engineering circuit analysis
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Engineering Books For Circuit Analysis AND electrical engineeringTRANSCRIPT
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Sinusoidal Steady State Analysis1 Characteristics of Sinusoidal 2 Phasors3 Phasor Relationships for R, L and C4 Impedance 5 Examples for Sinusoidal Circuits Analysis
References: Hayt-Ch7; Gao-Ch3; Engineering Circuit Analysis
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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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1 Characteristics of Sinusoidal Key Words: Period: T , Frequency: f , Radian frequency Phase angle Amplitude: Vm Im Sinusoidal Steady State Analysis
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1 Characteristics of Sinusoidal Both the polarity and magnitude of voltage are changing. Sinusoidal Steady State Analysis
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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 Sinusoidal Steady State Analysis
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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.Sinusoidal Steady State Analysis
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1 Characteristics of Sinusoidal Phase (angle)Sinusoidal Steady State Analysis
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1 Characteristics of Sinusoidal Phase difference Sinusoidal Steady State Analysis
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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.Sinusoidal Steady State Analysis
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2 PhasorsKey Words: Complex Numbers Rotating Vector PhasorsA sinusoidal voltage/current at a given frequency , is characterized by only two parameters :amplitude an phaseSinusoidal Steady State Analysis
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2 PhasorsE.g. voltage responseSinusoidal Steady State Analysis
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2 PhasorsRotating VectorIm Sinusoidal Steady State Analysisi(t1)Imag
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2 PhasorsRotating VectorSinusoidal Steady State Analysis
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2 PhasorsComplex Numbers Sinusoidal Steady State Analysis
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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. Sinusoidal Steady State Analysis
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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. Sinusoidal Steady State Analysis
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3 Phasor Relationships for R, L and CKey Words: I-V Relationship for R, L and C, Power conversionSinusoidal Steady State Analysis
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3 Phasor Relationships for R, L and C v~i relationship for a resistorWave and Phasor diagrams ResistorSupposeSinusoidal Steady State Analysis
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3 Phasor Relationships for R, L and CWith a resistor , v(t) and i(t) are in phase . Sinusoidal Steady State Analysis
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3 Phasor Relationships for R, L and CResistorSinusoidal Steady State Analysis
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3 Phasor Relationships for R, L and C v~i relationshipInductor Sinusoidal Steady State Analysis
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3 Phasor Relationships for R, L and C v~i relationshipInductorSinusoidal Steady State Analysis
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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.
Sinusoidal Steady State Analysis
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3. Phasor Relationships for R, L and C v ~ i relationshipInductorWave and Phasor diagrams Sinusoidal Steady State Analysis
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3 Phasor Relationships for R, L and CInductorP4.5L = 10mHv = 100sintFind iL when f = 50Hz and 50kHz. Sinusoidal Steady State Analysis
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3 Phasor Relationships for R, L and C v ~ i relationshipCapacitor Sinusoidal Steady State Analysis
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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. 3 Phasor Relationships for R, L and C v ~ i relationshipCapacitor Sinusoidal Steady State Analysis
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3 Phasor Relationships for R, L and C v ~ i relationshipCapacitorWave and Phasor diagrams Sinusoidal Steady State Analysis
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3 Phasor Relationships for R, L and CCapacitorP4.7Suppose C=20FAC source v=100sintFind XC and I for f = 50Hz, 50kHz Sinusoidal Steady State Analysis
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3 Phasor Relationships for R, L and CReview (v-I relationship)Sinusoidal Steady State Analysis
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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.Sinusoidal Steady State Analysis
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4 ImpedanceKey Words: complex currents and voltages. Impedance Phasor DiagramsSinusoidal 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 ImpedanceComplex voltage Complex current Complex ImpedanceSinusoidal Steady State Analysis
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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 frequencySinusoidal Steady State Analysis
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4 ImpedanceComplex ImpedanceSinusoidal Steady State Analysis
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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 Sinusoidal Steady State Analysis
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4 ImpedanceComplex ImpedanceNow use the voltage divider to find VC: Sinusoidal Steady State Analysis
*
*P4.2, vivi/3vivi /3 vi
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*P4.2, vivi/3vivi /3 vi j90*P4.2, vivi/3vivi /3 vi
*P4.2, vivi/3vivi /3 vi