29 Overview why how to use rms values

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29 Overview • why & how to use rms values • determine impedance of

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

sinusoidal current “ac” • I ~ sine, cosine variation with time: (I = Io

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

basic circuits with: 3

basic circuits with: 3

resistors: VR ~ I 4

resistors: VR ~ I 4

inductors: VL ~ d. I/dt voltage “leads” current 5

inductors: VL ~ d. I/dt voltage “leads” current 5

capacitors: VC ~ Q current “leads” voltage 6

capacitors: VC ~ Q current “leads” voltage 6

impedance Z = “ac R” 7

impedance Z = “ac R” 7

Example: 55 m. H Inductor, r = 0, connected to household 120 VAC (60

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

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

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

Example I(t) = 0. 577 Io 10

Example I(t) = 0. 577 Io 10

Summary • sine dependent I has I rms = 0. 707 Io • other

Summary • sine dependent I has I rms = 0. 707 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 11

exponential notation used to replace cosine or sine dependence 12

exponential notation used to replace cosine or sine dependence 12

exp derivatives 13

exp derivatives 13

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

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

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

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

Student Data (L ~ 1 m. H, f ~ 10, 000 Hz) V V-ind

Student Data (L ~ 1 m. H, f ~ 10, 000 Hz) V V-ind V-R 15 ohm 6. 7 6. 6 1. 0 60 ohm 6. 3 4. 8 4. 3 100 ohm 6. 5 3. 9 5. 4 angle 79 50 36 16

Trig Calculations 17

Trig Calculations 17

Phasor Calculation e phase e 2 f e 1 18

Phasor Calculation e phase e 2 f e 1 18

Phasor Calculation e phase e 2 f e 1 19

Phasor Calculation e phase e 2 f e 1 19

phasor 20

phasor 20

Exercise • Use trig identity & phasor method to show that • has amplitude

Exercise • Use trig identity & phasor method to show that • has amplitude 5. 66 and phase 45°. 21

Resonance in an RLC Circuit • • • min. Z: when XL = XC

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 22

Transformer 23

Transformer 23

AC Power average 24

AC Power average 24

AC Power 25

AC Power 25

An I(t) current source continuously repeats the following pattern: {1 seconds @ 3 ampere,

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

If a sinusoidal generator has a maximum voltage of 170 V, what is the

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

R setting Actual R 10 ohm 30 ohm 60 ohm 100 ohm Vapp(V) Vind(V)

R setting Actual R 10 ohm 30 ohm 60 ohm 100 ohm Vapp(V) Vind(V) VR(V) Table 2: Calculated Data cosf f(degrees) VL = Vsinf Vr = Vcosf - VR r = RVr/VR L = RVL/(w. VR) 28

Alternating Current Generators fm = NBAcosq. 29

Alternating Current Generators fm = NBAcosq. 29

Generators fm = NBAcosq: ( q = wt + d when rotating ) emf

Generators fm = NBAcosq: ( q = wt + d when rotating ) emf = -dfm/dt = -NBAw(-sin(wt + d)) emf = NBAw sin(wt + d) (emf)peak = NBAw. 30

AC Generator applied to Resistor 31

AC Generator applied to Resistor 31