PHY2054: Chapter 211AC Source and RLC Circuits21/2LCXfLXfCππ==maxmaxtanLCIZXXRεφ=−=Inductive reactanceCapacitive reactance()22LCZRXX=+−Total impedancePhase angleMaximum currentPHY2054: Chapter 212Pictorial Understanding of ReactancetanLCXXRφ−=()22LCZRXX=+−cosRZφ=PHY2054: Chapter 213Summary of Circuit Elements, Impedance, Phase Angles()22LCZRXX=+−tanLCXXRφ−=PHY2054: Chapter 214QuizÎThree identical EMF sources are hooked to a single circuit element, a resistor, a capacitor, or an inductor. The current amplitude is then measured as a function of frequency. Which one of the following curves corresponds to an inductive circuit? (1) a (2) b (3) c (4) Can’t tell without more infofImaxacbmax max2/LLXfLIXπε==For inductor, higher frequency gives higherreactance, therefore lower currentPHY2054: Chapter 215RLC Example 1ÎBelow are shown the driving emf and current vs time of an RLC circuit. We can conclude the following Current “leads” the driving emf (φ<0) Circuit is capacitive (XC> XL)εItPHY2054: Chapter 216RLC Example 2ÎR = 200Ω, C = 15μF, L = 230mH, εmax= 36v, f = 60 Hz Resonant frequency ()()601/ 2 0.230 15 10 85.6Hzfπ−=×=2 60 0.23 86.7LXπ=×× = Ω()61/ 2 60 15 10 177CXπ−=×××=Ω()22200 86.7 177 219Z =+−=Ωmax max/ 36 / 219 0.164AIZε===XC> XLCapacitive circuit186.7 177tan 24.3200φ−−⎛⎞==−°⎜⎟⎝⎠Current leads emf(as expected)PHY2054: Chapter 217Imaxvs Frequency and ResonanceÎCircuit parameters: C = 2.5μF, L = 4mH, εmax= 10v f0= 1 / 2π(LC)1/2= 1590 Hz Plot Imaxvs fR = 5ΩR = 10ΩR = 20ΩImaxResonance0ff=f / f0()22max10 / 2 1/ 2IRfLfCππ=+−PHY2054: Chapter 218Power in AC CircuitsÎRecall power formulaÎRewrite using Îcosφ is the “power factor” To maximize power delivered to circuit ⇒ make φ close to zero Max power delivered to load happens at resonance E.g., too much inductive reactance (XL) can be cancelled by increasing XC(e.g., circuits with large motors)2ave rmsPIR=rmsave rms rms rmscosZPIRIεεφ==ave rms rmscosPIεφ=cosRZφ=rmsrmsIZε=rms max/2II=PHY2054: Chapter 219Power Example 1ÎR = 200Ω, XC= 150Ω, XL= 80Ω, εrms= 120v, f = 60 Hz()22200 80 150 211.9Z =+−=Ω180 150tan 19.3200φ−−⎛⎞==−°⎜⎟⎝⎠cos 0.944φ=ave rms rmscos 120 0.566 0.944 64.1WPIεφ==××=rms rms/ 120 / 211.9 0.566AIZε== =22ave rms0.566 200 64.1WPIR== ×=Current leads emfCapacitive circuitSamePHY2054: Chapter 2110Power Example 1 (cont)ÎR = 200Ω, XC= 150Ω, XL= 80Ω, εrms= 120v, f = 60 HzÎHow much capacitance must be added to maximize the power in the circuit (and thus bring it into resonance)? Want XC= XLto minimize Z, so must decrease XCSo we must add 15.5μF capacitance to maximize power150 1/ 2 17.7μFCXfCCπ=Ω= =new new80 33.2μFCLXX C==Ω =PHY2054: Chapter 2111Power vs Frequency and ResonanceÎCircuit parameters: C = 2.5μF, L = 4mH, εmax= 10v f0= 1 / 2π(LC)1/2= 1590 Hz Plot Pavevs f for different R valuesR = 5ΩR = 10ΩR = 20Ω0ff=PaveR = 2ΩResonancef / f0PHY2054: Chapter 2112QuizÎA generator produces current at a frequency of 60 Hz with peak voltage and current amplitudes of 100V and 10A, respectively. What is the average power produced if they are in phase? (1) 1000 W (2) 707 W (3) 1414 W (4) 500 W (5) 250 W1ave peak peak rms rms2PIIεε==PHY2054: Chapter 2113QuizÎThe figure shows the current and emf of a series RLC circuit. To increase the rate at which power is delivered to the resistive load, which option should be taken? (1) Increase R (2) Decrease L (3) Increase L (4) Increase CCurrent lags applied emf (φ > 0), thus circuit is inductive. Either(1) Reduce XLby decreasing L or(2) Cancel XLby increasing XC(decrease C).tanLCXXRφ−=PHY2054: Chapter 2114Example: LR CircuitÎVariable frequency EMF source with εm=6V connected to a resistor and inductor. R=80Ω and L=40mH. At what frequency f does VR= VL? At that frequency, what is phase angle φ? What is the current amplitude? What is the rms current?2 318HzLXfLRfπ==⇒=tan / 1 45LXRφφ==⇒=°22max max/ 80 80 6 /113 0.053AIε=+==rms max/ 2 0.037 AII==PHY2054: Chapter 2115TransformersÎPurpose: change alternating (AC) voltage to a bigger (or smaller) valueppBVNtΔΦ=ΔBssVNtΔΦ=ΔInput AC voltagein the “primary”turns produces a fluxssppNVVN=Changing flux in“secondary” turnsinduces an emfPHY2054: Chapter 2116TransformersÎNothing comes for free, however! Increase in voltage comes at the cost of current. Output power cannot exceed input power! power in = power out (Losses usually account for 10-20%)ppssiV iV=ppsps sVNiiVN==PHY2054: Chapter 2117Transformers: Sample Problem ÎA transformer has 330 primary turns and 1240 secondary turns. The input voltage is 120 V and the output current is 15.0 A. What is the output voltage and input current?1240120 451V330ssppNVVN⎛⎞== =⎜⎟⎝⎠“Step-up”transformerpp ssiV iV=45115 56.4 A120spspViiV⎛⎞== =⎜⎟⎝⎠PHY2054: Chapter 2118¾ This is how first experiment by Faraday was done¾ He only got a deflection of the galvanometer when the switch is opened or closed¾ Steady current does notmake induced emf.TransformersPHY2054: Chapter 2119MicrophoneTape recorderApplicationsPHY2054: Chapter 2120ConcepTest: Power lines ÎAt large distances, the resistance of power lines becomes significant. To transmit maximum power, is it better to transmit (high V, low i) or (high i, low V)? (1) high V, low i (2) low V, high i (3) makes no differencePower loss is i2RPHY2054: Chapter 2121Electric Power Transmissioni2R: 20x smaller current ⇒ 400x smaller power
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