PowerPoint PresentationSlide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12PHASORS You can solve AC circuit analysis problems that involve Circuits with linear elements (R, C, L) plus independent and dependent voltage and/or current sources operating at a single angular frequency w = 2pf (radians/s) such as v(t) = V0cos(wt) or i(t) = I0cos(wt). By using any of Ohm’s Law, KVL and KCL equations, doing superposition analysis, nodal analysis or mesh analysis, AND Using instead of the terms below on the left (general excitation), the terms below on the right (sinusoidal excitation):Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Apply this approach to the capacitor circuit above, where the voltage source has the value vS(t) = 4 cos(wt) volts. The phasor voltage VS is then purely real: VS = 4. The phasor current is I = VS/ZC = jwCVS = (wC)VSejp/2, where we use the fact that j = (-1)1/2 = ejp/2; thus, the current in a capacitor leads the capacitor voltage by p/2 radians (90o). Note: Often (especially in this class) we may not care about the phase angle, and will focus just on the amplitude of the voltage or current that we obtain. This will be particularly true of filters and amplifiers.Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Week 5a1EE42: Running Checklist of Electronics Terms 14.02.05 – Dick White Terms are listed roughly in order of their introduction. Most definitions can be found in your text. TERM Charge, current, voltage, resistance , conductance, energy, power Coulomb, ampere, volt, ohm, siemen (mho), joule, watt Reference directions Kirchhoff’s Current Law (KCL), Kirchhoff’s Voltage Law (KVL), Ohm’s Law Series connection, parallel connection DC (steady), AC (time-varying) Independent and dependent ideal voltage and current source Voltage divider, current divider Analog (A/D), Digital (D/A) Multimeter (DMM), Oscilloscope Prefixes (milli-, etc.) Linear, nonlinear elements Superposition (analysis) Nodal analysis (node, supernode) Loop analysis (mesh, branch) Power delivery, dissipation, storage, maximum power transfer Equivalent circuits (Rs, Cs or Ls in series/parallel; Thevenin, Norton) Frequency; angular frequency; period; phase (Hz; radian/s) Capacitor, inductor, transformer Phasor, impedance, reactance Amplifier, filter, transfer function Steady-state, transient, sinusoidal excitation Terms2Week 5a2Lecture 5a Review: Types of Circuit ExcitationWhy Sinusoidal Excitation?PhasorsWeek 5a3Types of Circuit ExcitationLinear Time- Invariant CircuitSteady-State ExcitationLinear Time- Invariant CircuitORLinear Time- Invariant CircuitDigitalPulseSourceTransient ExcitationLinear Time- Invariant CircuitSinusoidal (Single-Frequency) ExcitationWeek 5a4Why is Sinusoidal Single-Frequency Excitation Important?1. Some circuits are driven by a single-frequency sinusoidal source. Example: The electric power system at frequency of60+/-0.1 Hz in U. S. Voltage is a sinusoidal function of time because it is produced by huge rotating generators powered by mechanical energy source such as steam (produced by heat from natural gas, fuel oil, coal or nuclear fission) or by falling water from a dam (hydroelectric).Week 5a5Bonneville Dam (Columbia River) Where Much of California’s Electric Power Comes FromWeek 5a6Turbine-generator sets at Bonneville DamWeek 5a7VoltageTimeA B CTime for which rotor position is shownCoil APlane of Coil CRotation of RotorPlane ofCoil APlane of Coil BWhere 3-Phase Electricity Comes FromGeneratordriven byfallingwater has3 separatecoilsOutput voltagesfrom the 3 coils(they leave thegenerating planton 3 separatecables)Direct currentin the rotor(rotating coil)produces amagnetic fieldthat generatescurrents in stationary coilsA, B and CWeek 5a82. Some circuits are driven by sinusoidal sources whose frequency changes slowly over time.Example: Music reproduction system (different notes).Why Sinusoidal Excitation? (continued)3. And, you can express any periodic electrical signal as asum of single-frequency sinusoids – so you cananalyze the response of the (linear, time-invariant) circuit to each individual frequency component and then sum the responses to get the total response.Week 5a9abc dTime (ms)Frequency (Hz)Signal (V)Relative AmplitudeSignal (V)Signal (V) Representing a Square Wave as a Sum of Sinusoids(a) Square wave with 1-second period. (b) Fundamental compo-nent (dotted) with 1-second period, third-harmonic (solid black)with1/3-second period, and their sum (blue). (c) Sum of first tencomponents. (d) Spectrum with 20 terms.Week 5a10Single-frequency sinusoidal-excitation AC circuit problems1. The technique we’ll show works on circuits composed of linear elements (R, C, L) that don’t change with time “linear time-invariant circuits”. 2. The circuit is driven with independent voltage and/or current sources whose voltages or currents vary at a single frequency, f, measured in Hertz (abbreviated Hz) this is the number of cycles the voltages or currents execute per second. We can represent thesource voltages or currents as functions of time as v(t) = V0cos(t) or i(t) = I0cos(t),where f is the angular frequency in radians per second. Example: In the U. S. the AC power frequency, f, is 60 Hz and the peak voltage V0 is 170 V, so = 377 radians/s and v(t) = 170cos(377t) V. More generally, we might have sources v(t) = V0sin(t) or i(t) = I0cos(t = ), where is a phase angle.Week 5a11We could solve our circuit equations using such functions of time, but we’d have to do a lot of tedious trigonometric transformations. Instead we use a mathematical trick to eliminate time dependence from our equations!The trick is based on a fundamental fact about linear, time-invariant circuits excited with sinusoidal sources: the frequencies of all the voltages and currents in the circuit are identical.Week 5a12 SAMERULE: “Sinusoid in”-- “Same-frequency sinusoid out” is true for linear time-invariant circuits. (The term “sinusoid” is intended to include both sine and cosine functions of time.)Intuition: Think of sinusoidal excitation (vibration) of a linear mechanical system – every part vibrates at the same frequency, even though perhaps at different phases.SAME Circuit of linear elements (R, L, C)Excitation: Output: vS(t) = VScos(t + )Iout(t) = I0cost(t + )GivenGivenGiven? ?Week 5a13
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