E p,rms = 240 V E rms = 120 V N p N s C. f = 60 Hz R = 3.8 L

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Roland Holt
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Discussion Question 1A P1, Week 1 Power in AC Circuits An electronic device, consisting of a simle C circuit, is designed to be connected to an American-standard ower outlet delivering an EMF of 1 V

1 Discussion Question 1A P1, Week 1 Power in AC Circuits An electronic device, consisting of a simle C circuit, is designed to be connected to an American-standard ower outlet delivering an EMF of 1 V at 6 Hz frequency. However, you have transorted this device to Guyana, where standard AC ower outlets deliver a blazing 4 V. To overcome this roblem, you connect your device to the Guyanese wall outlet through a ste-down transformer with N turns in the rimary coil and N s turns in the secondary. The values of the resistance, caacitance, and inductance in your device are given in the figure. E, = 4 V E E = 1 V N N s C f = 6 Hz = 3.8 =.5 H C = 177 F (a) First, let s design the transformer. What relation must you imose on N and N s to dro the Guyanese line voltage E down to E = 1 V? Try thinking hysically: As is tyical, your transformer has an iron core which ensures that the magnetic field roduced in the rimary coil is transferred exactly to the secondary coil. Now E = -d/dt to ste down the voltage, which coil needs to have more turns? The field in the rimary will be the same as in the secondary. The flux on the rimary will be BN A. The flux on the secondary will be BN A. P s s d db db By Faraday's law: NPA ; s NsA dt dt dt We want s 1 Ns 1 N s N N (b) et the time-deendent EMF roduced by the Euro generator be E (t) = E, sin(t) which means t 1 sint where t EMF delivered to the secondary side of the transformer. What is the time-deendent current I(t) roduced in the load? Write your answer in the form I(t) = I sin(t-), and determine the numerical values of I and. s Page 1 of 4

2 We begin by comuting the reactances: rad/s C C C º ere is a hasor diagram for reactances. Z = I / Z. t. ) A Hence I=31.3 sin( 45 4 since lags I (c) What is the average ower delivered to the load? Your formula sheet has a coule of cookbook formulas you can use. P I cos cos( ) kw (d) Obtain an algebraic exression for the delivered to the load in te of,, and - C. Hint -write a hasor diagram for Z in te of and and use trig to find the required hase angle. P I cos cos. From triangle Z=/cos hence P Z P From triangle cos cos Z (e) Suose your Guyanese AC generator has a variable frequency f, though the EMF it sulies is fixed at 4 V. What f would you select so that the imum ossible ower is delivered to your device? Hint- sketch the exression for you obtained in art (d) as a function of. Which value of imizes ? What frequency gives you that value for? The formula says P monotonically decreases with. So the imal P is 1 1 when = or C f Hz C C Page of 4

3 (f) What is the imum average ower that you can deliver to your device? From our exression P the imum ower is just P. This makes sense since at resonance the load is urely resistive and this is the familar result for a resistor,. P kw Consider a slightly different situation. et s say you built your device to be urely inductive it contains nothing more than a coil you wound yourself, giving =.5 H. However, no device is erfect: every electronic circuit element contains some residual inductance, caacitance, and resistance. You connect your device to the Guyanese wall outlet and measure the resonse in the load. You find a current with an value of I = 8.5 A and lags the transformer EMF by 3. E, = 4 V E E = 1 V N N s C f = 6 Hz =.5 H I = 8.5 A = 3º (g) What is the resistance of your device? You will need a coule of equations to reverse engineer this circuit: you measured the voltage, current, and hase and now you have to figure out. Big hint: obtain an equation for in te of the imedance Z and the hase. (It's a useful equation, you might want to remember it!) From triangle Z /cos or Z cos. Z = 1 cos = cos I 8.5 (h) What is the caacitance C of your device? I Z 1 From triangle Zsin = sin = I C 1 1 C 1 o sin ( (. 5 sin3 I 85. C 5F Z Page 3 of 4

4 (i) In art (c), you robably used the average-ower formula =E I cos() from your formula sheet (the others, like =I, are equivalent). et s derive that result! First, what is the instantaneous ower P(t) delivered to the load? Give an algebraic answer (no numbers) in te of E, I,,, and t. Instantaneous ower is P = IV, forever and ever, under all circumstances. emember the time-deendent exressions you found in art (b) and in rearation for the next question, you should massage your result using sin( a b ) sin a cosb cos a sinb. P I cossin t sinsint cos et sint and I I sin t where is the hase lag of the current relative to the voltage. P = I I sint sint P I sint sint cos cost sin t (j) Now calculate the time-average of your result for P(t) over one cycle to obtain the average ower delivered to the load. emember, the average of sin and cos is 1/, while the average of (sincos) is zero. 1 P I cossin t sinsint cost P I cos sin t sin sint cost As suggested by the figure sin t (solid) averages to 1/ and sint cost (dashed curve) averages to zero over one cycle. I Hence P cos I cos Page 4 of 4

5 Discussion Question 1B P1, Week 1 Electromagnetic Waves A laser beam travels through vacuum. The electric field of the lane electromagnetic wave roduced by the laser has the form given below. The wavelength of the beam is = 514 nm, and the amlitude of the electric field is E =.5 x 1 4 N/C. E (x, y,z,t) y ˆ E cos(kz t 45 ) (a) In what direction is this wave roagating? What does roagating wave mean anyway? It means that the shae of the wave remains the same as a function of time, it just moves. The shae of the wave is determined by that cosine. As time rolls forward, how do you have to change osition so that the shae looks the same? et's follow a sot on the wave corresonding to E ye ˆ cos( 45 ). This sot obeys kz t or z - t. Hence this sot moves along the -z ˆ axis and evidently c k k (b) What are the magnitudes of the wave number k and angular frequency? c 15 k 1.1 m ; ck rad/sec (c) Make a sketch of what the E field looks like at some moment in time. At t, we have E y Ecos kz which is a cosine curve that imizes at z = + k E y t = z Page 1 of

6 (d) Write down an exression for the magnetic field B(x,y,z,t). Exress your answer algebraically (i.e. no numbers) in te of the symbols k,, and B (the latter being the magnetic field amlitude). Be sure to indicate the direction of the magnetic field. The E and B fields of an electromagnetic wave have a fixed relationshi to the direction of roagation of the wave. You know the direction of the wave, and the direction of E, so you can figure out the direction of B with the hel of your sketch. Add the B-field to your drawing in art (c). The B field has the same osition and time deendence as the E field. It is directed such that E B is along the direction of roagation (marked S in the figure) Bxyzt (,,, ) xb ˆ cos( kzt45 ) z y x S E B (e) What is the amlitude B of the magnetic field? B E.51 c T Page of

EKT103 ELECTRICAL ENGINEERING

EKT103 ELECTRICAL ENGINEERING EKT13 EECTRCA ENGNEERNG Chater 1 Three-Phase System 1 COURSE OUTCOME (CO) CO1: Ability to define and exlain the concet of single-hase and threehase system. 2 Revision A sinusoid is a signal that has the