1 29 Overview why & how to use rms values determine impedance of L & C why & how: phase relationships in ac circuits

2 sinusoidal current “ac” I ~ sine, cosine variation with time: (I = Io cos(wt + phi)) w = 2pf, e.g. US grid uses 60 cycles/sec, w = 2p(60) = 377 rad/s

3 basic circuits with:

4 resistors: V R ~ I

5 inductors: V L ~ dI/dt voltage “leads” current

6 capacitors: V C ~ Q current “leads” voltage

7 impedance Z = “ac R”

8 Example: 55mH Inductor, r = 0, connected to household 120VAC (60 hertz).

9 Example: 10  F capacitor: connected to household 120VAC (60 hertz).

10 Example I(t) = I o

11 Summary sine dependent I has I rms = Io other rms values from direct calculation phase relations: R: phi = 0 L: voltage on inductor leads I. C: I to capacitor leads voltage. impedance & resonance in RLC circuit

12 exponential notation used to replace cosine or sine dependence

13 exp derivatives

14 RLC exp application: From dx/dt = I, Z and phase are:

15 ac LR lab measure: voltages calculate: L & phase angle

16 Student Data (L ~ 1mH, f ~ 10,000Hz) 15ohm60ohm100ohm V V-ind V-R angle795036

17 Trig Calculations

18 Phasor Calculation     phase

19 Phasor Calculation     phase

20 phasor

21 Exercise Use trig identity & phasor method to show that has amplitude 5.66 and phase 45°.

22 Resonance in an RLC Circuit min. Z: when XL = XC result: large currents application: radio tuner hi power at tuned freq. low power at other f’s Ex. calc LC for f = 10,000

23 Transformer

24 AC Power average

25 AC Power

26 An I(t) current source continuously repeats the following pattern: {1 3 ampere, 1 0 ampere} Calculate average, rms I.

27 If a sinusoidal generator has a maximum voltage of 170V, what is the root- mean-square voltage of the generator?

28 R setting Actual R  10 ohm30 ohm60 ohm100 ohm V app (V) V ind (V) V R (V) Table 2: Calculated Data cosf f(degrees) V L = Vsinf V r = Vcosf - V R r = RV r /V R L = RV L /(wV R )

29 Alternating Current Generators  m = NBAcos .

30 Generators  m = NBAcos  : (  =  t +  when rotating ) emf = -d  m /dt = -NBA  (-sin(  t +  )) emf = NBA  sin(  t +  ) (emf) peak = NBA .

31 AC Generator applied to Resistor