alternating currents & electromagnetic waves

Transcription

1 alternating currents & electromagnetic waves PHY232 Remco Zegers Room W109 cyclotron building

2 quiz (extra credit) R L I V At t=0, the switch is closed. Shortly after that: a) the current slowly increases from I=0 to I=V/R b) the current slowly decreases from I=V/R to I=0 c) the current is a constant I=V/R The coil opposes the flow of current due to self-inductance, so the current cannot immediately become the maximum I=V/R. It will slowly rise to this value PHY232 - Remco Zegers - alternating currents and electromagnetic waves 2

3 Alternating current circuits R R I I V previously, we look at DC circuits (the voltage delivered by the source was constant). Now, we look at AC circuits, in which case the source is sinusoidal. A is used in circuits to denote the difference V PHY232 - Remco Zegers - alternating currents and electromagnetic waves 3

4 A circuit with a resistor I R V(t)=V 0 sinωt I R (A) V 0 =10 V R=2 Ohm ω=1 rad/s The voltage over the resistor is the same as the voltage delivered by the source: V R (t)=v 0 sinωt=v 0 sin(2πft) The current through the resistor is: I R (t)= V 0 /R sinωt Since V(t) and I(t) have the same behavior as a function of time, they are said to be in phase. V 0 is the maximum voltage V(t) is the instantaneous voltage ω is the angular frequency; ω=2πf f: frequency (Hz) SET YOUR CALCULATOR TO RADIANS WHERE NECESSARY PHY232 - Remco Zegers - alternating currents and electromagnetic waves 4

5 lon-capa you should now do problem 1 from set 7. PHY232 - Remco Zegers - alternating currents and electromagnetic waves 5

6 rms currents/voltages I R (A) I R (A) V R (V) I rms V rms To understand energy consumption by the circuit, it doesn t matter what the sign of the current/voltage is. We need the absolute average currents and voltages (root-mean-square values) : V rms =V max / 2 I rms =I max / 2 The following hold: V rms =I rms R V max =I max R PHY232 - Remco Zegers - alternating currents and electromagnetic waves 6

7 power consumption by an AC circuit I R (A) V R (V) I rms V rms We already saw (DC): P=VI=V 2 /R=I 2 R For AC circuits with a single resistor: P(t)=V(t)*I(t)=V 0 I 0 sin 2 ωt P(W) The average power consumption: P ave =V rms *I rms =V 2 rms/r=i 2 rmsr P ave =(V max / 2)( I max / 2)= I max V max /2 PHY232 - Remco Zegers - alternating currents and electromagnetic waves 7

8 A circuit with a single capacitor C I V(t)=V 0 sinωt I C (A) V c = V 0 sinωt Q c =CV c =CV 0 sinωt I c =ΔQ c /Δt= ωcv 0 cosωt= ωcv 0 sin(ωt+π/2) So, the current peaks ahead in time (earlier) of the voltage There is a difference in phase of π/2 (90 0 ) why? When there is not much charge on the capacitor it readily accepts more and current easily flows. However, the E-field and potential between the plates increase and consequently it becomes more difficult for current to flow and the current decreases. If the potential over C is maximum, the current is zero. PHY232 - Remco Zegers - alternating currents and electromagnetic waves 8

9 capacitive reactance I C (A) V c = V 0 sinωt Q c =CV c =CV 0 sinωt I c =ΔQ c /Δt= ωcv 0 cosωt= ωcv 0 sin(ωt+π/2) Note: I max = ωcv 0 For a resistor we have I=V 0 /R so 1/ωC is similar to R And we write: I=V/X c with X c = 1/ωC the capacitive reactance Units of X c are Ohms. The capacitive reactance acts as a resistance in this circuit. PHY232 - Remco Zegers - alternating currents and electromagnetic waves 9

10 power consumption in a capacitive circuit There is no power consumption in a purely capacitive circuit: Energy (1/2CV 2 ) gets stored when the (absolute) voltage over the capacitor is increasing, and released when it is decreasing. P ave = 0 for a purely capacitive circuit PHY232 - Remco Zegers - alternating currents and electromagnetic waves 10

11 A circuit with a single inductor L I I L (A) V(t)=V 0 sinωt V L = V 0 sinωt=lδi/δt I L =-V 0 /(ωl)cosωt= V 0 /(ωl)sin(ωt-π/2) (no proof here: you need calculus ) So, the current peaks later in time than the voltage There is a difference in phase of π/2 (90 0 ) why? As the potential over the inductor rises, the magnetic flux produces a current that opposes the original current. The voltage across the inductor peaks when the current is just beginning to rise, due to this tug of war. PHY232 - Remco Zegers - alternating currents and electromagnetic waves 11

12 inductive reactance I L (A) V L = V 0 sinωt=lδi/δt I L =-V 0 /(ωl)cosωt= V 0 /(ωl)sin(ωt-π/2) Note: I max = V 0 /(ωl) For a resistor we have I=V 0 /R so ωl is similar to R And we write: I=V/X L with X L = ωl the inductive reactance Units of X L are Ohms. The inductive reactance acts as a resistance in this circuit. PHY232 - Remco Zegers - alternating currents and electromagnetic waves 12

13 power consumption in an inductive circuit There is no power consumption in a purely inductive circuit: Energy (1/2LI 2 ) gets stored when the (absolute) current through the inductor is increasing, and released when it is decreasing. P ave = 0 for a purely inductive circuit PHY232 - Remco Zegers - alternating currents and electromagnetic waves 13

14 Combining the three: the LRC circuit L C R I V(t)=V 0 sinωt Things to keep in mind when analyzing this system: 1) The current in the system has the same value everywhere I=I 0 sin(ωt-φ) 2) The voltage over all three components is equal to the source voltage at any point in time: V(t)=V 0 sin(ωt) PHY232 - Remco Zegers - alternating currents and electromagnetic waves 14

15 An LRC circuit I V tot V R V C V L For the resistor: V R =I R R and V R and I R =I are in phase For the capacitor: V c =IX c and V c lags I c =I by 90 0 For the inductor: V L =IX L and V L leads I L =I by 90 0 at any instant: V L +V c +V R =V 0 sin(ωt), that is the total voltage V tot is the vector addition of the three individual components, it peaks at a different time as the current I PHY232 - Remco Zegers - alternating currents and electromagnetic waves 15

16 impedance V tot = V L (t)+v c (t)+v R (t) Remember V R, V c and V L peak at different times V tot = [V R2 +( V L - V C ) 2 ]= (you need vector to derive this...) [ (IR) 2 +(IX L -IX C ) 2 ]=I [R 2 +(X L -X c ) 2 ] define X=X L -X c : reactance of an RLC circuit define Z= [R 2 +(X L -X c ) 2 ]= [R 2 +X 2 ] : impedance of RLC circ. V tot =IZ & I=V tot /Z looks like Ohms law The current I and the voltage V tot are out of phase by an angle φ. This angle can be calculated with: tanφ=opposite/adjacent=( V L - V c )/V R =X/R The time difference between the peak current and peak voltage V tot can be calculated with φ/ω PHY232 - Remco Zegers - alternating currents and electromagnetic waves 16

17 phase angle The current I and the voltage V tot are out of phase by an angle φ. This angle can be calculated with: tanφ=( V L - V c )/V R =X/R difference in peak time between I and V tot is equal φ/ω I V tot V R V C V L PHY232 - Remco Zegers - alternating currents and electromagnetic waves 17

18 power consumption by an LRC circuit Even though the capacitor and inductor do not consume energy on the average, they affect the power consumption since the phase between current and voltage is modified. P=I 2 rms R=I rms V R V R =V rms cosφ: the cosφ is due to the difference in phase between current and voltage So: P=V rms I rms cosφ cosφ: power factor of a circuit PHY232 - Remco Zegers - alternating currents and electromagnetic waves 18

19 lon-capa you should now do problem 4 from LON-CAPA 7 PHY232 - Remco Zegers - alternating currents and electromagnetic waves 19

20 Given: R=250 Ohm L=0.6 H C=3.5 μf f=60 Hz V 0 =150 V example I L C R questions: V(t)=V 0 sinωt a) what is the angular frequency of the system? b) what are the inductive and capacitive reactances? c) what is the impedance, what is the phase angle φ d) what is the maximum current and peak voltages over each element Compare the algebraic sum of peak voltages with V 0. Does this make sense? e) what are the instantaneous voltages and rms voltages over each element. Consider V tot to have zero phase. f) power consumed by each element and total power consumption PHY232 - Remco Zegers - alternating currents and electromagnetic waves 20

21 answers a) angular frequency ω of the system? ω=2πf=2π60=377 rad/s b) Reactances? X C =1/ωC=1/(377 x 3.5x10-6 )=758 Ohm X L = ωl=377x0.6=226 Ohm c) Impedance and phase angle Given: R=250 Ohm L=0.6 H C=3.5 μf f=60 Hz V 0 =150 V Z= [R 2 +(X L -X c ) 2 ]= [ ( ) 2 ]=588 Ohm φ=tan -1 [(X L -X C )/R)=tan -1 [( )/250]= (or 1.13 rad) d) Maximum current and maximum component voltages: I max =V max /Z=150/588=0.255 A V R =I max R=0.255x250=63.8 V V C =I max X C =0.255x758=193 V V L =I max X L =0.255x266=57.6 V Sum: V R +V C +V L =314 V. This is larger than the maximum voltage delivered by the source (150 V), which seems odd. But remember, the voltages over the three components peak at different times, so the sum will in reality never exceed 150 V! PHY232 - Remco Zegers - alternating currents and electromagnetic waves 21

22 I max =V max /Z=0.255 A V R =I max R=63.8 V V C =I max X C =193 V answers V L =I max X L =57.6 V φ= (or 1.13 rad) V tot =150 V e) instantaneous voltages over each element (V tot has 0 phase)? start with the driving voltage V=V 0 sinωt=v tot V R (t)=63.8sin(ωt+1.13) (note the phase relative to V tot ) V C (t)=193sin(ωt-0.44) phase angle : 1.13-π/2=-0.44 V L (t)=57.6sin(ωt+2.7) phase angle : 1.13+π/2=2.7 rms voltages over each element? V R, rms =63.8/ 2=45.1 V V C,rms =193/ 2=136 V V L,rms =57.6/ 2=40.7 V PHY232 - Remco Zegers - alternating currents and electromagnetic waves 22

23 answers g) power consumed by each element and total power consumed? P C =P L =0 no energy is consumed by the capacitor or inductor P R =I rms2 R=(I max / 2) 2 R= R/2= *250/2)=8.13 W or: P R =V rms2 /R=(45.1) 2 /250=8.13 W (don t use V rms =V 0 / 2!!) or: P R =V rms I rms cosφ=(150/ 2)(0.255/ 2)cos( )=8.13 W total power consumed=power consumed by resistor! PHY232 - Remco Zegers - alternating currents and electromagnetic waves 23

24 lon-capa you should now try problem 6 of lon-capa set 7, except for the last part PHY232 - Remco Zegers - alternating currents and electromagnetic waves 24

25 LRC circuits: an overview Reactance of capacitor: X c = 1/ωC Reactance of inductor: X L = ωl Current through circuit: same for all components Ohms law for LRC circuit: V tot =IZ Impedance: Z= [R 2 +(X L -X c ) 2 ] phase angle between current and source voltage: tanφ=( V L - V c )/V R =(X L -X c )/R Power consumed (by resistor only): P=I 2 rmsr=i rms V R P=V rms I rms cosφ V R =I max R in phase with current I, out of phase by φ with V tot V C =I max X C behind by 90 0 relative to I (and V R ) V L =I max X L ahead of 90 0 relative to I (and V R ) PHY232 - Remco Zegers - alternating currents and electromagnetic waves 25

26 Resonances in an RLC circuit If we chance the (angular) frequency the reactances will change since: Reactance of capacitor: X c = 1/ωC Reactance of inductor: X L = ωl Consequently, the impedance Z= [R 2 +(X L -X c ) 2 ] changes Since I=V tot /Z, the current through the circuit changes If X L =X C (I.e. 1/ωC= ωl or ω 2 =1/LC), Z is minimal, I is maximum) ω= (1/LC) is the resonance angular frequency At the resonance frequency φ=0 Z I 0 φ ω PHY232 - Remco Zegers - alternating currents and electromagnetic waves 26

27 example Using the same given parameters as the earlier problem, what is the resonance frequency? Given: R=250 Ohm L=0.6 H C=3.5 μf f=60 Hz V 0 =150 V ω= (1/LC)=690 rad/s f= ω/2π=110 Hz PHY232 - Remco Zegers - alternating currents and electromagnetic waves 27

28 question An LRC circuit has R=50 Ohm, L=0.5 H and C=5x10-3 F. An AC source with V max =50V is used. If the resistance is replaced with one that has R=100 Ohm and the V max of the source is increased to 100V, the resonance frequency will: a) increase b)decrease c) remain the same answer c) the resonance frequency only depends on L and C PHY232 - Remco Zegers - alternating currents and electromagnetic waves 28

29 conceptual problem about LRC which of the following is not correct in case of an AC LRC circuit? a) in case the inductive (X L ) and capacitive reactances (X C ) are equal in size, the rms current through the circuit is largest. b) The current through and voltage over the resistor are in always in phase c) if the LRC circuit is in resonance, both the capacitive reactance and inductive reactance must be zero d) the rms current at resonance (I.e. when it is largest) is independent of the frequency. PHY232 - Remco Zegers - alternating currents and electromagnetic waves 29

30 quiz (extra credit) The sum of maximum voltages over the resistor, capacitor and inductor in an LRC circuit cannot be higher than the maximum voltage delivered by the source since it violates Kirchhoff s 2 nd rule (sum of voltage gains equals the sum of voltage drops). a) true b) false answer: false The maximum voltages in each component are not achieved at the same time! PHY232 - Remco Zegers - alternating currents and electromagnetic waves 30

31 loncapa You should now try question 6, part 7 and question 5 of lon-capa set 7. PHY232 - Remco Zegers - alternating currents and electromagnetic waves 31

32 transformers transformers are used to convert voltages to lower/higher levels PHY232 - Remco Zegers - alternating currents and electromagnetic waves 32

33 transformers primary circuit with N p loops in coil V p V s secondary circuit with N s loops in coil iron core If an AC current is applied to the primary circuit: V p =-N p ΔΦ B /Δt The magnetic flux is contained in the iron and the changing flux acts in the secondary coil also: V s =-N s ΔΦ B /Δt Therefore: V s =(N s /N p )V p if N s <N p then V s <V p A perfect transformer is a pure inductor (no resistance), so no power loss: P p =P S and V p I p =V s I s ; if N s <N p then V s <V p and I S >I p PHY232 - Remco Zegers - alternating currents and electromagnetic waves 33

34 question a transformer is used to bring down the high-voltage delivered by a powerline (10 kv) to 120 V. If the primary coil has windings, a) how many are there in the secondary coil? b) If the current in the powerline is 0.1 A, what is the maximum current at 120 V? a) V s =(N s /N p )V p or N s =(V s /V p )N p = 120 windings b) V p I p =V s I s so I s =V p I p /V s =8.33 A PHY232 - Remco Zegers - alternating currents and electromagnetic waves 34

35 question Is it more economical to transmit power from the power station to homes at high voltage or low voltage? a) high voltage b) low voltage answer: high voltage If the voltage is high, the current is low If the current is low, the voltage drop over the power line (with resistance R) is low, and thus the power dissipated in the line ([ΔV] 2 /R=I 2 R) also low PHY232 - Remco Zegers - alternating currents and electromagnetic waves 35

36 electromagnetic waves James Maxwell formalized the basic equations governing electricity and magnetism ~1870: Coulomb s law Magnetic force Ampere s Law (electric currents make magnetic fields) Faraday s law (magnetic fields make electric currents) Since changing electric fields produce magnetic fields and vice versa, he concluded: electricity and magnetism are two aspects of the same phenomenon. They are unified under one set of laws: the laws of electromagnetism PHY232 - Remco Zegers - alternating currents and electromagnetic waves 36

37 electromagnetic waves Maxwell found that electric and magnetic waves travel together through space with a velocity of 1/ (μ 0 ε 0 ) v=1/ (μ 0 ε 0 )=1/ (4πx10-7 x 8.85x10-12 )=2.998x10 8 m/s which is just the speed of light (c) PHY232 - Remco Zegers - alternating currents and electromagnetic waves 37

38 electromagnetic waves can be used to broadcast Consider the experiment performed by Herz (1888) I Herz made an RLC circuit with L=2.5 nh, C=1.0nF The resonance frequency is ω= (1/LC)=6.32x10 8 rad/s f= ω/2π=100 MHz. Recall that the wavelength of waves λ=v/f=c/f=3x10 8 /100x10 6 =3.0 m wavelength: λ=v/f PHY232 - Remco Zegers - alternating currents and electromagnetic waves 38

39 He then constructed an antenna dipole antenna charges and currents vary sinusoidally in the primary and secondary circuits. The charges in the two branches also oscillate at the same frequency f I PHY232 - Remco Zegers - alternating currents and electromagnetic waves 39

40 producing the electric field wave antenna PHY232 - Remco Zegers - alternating currents and electromagnetic waves 40

41 producing the magnetic field wave E and B are in phase and E=cB with c: speed of light The power/m 2 =0.5E max B max /μ 0 antenna II II The energy in the wave is shared between the E-field and the B-field PHY232 - Remco Zegers - alternating currents and electromagnetic waves 41

42 c=fλ PHY232 - Remco Zegers - alternating currents and electromagnetic waves 42

43 lon-capa now try questions 2 and 7 from set 7. PHY232 - Remco Zegers - alternating currents and electromagnetic waves 43