alternating currents & electromagnetic wavesQuestionAnswerAlternating current circuitsA circuit with a resistorrms currents/voltagespower consumption in an AC circuitvector representationAC circuit with a single capacitorCapacitive circuit - continuedPower consumption in a capacitive circuitSlide 12Slide 13Slide 14ReactanceCombining the three: the LRC circuitAn LRC circuitimpedanceSlide 19Power consumption by an LRC circuitExampleanswersSlide 23Slide 24LRC circuits: an overviewQuestionResonances in an RLC circuitexamplequestiontransformersSlide 31Slide 32Slide 33electromagnetic wavesSlide 35electromagnetic waves can be used to broadcast…He then constructed an antennaproducing the electric field waveproducing the magnetic field waveSlide 40Slide 41alternating currents & electromagnetic wavesPHY232 – Spring 2007Jon Pumplinhttp://www.pa.msu.edu/~pumplin/PHY232(Ppt courtesy of Remco Zegers)PHY232 - Pumplin - alternating currents and electromagnetic waves 2QuestionAt 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/RLRVIPHY232 - Pumplin - alternating currents and electromagnetic waves 3Answer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/RThe coil opposes the flow of current due to self-inductance, so thecurrent cannot immediately become the maximum I=V/R. It willslowly rise to this value (characteristic time Tau = L/R).LRVIPHY232 - Pumplin - alternating currents and electromagnetic waves 4Alternating current circuits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.RV VRIIPHY232 - Pumplin - alternating currents and electromagnetic waves 5A circuit with a resistor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 NECESSARYIV(t)=V0sintRIR(A)V0=10 VR=2 Ohm=1 rad/sPHY232 - Pumplin - alternating currents and electromagnetic waves 6rms 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=ImaxRIR(A)|IR|(A) |VR|(V)VrmsIrmsPHY232 - Pumplin - alternating currents and electromagnetic waves 7power 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|IR|(A) |VR|(V)VrmsIrmsP(W)PHY232 - Pumplin - alternating currents and electromagnetic waves 8vector representationtime (s)V0-V0VThe voltage or current as a function of time can bedescribed by the projection of a vector rotating with constant angular velocity on one of the axes (x or y).=tPHY232 - Pumplin - alternating currents and electromagnetic waves 9AC circuit with a single capacitorIV(t)=V0sintCVc = V0sintQc = CVc= C V0 sintIc = Qc/t =  C V0 costSo, the current peaks ahead of the voltage:There is a difference in phase of /2 (900).I (A)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 andthe current decreases. If the potential over C is maximum, the current is zero.PHY232 - Pumplin - alternating currents and electromagnetic waves 10Capacitive circuit - continuedI (A)Note: Imax=  C V0For a resistor we have I = V0/R so ‘1/C’ is similar to ‘R’And we write: I=V/Xc with Xc= 1/C the capacitive reactanceUnits of Xc are Ohms. The capacitive reactance acts like a resistancein this circuit.IV(t) = V0 sintCPHY232 - Pumplin - alternating currents and electromagnetic waves 11Power consumption in a capacitive circuitThere is no power consumption in a purely capacitive circuit:Energy (1/2 C V2) gets stored when the (absolute) voltage over thecapacitor is increasing, and released when it is decreasing.Pave = 0 for a purely capacitive circuitPHY232 - Pumplin - alternating currents and electromagnetic waves 12AC circuit with a single inductorIV(t) = V0 sintLVL= V0 sint = L I/tI = -(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)I (A)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.PHY232 - Pumplin - alternating currents and electromagnetic waves 13Inductive circuit - continuedNote: 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 the inductive reactanceUnits of XL are Ohms. The inductive reactance acts as a resistancein this circuit.IL(A)IV(t) = V0 sintLI(A)PHY232 - Pumplin - alternating currents and electromagnetic waves 14Power consumption in an inductive circuitThere is no power consumption in a purely inductive circuit:Energy (1/2 L I2) gets stored when the (absolute) current through theinductor is increasing, and released when it is decreasing.Pave = 0 for a purely inductive circuitPHY232 - Pumplin - alternating currents and electromagnetic waves 15Reactance 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