PHYS Fields and Waves

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1 PHYS Fields and Waves

2 Idea: how to deal with alternating currents Edison wanted direct current, Tesla alternating current, who won? Mathematically: Instantaneous current: Instantaneous voltage: w is the angular frequency i I cost v V cost w=2pf, f = 60 Hertz (50 Hertz in Europe) V=120 volts (220 in Europe, South America) Symbol in circuits: Average I?: i 0 Ave How to measure a changing current? There are several ways.

3 How to characterize alternating currents? Method 1: Rectified average i I cost Use average of i Average of rectified I?: I rav t2 / t / 2 I cos dt 2I d t0 t / 2 t2 dt t cos 2 2I 0.637I t0

4 How to characterize alternating currents? Method 2: Root mean square average i I cost Use average of i I 2 Average of i?: 2 Ave Irms rms 1 2 i 2 Ave t2 t2 / 2 I dt dt 2 i t0 t0 t2 t0 i I cos t I 1cos2t dt cos2t I I Irms I ompare to I rav I

5 How to characterize alternating currents? Method 3: Phasor i I cost Take instantaneous current as component of a vector Not a Mexican Wrestler! et vector rotate with angular speed. Useful for comparing times et us see some examples

6 Given: I rms 2.7 A a) I Ave 0 b) c) I i 2 i I A 7.3 A rms Ave Ave rms I I I I rms 2.7 A rms A 3.8 A 2 Hmwk: Probls 11 th ed.: 31.1 and 2 (soln.: 2.97 A and 1.89 A) or 12 th Ed.: 31.3 and 2

7 Summary of Section 31.1 i I cost Symbol in circuits: v V cost Average I?: i 0 Ave 2 I rav I I rms I 2 Phasor Hmwk Sect. 31.1: Probls 11 th ed.: 31.1 and 2 or 12 th Ed.: 31.3 and 2

8 Resistor in an ac circuit i I cost Voltage drop across R? v R ir IRcost V R cost Phasor diagram

9 di dt v d I Isint Inductor in an ac circuit cos dt Icos t 90 t o Voltage drop across? i I cost V cos t90 Phasor diagram o V I is like a resistance, call it "Reactance" X Inductive reactance Reactance

10 a) Want X to have i=250 ma when V=3.6 V 6 V IX X V / I 3.6 V / A 14.4 k X 2 f 3 6 X / 2 f 14.4 / 2 Now want mH b) Want I for 16 MHz and 160 khz I V / X V / 2 f As f increases by 10, I decreases by 10 I=25mA As f decreases by 10, I increases by 10 I=2500mA Hmwk: Probls 31.3 and 5 (11 th Ed) or 5 and 9 (12 th Ed.)

11 i dq dt q apacitor in an ac circuit I I cost sint I q v v sin t cos t 90 o I i I cost Voltage across? Phasor diagram X 1 Reactance Take V I IX is capacitive reactance

12 v 1.2 Acos 2500t a) i 0.006cos 2500t A R 200 b) X rad/s 510 F 80 v V cos cos c) t / V 2500 rad/s t / 2 Hmwk: Probls 11th Ed.: 31.7 & 10 Soln: a) 1736 W, vr=1.10cos [120t] 12 th Ed.: & 12 Soln: a) 1736, v R =1.10cos [120t]

13 Summary of Section 31.2 R in an ac circuit i I cost v V cost R R in an ac circuit v V cos t 90 o Reactance X in an ac circuit v I cos Reactance t90 o X 1

14 Summary of Section 31.2 Homework for section 31.2: 11th Ed.: Probls 3, 5, 7 and th Ed.: Probls 5, 9, 11 and 12

15 i and v R are in phase i I cost v V cost R R v I cos t90 o v is behind 90 o v V cos t 90 o v is ahead 90 o At this point phasors become useful

16 v R is in phase with i v is behind 90 o v across source? v v v v v is ahead 90 o R cos cos 90 I o cos o 90 R v V t V t t Horizontal projection Horizontal projection Horizontal projection

17 Sum of horizontal projections of vectors = horizontal projection of sum of vectors Magnitude of sum : Sum voltage vectors: V 2 2 V V V IR IX IX I R X X R IZ; " impedance" is Z R X X R 1/ V V IX IX X X Phase angle : tan In conclusion, if the current is : The source voltage is : V IR R R i I cost v IZ cost

18 Formalism is still valid if an element is missing If R is missing, set R=0 If is missing, set =0 If is missing, set 1/ Z R X X R et us see an example!

19 X 10, 000 rad/s 60 mh X 1/ 1/ 10, 000 rad/s F Z R X X V 50 V I 0.1 A Z 500 V X X tan tan 53 R V IX V IX 0.1 A V IR 0.1 A V; 0.1 A V R Hmwk: Problem (11 th Ed.) or 14 (12 th Ed.)

20 A rads/s i I cos t 0.1 cos 10, 000 t vr ir IR cos t 30 V cos 10, 000 rads/s t v ix V cos t 90 V sin t 60 V sin 10, 000 rads/s t 0 v ix V cos t 90 V sin t 20 V sin 10, 000 rads/s t 0 Instantaneous source voltage v V cos t 50 cos 10, 000 t V rads/s Hmwk: Problem (11 th Ed.) or 16 (12 th Ed.)

21 Summary of Section 31.3 If the current is : The source voltage is : I mpedance is i I cost v IZ cost /, Z R X X R X X V V IX IX X X Phase angle : tan VR IR R Voltages across elements: vr IR cost v V cos t 90 IX cos t v cos 90 cos 90 V t IX t Instantaneous source voltage v V cost Hmwk Sect. 31.3: Probls & (11 th Ed.) or 14 & 16 (12 th Ed.)

22

23 Idea: vary to increase current I increases as Z decreases I V V Z 2 R X X 2 Z varies with 1/ 2 2 Z R X & X vary with X X 1/ Minimum of Z when X =X 0 1 Max. I at 0 Resonanting frequency

24 1 1 a) 0 5,000,000 rad/s 6 3 b) X 510 rad/s H 2000 X H F 6 rad/s 12 F 1/ 1/ Z R X X R R 500 Vrms 1 V c) irms 2 ma Z 500 d) Vrms R IrmsR A V Vrms Irms X 4 V Vrms Irms X 4 V

25 Summary of Section 31.5 I output is max at resonance X X Z R 0 1 Max. I at 0 Vrms irms Z V I R rmsr rms V I X rms rms V I X rms rms Hwk for sect. 31.5: Probs 27 a)& b) and 32 (11 th Ed.) or 31a)& b) and 36 (12 th Ed.) Solution of 32/36: 945 rad/s, 70.6, 450 V

26 Idea: Use an ac circuit at a given v and i to induce a different i and v in another circuit Start with a circuit with i and v Add a magnetic core Add a secondary loop A changing current I 1 induces a changing current I 2 I B B I

27 N Relations between currents and voltages 1 1 d dt B B N 2 2 d dt where is the flux per turn (equal on both sides of core) d B V2 V1 dt N1 N2 N1 Notice that the frequency is the same on both circuits Assuming no resistance in wires Power-in in circuit 1 = power-out in circuit 2 V N P I V P I V I I or I I V2 N2 B N

28 N2 V2 240 V a) Use a tranformer with 2 N1 V1 120 V V1 b) urrent: cannot use I2 I1 as I1 is not known V2 Must use P I V I V P 960 W P 960 W I2 4 A or I1 8 V2 240 V V1 120 V c) Resistance: take V RI R V / I 240 V / 4 A 60 Hmwk: Probl 33 (11 th Ed.) or 37 (12 th Ed.) A

29 Summary of Section 31.6 V N V N1 V N I I or I I V2 N2 P1 I1V 1 P2 I2V2 Frequency is the same on both circuits Power-in in circuit 1 = power-out in circuit 2 Hmwk Section 31.6: Probls 32 and 33 (11 th Ed.) or 36 and 37 (12 th Ed.)

30 PHYS Fields and Waves

31

32 You got what grade?

33

34 You got what grade?

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