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alternating currents & electromagnetic waves

alternating currents & electromagnetic waves. R. L. V. Question. At t=0, the switch is closed. 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. I. R. L. V.

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alternating currents & electromagnetic waves

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  1. alternating currents & electromagnetic waves

  2. R L V Question At t=0, the switch is closed. 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 I alternating currents and electromagnetic waves

  3. R L V 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 (characteristic time Tau = L/R). Answer • At t=0, the switch is closed. 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 I alternating currents and electromagnetic waves

  4. Alternating current circuits R R • Previously, we look at DC circuits: the voltage delivered by the source is constant, as on the left. • Now, we look at AC circuits, in which case the source is sinusoidal. A is used in circuits to denote this. I I V V alternating currents and electromagnetic waves

  5. A circuit with a resistor R • The voltage over the resistor is the same as the voltage delivered by the source: VR(t) = V0 sint = V0 sin(2ft) • The current through the resistor is: IR(t)= (V0/R)sint • Since V(t) and I(t) have the same behavior as a function of time, they are said to be ‘in phase’. • V0 is the maximum voltage • V(t) is the instantaneous voltage •  is the angular frequency; =2f f: frequency (Hz) • SET YOUR CALCULATOR TO RADIANS WHERE NECESSARY IR(A) I V0=10 V R=2 Ohm =1 rad/s V(t)=V0sint alternating currents and electromagnetic waves

  6. rms currents/voltages • 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) : • Vrms=Vmax/2 • Irms=Imax/2 • The following hold: • Vrms=IrmsR • Vmax=ImaxR IR(A) Vrms |IR|(A) |VR|(V) Irms alternating currents and electromagnetic waves

  7. power consumption in an AC circuit • We already know for DC P = V I = V2/R = I2 R • For AC circuits with a single resistor: P(t) = V(t) * I(t) = V0 I0 (sint)2 • Average power consumption: Pave= Vrms* Irms = V2rms/R = I2rms R where Vrms = Vmax/2) Irms = Imax/2 Vrms |IR|(A) |VR|(V) Irms P(W) alternating currents and electromagnetic waves

  8. vector representation V0 =t V time (s) -V0 The voltage or current as a function of time can be described by the projection of a vector rotating with constant angular velocity on one of the axes (x or y). alternating currents and electromagnetic waves

  9. AC circuit with a single capacitor C I I(A) V(t)=V0sint Vc = V0sint Qc = CVc= C V0 sint Ic = Qc/t =  C V0 cost So, the current peaks ahead of the voltage: There is a difference in phase of /2 (900). 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. alternating currents and electromagnetic waves

  10. Capacitive circuit - continued C I I(A) V(t) = V0 sint Note: Imax=  C V0 For a resistor we have I = V0/R so ‘1/C’ is similar to ‘R’ And we write: I=V/Xc with Xc= 1/C thecapacitive reactance Units of Xc are Ohms. The capacitive reactance acts like a resistance in this circuit. alternating currents and electromagnetic waves

  11. Power consumption in a capacitive circuit There is no power consumption in a purely capacitive circuit: Energy (1/2 C V2) gets stored when the (absolute) voltage over the capacitor is increasing, and released when it is decreasing. Pave = 0 for a purely capacitive circuit alternating currents and electromagnetic waves

  12. AC circuit with a single inductor L I I(A) V(t) = V0 sint VL= V0 sint = L I/t I= -(V0/(L)) cost (no proof here: you need calculus…) the current peaks later in time than the voltage: there is a difference in phase of /2 (900) 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. alternating currents and electromagnetic waves

  13. Inductive circuit - continued L IL(A) I I(A) V(t) = V0 sint Note: Imax= V0/(L) For a resistor we have I = V0/R so ‘L’ is similar to ‘R’ And we write: I = V/XL with XL = L theinductive reactance Units of XL are Ohms. The inductive reactance acts as a resistance in this circuit. alternating currents and electromagnetic waves

  14. Power consumption in an inductive circuit There is no power consumption in a purely inductive circuit: Energy (1/2 L I2) gets stored when the (absolute) current through the inductor is increasing, and released when it is decreasing. Pave = 0 for a purely inductive circuit alternating currents and electromagnetic waves

  15. Reactance The inductive reactance (and capacitive reactance) are like the resistance of a normal resistor, in that you can calculate the current, given the voltage, using I = V/XL (or I = V/XC ). This works for the Maximum values, or for the RMS average values. But I and V are “out of phase”, so the maxima occur at different times. alternating currents and electromagnetic waves

  16. Combining the three: the LRC circuit L C R • Things to keep in mind when analyzing this system: • 1) The current in the system has the same value everywhere I = I0 sin(t-) • 2) The voltage over all three components is equal to the source voltage at any point in time: V(t) = V0 sin(t) I V(t)=V0sint alternating currents and electromagnetic waves

  17. An LRC circuit L C R I • For the resistor: VR = IR and VR and I are in phase • For the capacitor: Vc = I Xc (“Vc lags I by 900”) • For the inductor: VL= I XL (“VL leads I by 900”) • at any instant: VL+Vc+VR=V0 sin(t). But the maximum values of VL+Vc+VR do NOT add up to V0 because they have their maxima at different times. VR VC I VL V(t)=V0sint alternating currents and electromagnetic waves

  18. impedance L C R • Define X = XL-Xc = reactance of RLC circuit • Define Z = [R2+(XL-Xc)2]= [R2+X2] = impedance of RLC cir • Then Vtot = I Z looks like Ohms law! I V(t)=V0sint alternating currents and electromagnetic waves

  19. Resonance • If the maximum voltage over the capacitor equals the maximum voltage over the inductor, the difference in phase between the voltage over the whole circuit and the voltage over the resistor is: • a) 00 • b)450 • c)900 • d)1800 In this case, XL alternating currents and electromagnetic waves

  20. 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 = I2rms R alternating currents and electromagnetic waves

  21. Given: R=250 Ohm L=0.6 H C=3.5 F f=60 Hz V0=150 V Example L C R • questions: • what is the angular frequency of the system?what are the inductive and capacitive reactances? • what is the impedance, what is the phase angle  • what is the maximum current and peak voltages over each element • compare the algebraic sum of peak voltages with V0. Does this make sense? • what are the instantaneous voltages and rms voltages over each element? • what is power consumed by each element and total power consumption I V(t)=V0sint alternating currents and electromagnetic waves

  22. Given: R=250 Ohm L=0.6 H C=3.5 F f=60 Hz V0=150 V answers • a) angular frequency  of the system? • =2f=260=377 rad/s • b) Reactances? • XC=1/C=1/(377 x 3.5x10-6)=758 Ohm • XL= L=377x0.6=226 Ohm • c) Impedance and phase angle • Z=[R2+(XL-Xc)2]=[2502+(226-758)2]=588 Ohm • =tan-1[(XL-XC)/R)=tan-1[(226-758)/250]=-64.80 (or –1.13 rad) • d) Maximum current and maximum component voltages: • Imax=Vmax/Z=150/588=0.255 A • VR=ImaxR=0.255x250=63.8 V • VC=ImaxXC=0.255x758=193 V • VL=ImaxXL=0.255x266=57.6 V • Sum: VR+VC+VL=314 V. This is larger than the maximum voltage delivered by the source (150 V). This makes sense because the relevant sum is not algebraic: each of the voltages are vectors with different phases. alternating currents and electromagnetic waves

  23. Imax=Vmax/Z=0.255 A • VR=ImaxR=63.8 V • VC=ImaxXC=193 V • VL=ImaxXL=57.6 V • =-64.80 (or –1.13 rad) • Vtot=150 V answers • f) instantaneous voltages over each element (Vtot has 0 phase)? • start with the driving voltage V=V0sint=Vtot • VR(t)=63.8sin(t+1.13) (note the phase relative to Vtot) • VC(t)=193sin(t-0.44) phase angle : 1.13-/2=-0.44 • VL(t)=57.6sin(t+2.7) phase angle : 1.13+/2=2.7 • rms voltages over each element? • VR,rms=63.8/2=45.1 V • VC,rms=193/2=136 V • VL,rms=57.6/2=40.7 V alternating currents and electromagnetic waves

  24. answers • g) power consumed by each element and total power consumed? • PC=PL=0 no energy is consumed by the capacitor or inductor • PR=Irms2R=(Imax/2)2R=0.2552R/2=0.2552*250/2)=8.13 W • or: PR=Vrms2/R=(45.1)2/250=8.13 W (don’t use Vrms=V0/2!!) • or: PR=VrmsIrmscos=(150/2)(0.255/2)cos(-64.80)=8.13 W • total power consumed=power consumed by resistor! alternating currents and electromagnetic waves

  25. LRC circuits: an overview • Reactance of capacitor: Xc= 1/C • Reactance of inductor: XL= L • Current through circuit: same for all components • ‘Ohms’ law for LRC circuit: Vtot=I Z • Impedance: Z=[R2+(XL-Xc)2] • phase angle between current and source voltage: tan=(|VL|-|Vc|)/VR=(XL-Xc)/R • Power consumed (by resistor only): P=I2rmsR=IrmsVR P=VrmsIrmscos • VR=ImaxR in phase with current I, out of phase by  with Vtot • VC=ImaxXC behind by 900 relative to I (and VR) • VL=ImaxXL ahead of 900 relative to I (and VR) alternating currents and electromagnetic waves

  26. Question • 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 2nd 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! alternating currents and electromagnetic waves

  27. Resonances in an RLC circuit • If we chance the (angular) frequency the reactances will change since: • Reactance of capacitor: Xc= 1/C • Reactance of inductor: XL= L • Consequently, the impedance Z=[R2+(XL-Xc)2] changes • Since I=Vtot/Z, the current through the circuit changes • If XL=XC (I.e. 1/C= L or2=1/LC), Z is minimal, I is maximum) • = (1/LC) is the resonance angular frequency • At the resonance frequency =0 alternating currents and electromagnetic waves

  28. 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 F f=60 Hz V0=150 V = (1/LC)=690 rad/s f= /2=110 Hz alternating currents and electromagnetic waves

  29. question • An LRC circuit has R=50 Ohm, L=0.5 H and C=5x10-3 F. An AC source with Vmax=50V is used. If the resistance is replaced with one that has R=100 Ohm and the Vmax 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 alternating currents and electromagnetic waves

  30. transformers transformers are used to convert voltages to lower/higher levels alternating currents and electromagnetic waves

  31. transformers primary circuit with Np loops in coil secondary circuit with Ns loops in coil Vp Vs iron core If an AC current is applied to the primary circuit: Vp=-NpB/t The magnetic flux is contained in the iron and the changing flux acts in the secondary coil also: Vs=-NsB/t Therefore: Vs=(Ns/Np)Vp if Ns<Np then Vs<Vp A perfect transformer is a pure inductor (no resistance), so no power loss: Pp=PS and VpIp=VsIs ; if Ns<Np then Vs<Vp and IS>Ip alternating currents and electromagnetic waves

  32. 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 10000 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? • Vs=(Ns/Np)Vp or Ns=(Vs/Vp)Np = 120 windings • VpIp=VsIs so Is=VpIp/Vs=8.33 A alternating currents and electromagnetic waves

  33. 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 ([V]2/R=I2R) also low alternating currents and electromagnetic waves

  34. 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 fields 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 alternating currents and electromagnetic waves

  35. electromagnetic waves Maxwell found that electric and magnetic waves travel together through space with a velocity of 1/(00) v=1/(00)=1/(4x10-7 x 8.85x10-12)=2.998x108 m/s which is just the speed of light (c) alternating currents and electromagnetic waves

  36. 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 = (1/LC)=6.32x108 rad/s f= /2=100 MHz. Recall that the wavelength of waves =v/f=c/f=3x108/100x106=3.0 m wavelength: =v/f alternating currents and electromagnetic waves

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

  38. ++++++ ---------- ---------- ++++++ producing the electric field wave antenna alternating currents and electromagnetic waves

  39. ++++++ ---------- I I I ---------- ++++++ I producing the magnetic field wave E and B are in phase and E=cB with c: speed of light The power/m2=0.5EmaxBmax/0 The energy in the wave is shared between the E-field and the B-field antenna alternating currents and electromagnetic waves

  40. question Can a single wire connected to the + and – poles of a DC battery act as a transmitter of electromagnetic waves? • yes • no answer: no: there is no varying current and hence no wave can be made. alternating currents and electromagnetic waves

  41. c=f  alternating currents and electromagnetic waves

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