engineering circuit analysis-ch4
DESCRIPTION
Circuit Analysis techniquesTRANSCRIPT
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Ch4 Sinusoidal Steady State Analysis4.1 Characteristics of Sinusoidal 4.2 Phasors4.3 Phasor Relationships for R, L and C4.4 Impedance 4.5 Parallel and Series Resonance4.6 Examples for Sinusoidal Circuits Analysis
References: Hayt-Ch7; Gao-Ch3; Engineering Circuit Analysis
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Ch4 Sinusoidal Steady State AnalysisAny steady state voltage or current in a linear circuit with a sinusoidal source is a sinusoidAll steady state voltages and currents have the same frequency as the sourceIn order to find a steady state voltage or current, all we need to know is its magnitude and its phase relative to the source (we already know its frequency)We do not have to find this differential equation from the circuit, nor do we have to solve itInstead, we use the concepts of phasors and complex impedancesPhasors and complex impedances convert problems involving differential equations into circuit analysis problems Focus on steady state; Focus on sinusoids.
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4.1 Characteristics of Sinusoidal Key Words: Period: T , Frequency: f , Radian frequency Phase angle Amplitude: Vm ImCh4 Sinusoidal Steady State Analysis
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4.1 Characteristics of Sinusoidal Both the polarity and magnitude of voltage are changing.Ch4 Sinusoidal Steady State Analysis
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4.1 Characteristics of Sinusoidal Radian frequency(Angular frequency): = 2f = 2/T (rad/s Period: T Time necessary to go through one cycle. (s)Frequency: f Cycles per second. (Hz)f = 1/TAmplitude: Vm Im i = Imsint v =Vmsint Ch4 Sinusoidal Steady State Analysis
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4.1 Characteristics of Sinusoidal Effective Roof Mean Square (RMS) Value of a Periodic Waveform is equal to the value of the direct current which is flowing through an R-ohm resistor. It delivers the same average power to the resistor as the periodic current does.Ch4 Sinusoidal Steady State Analysis
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4.1 Characteristics of Sinusoidal Phase (angle)Ch4 Sinusoidal Steady State Analysis
Chart3
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6.9951425490.26074040623.2717769704
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4.1 Characteristics of Sinusoidal Phase differenceCh4 Sinusoidal Steady State Analysis
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4.1 Characteristics of Sinusoidal Review The sinusoidal waves whose phases are compared must: Be written as sine waves or cosine waves. With positive amplitudes. Have the same frequency.Ch4 Sinusoidal Steady State Analysis
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4.1 Characteristics of Sinusoidal Phase differenceP4.1,Ch4 Sinusoidal Steady State AnalysisIf
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4.1 Characteristics of Sinusoidal Phase differenceP4.2,Ch4 Sinusoidal Steady State Analysis
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4.2 PhasorsKey Words: Complex Numbers Rotating Vector PhasorsA sinusoidal voltage/current at a given frequency , is characterized by only two parameters :amplitude an phaseCh4 Sinusoidal Steady State Analysis
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4.2 PhasorsE.g. voltage responseCh4 Sinusoidal Steady State Analysis
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4.2 PhasorsRotating VectorImCh4 Sinusoidal Steady State Analysisi(t1)Imag
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4.2 PhasorsRotating VectorCh4 Sinusoidal Steady State Analysis
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4.2 PhasorsComplex NumbersCh4 Sinusoidal Steady State Analysis
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4.2 PhasorsComplex NumbersArithmetic With Complex NumbersAddition: A = a + jb, B = c + jd, A + B = (a + c) + j(b + d)Ch4 Sinusoidal Steady State Analysis
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4.2 PhasorsComplex NumbersArithmetic With Complex NumbersSubtraction : A = a + jb, B = c + jd, A - B = (a - c) + j(b - d)Ch4 Sinusoidal Steady State Analysis
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4.2 PhasorsComplex NumbersArithmetic With Complex NumbersMultiplication : A = Am A, B = Bm BA B = (Am Bm) (A + B)Division: A = Am A , B = Bm B A / B = (Am / Bm) (A - B)Ch4 Sinusoidal Steady State Analysis
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4.2 PhasorsPhasors A phasor is a complex number that represents the magnitude and phase of a sinusoid:Phasor DiagramsA phasor diagram is just a graph of several phasors on the complex plane (using real and imaginary axes).A phasor diagram helps to visualize the relationships between currents and voltages.Ch4 Sinusoidal Steady State Analysis
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4.2 PhasorsComplex Exponentials A real-valued sinusoid is the real part of a complex exponential. Complex exponentials make solving for AC steady state an algebraic problem.Ch4 Sinusoidal Steady State Analysis
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4.3 Phasor Relationships for R, L and CKey Words: I-V Relationship for R, L and C, Power conversionCh4 Sinusoidal Steady State Analysis
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4.3 Phasor Relationships for R, L and C v~i relationship for a resistorWave and Phasor diagrams ResistorSupposeCh4 Sinusoidal Steady State Analysis
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EMBED Equation.3
EMBED Equation.3
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_1202715897.unknown
_1202715928.unknown
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4.3 Phasor Relationships for R, L and CWith a resistor , v(t) and i(t) are in phase .Ch4 Sinusoidal Steady State Analysis
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4.3 Phasor Relationships for R, L and C PowerResistorp0 Transient PowerCh4 Sinusoidal Steady State AnalysisNote: I and V are RMS values.
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EMBED Equation.3
EMBED Equation.3
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_1202715928.unknown
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4.3 Phasor Relationships for R, L and CResistorCh4 Sinusoidal Steady State Analysis
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4.3 Phasor Relationships for R, L and C v~i relationshipInductorCh4 Sinusoidal Steady State Analysis
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4.3 Phasor Relationships for R, L and C v~i relationshipInductorCh4 Sinusoidal Steady State Analysis
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4.3 Phasor Relationships for R, L and C v ~ i relationshipInductor The derivative in the relationship between v(t) and i(t) becomes a multiplication by j in the relationship between and . The time-domain differential equation has become the algebraic equation in the frequency-domain. Phasors allow us to express current-voltage relationships for inductors and capacitors in a way such as we express the current-voltage relationship for a resistor.
Ch4 Sinusoidal Steady State Analysis
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4.3 Phasor Relationships for R, L and C v ~ i relationshipInductorWave and Phasor diagrams Ch4 Sinusoidal Steady State Analysis
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4.3 Phasor Relationships for R, L and C PowerInductorCh4 Sinusoidal Steady State Analysis
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4.3 Phasor Relationships for R, L and CInductorP4.5L = 10mHv = 100sintFind iL when f = 50Hz and 50kHz.Ch4 Sinusoidal Steady State Analysis
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4.3 Phasor Relationships for R, L and C v ~ i relationshipCapacitorCh4 Sinusoidal Steady State Analysis
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EMBED Equation.3
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v(t) = Vm ejtRepresent v(t) and i(t) as phasors: The derivative in the relationship between v(t) and i(t) becomes a multiplication by j in the relationship between and . The time-domain differential equation has become the algebraic equation in the frequency-domain. Phasors allow us to express current-voltage relationships for inductors and capacitors much like we express the current-voltage relationship for a resistor. 4.3 Phasor Relationships for R, L and C v ~ i relationshipCapacitorCh4 Sinusoidal Steady State Analysis
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EMBED Equation.3
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4.3 Phasor Relationships for R, L and C v ~ i relationshipCapacitorWave and Phasor diagrams Ch4 Sinusoidal Steady State Analysis
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4.3 Phasor Relationships for R, L and CPowerCapacitorAverage Power: P=0Ch4 Sinusoidal Steady State Analysis
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4.3 Phasor Relationships for R, L and CCapacitorP4.7Suppose C=20FAC source v=100sintFind XC and I for f = 50Hz, 50kHzCh4 Sinusoidal Steady State Analysis
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4.3 Phasor Relationships for R, L and CReview (v-I relationship)Ch4 Sinusoidal Steady State Analysis
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4.3 Phasor Relationships for R, L and CSummary Frequency characteristics of an Ideal Inductor and Capacitor: A capacitor is an open circuit to DC currents; A Inducter is a short circuit to DC currents.Ch4 Sinusoidal Steady State Analysis
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4.4 ImpedanceKey Words: complex currents and voltages. Impedance Phasor DiagramsCh4 Sinusoidal Steady State Analysis
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AC steady-state analysis using phasors allows us to express the relationship between current and voltage using a formula that looks likes Ohms law:4.4 ImpedanceComplex voltage Complex current Complex ImpedanceCh4 Sinusoidal Steady State Analysis
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4.4 ImpedanceComplex ImpedanceComplex impedance describes the relationship between the voltage across an element (expressed as a phasor) and the current through the element (expressed as a phasor)Impedance is a complex number and is not a phasor (why?).Impedance depends on frequencyCh4 Sinusoidal Steady State Analysis
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4.4 ImpedanceComplex ImpedanceCh4 Sinusoidal Steady State Analysis
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4.4 ImpedanceComplex ImpedanceImpedance in series/parallel can be combined as resistors.Ch4 Sinusoidal Steady State Analysis
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4.4 ImpedanceComplex ImpedanceP4.8,Ch4 Sinusoidal Steady State Analysis
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EMBED Equation.3
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4.4 ImpedanceComplex ImpedancePhasors and complex impedance allow us to use Ohms law with complex numbers to compute current from voltage and voltage from currentHow do we find VC?First compute impedances for resistor and capacitor:ZR = 20kW = 20kW 0 ZC = 1/j (377 *1mF) = 2.65kW -90 Ch4 Sinusoidal Steady State Analysis
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4.4 ImpedanceComplex ImpedanceNow use the voltage divider to find VC:Ch4 Sinusoidal Steady State Analysis
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4.4 Impedance Impedance allows us to use the same solution techniques for AC steady state as we use for DC steady state.All the analysis techniques we have learned for the linear circuits are applicable to compute phasorsKCL & KVLnode analysis / loop analysissuperpositionThevenin equivalents / Norton equivalentssource exchangeThe only difference is that now complex numbers are used.
Complex ImpedanceCh4 Sinusoidal Steady State Analysis
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4.4 ImpedanceKirchhoffs LawsKCL and KVL hold as well in phasor domain.Ch4 Sinusoidal Steady State Analysis
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4.4 ImpedanceAdmittanceI = YV, Y is called admittance, the reciprocal of impedance, measured in siemens (S)Resistor:The admittance is 1/RInductor:The admittance is 1/jLCapacitor:The admittance is j C
Ch4 Sinusoidal Steady State Analysis
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4.4 ImpedancePhasor DiagramsA phasor diagram is just a graph of several phasors on the complex plane (using real and imaginary axes).A phasor diagram helps to visualize the relationships between currents and voltages.I = 2mA 40, VR = 2V 40 VC = 5.31V -50, V = 5.67V -29.37Ch4 Sinusoidal Steady State Analysis
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4.5 Parallel and Series ResonanceKey Words: RLC Circuit, Series Resonance Parallel ResonanceCh4 Sinusoidal Steady State Analysis
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4.5 Parallel and Series Resonance(2nd Order RLC Circuit )Series RLC CircuitCh4 Sinusoidal Steady State Analysis
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4.5 Parallel and Series Resonance(2nd Order RLC Circuit )Series RLC CircuitCh4 Sinusoidal Steady State Analysis
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4.5 Parallel and Series Resonance(2nd Order RLC Circuit )Series RLC CircuitCh4 Sinusoidal Steady State Analysis
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4.5 Parallel and Series ResonanceSeries RLC Circuit(2nd Order RLC Circuit )P4.9, R. L.C Series CircuitR = 30L = 127mHC = 40FSource . Find 1) XLXCZ2) and i; 3) and vR; and vL; and vC; 4) Phasor diagrams;Ch4 Sinusoidal Steady State Analysis
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4.5 Parallel and Series ResonanceSeries Resonance(2nd Order RLC Circuit )Ch4 Sinusoidal Steady State Analysis
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4.5 Parallel and Series ResonanceSeries Resonance(2nd Order RLC Circuit )Zminwhen V=constant, I=Imax=I0Ch4 Sinusoidal Steady State Analysis
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4.5 Parallel and Series ResonanceSeries Resonance(2nd Order RLC Circuit )Ch4 Sinusoidal Steady State Analysis
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4.5 Parallel and Series ResonanceSeries Resonance(2nd Order RLC Circuit )Ch4 Sinusoidal Steady State Analysis
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4.5 Parallel and Series ResonanceSeries Resonance(2nd Order RLC Circuit )Ch4 Sinusoidal Steady State Analysis
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4.5 Parallel and Series ResonanceSeries Resonance(2nd Order RLC Circuit )Ch4 Sinusoidal Steady State Analysis
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4.5 Parallel and Series ResonanceSeries Resonance(2nd Order RLC Circuit )Ch4 Sinusoidal Steady State Analysis
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4.5 Parallel and Series ResonanceSeries Resonance(2nd Order RLC Circuit )Ch4 Sinusoidal Steady State Analysis
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4.5 Parallel and Series ResonanceParallel RLC CircuitCh4 Sinusoidal Steady State Analysis
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4.5 Parallel and Series ResonanceParallel RLC CircuitCh4 Sinusoidal Steady State Analysis
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4.5 Parallel and Series ResonanceParallel RLC CircuitP4.10,Find i1 i2 iCh4 Sinusoidal Steady State Analysis
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4.5 Parallel and Series ResonanceParallel RLC CircuitReviewFor sinusoidal circuit Series Two Simple Methods: Phasor Diagrams and Complex Numbers Parallel :Ch4 Sinusoidal Steady State Analysis
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4.6 Examples for Sinusoidal Circuits AnalysisKey Words: Bypass Capacitor RC Phase Difference Low-Pass and High-Pass FilterCh4 Sinusoidal Steady State Analysis
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P4.11, Let4.6 Examples for Sinusoidal Circuits AnalysisBypass Capacitor f 500HzDetermine VAB before the C is connected . And VAB after parallel C = 30FCh4 Sinusoidal Steady State Analysis
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P4.12,4.6 Examples for Sinusoidal Circuits AnalysisRC Phase Differencef = 300Hz, R = 100 If vo - vi= /4C =Ch4 Sinusoidal Steady State Analysis
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EMBED Equation.3
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