws current [eqn: hatI _c]. Find the volt-ampere rating required for the synchronous condenser to make the ratio [eqn: LEFT | hat V_``` RIGHT | / LEFT | hat V_s RIGHT |] equal to unity for each case in part (a). Compare each with the real power drawn by the load.

5

10. An automobile speedometer consist of a permanent magnet mounted on a rotating shaft connected to the automobile transmission. An aluminum "drag cup" with a pointer mounted on it is placed around this rotating magnet. The cup is free to rotate through an angle [eqn: psi] but is restrained by a torsion spring that provides a torque [eqn: T_s = -K psi]. The angular position of the cup can be used to determine the angular velocity of the shaft connected to the magnet and therefore the speed of the automobile. The model to be used in analyzing the speedometer is illustrated in figure. The permanent magnet is represented by a coil excited by a constant-current source. The drag cup is simulated by two coils shunted by resistances. These coils are attached to a rotatable frame, which in turn is restrained by torsion spring. An appropriate electrical model of the coupling field is

[eqn: lambda_1 = Mi_3 cos(phi-psi)+Li_1],

[eqn: lambda_2 = Mi_3 sin(phi-psi)+Li_2],

[eqn: lambda_3 = L_3 i_3 +Mi_1 cos(phi-psi) + Mi_2 sin(phi-psi)].

Assuming that the rotational velocity of the shaft is constant (i.e., the speed of the car is constant), find the deflection of the rotatable frame (of the speedometer pointer) as a function of the shaft rotational velocity [eqn: phi]. You may assume that the device is designed in such a way that

[eqn: left | L di over dt right | << left | Ri right |].

Good Luck !

6