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Ch4 Sinusoidal Steady State Analysis 4.1 Characteristics of Sinusoidal 4.2 Phasors 4.3 Phasor Relationships for R, L and C 4.4 Impedance 4.5 Parallel and Series Resonance 4.6 Examples for Sinusoidal Circuits Analysis 4.7 Magnetically Coupled Circuits References References: Hayt-Ch7; Gao-Ch3; Circuits and Analog Electronics

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Ch4 Sinusoidal Steady State Analysis Any steady state voltage or current in a linear circuit with a sinusoidal source is a sinusoid –All steady state voltages and currents have the same frequency as the source In 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 it Instead, we use the concepts of phasors and complex impedances Phasors 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 Key Words: Period: T, Frequency: f, Radian frequency Phase angle Amplitude: V m I m Ch4 Sinusoidal Steady State Analysis

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4.1 Characteristics of Sinusoidal v、iv、i t t1t1 t2t2 0 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): = 2 f = 2 /T (rad/s ） Period: T — Time necessary to go through one cycle. (s) Frequency: f — Cycles per second. (Hz) f = 1/T Amplitude: V m I m i = I m sin t ， v =V m sin t v、iv、i tt 22 0 Vm、ImVm、Im 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. Effective Value of a Periodic Waveform Ch4 Sinusoidal Steady State Analysis

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4.1 Characteristics of Sinusoidal Phase (angle) Phase angle <0 00 Ch4 Sinusoidal Steady State Analysis

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4.1 Characteristics of Sinusoidal Phase difference — v(t) leads i(t) by ( 1 - 2 ), or i(t) lags v(t) by ( 1 - 2 ) v、iv、i tt v i Out of phase 。 tt v、iv、i v i v、iv、i tt v i In phase. — v(t) lags i(t) by ( 2 - 1 ), or i(t) leads v(t) by ( 2 - 1 ) Ch4 Sinusoidal Steady State Analysis

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4.1 Characteristics of Sinusoidal Review The sinusoidal means whose phases are compared must: ① Be written as sine waves or cosine waves. ② With positive amplitudes. ③ Have the same frequency. 360°—— does not change anything. 90° —— change between sin & cos. 180°—— change between + & - Ch4 Sinusoidal Steady State Analysis

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4.1 Characteristics of Sinusoidal Phase difference P4.1, Find Ch4 Sinusoidal Steady State Analysis

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4.1 Characteristics of Sinusoidal Phase difference P4.2, v、iv、i tt v i - /3 /3 Ch4 Sinusoidal Steady State Analysis

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4.2 Phasors Key Words Key Words: Complex Numbers Rotating Vector Phasors A sinusoidal voltage/current at a given frequency, is characterized by only two parameters :amplitude an phase Ch4 Sinusoidal Steady State Analysis

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4.2 Phasors E.g. voltage response A sinusoidal v/i Complex transform Phasor transform By knowing angular frequency ω rads/s. Time domain Frequency domain Complex form: Phasor form: Angular frequency ω is known in the circuit. Ch4 Sinusoidal Steady State Analysis

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4.2 Phasors Rotating Vector i ImIm i (t1) t1t1 i tt ImIm tt x y A complex coordinates number: Real value: Ch4 Sinusoidal Steady State Analysis

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4.2 Phasors Rotating Vector VmVm x y 0 Ch4 Sinusoidal Steady State Analysis

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4.2 Phasors Complex Numbers — Rectangular Coordinates — Polar Coordinates conversion ： |A| a b real axis imaginary axis Ch4 Sinusoidal Steady State Analysis

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4.2 Phasors Complex Numbers Arithmetic With Complex Numbers Addition: A = a + jb, B = c + jd, A + B = (a + c) + j(b + d) Real Axis Imaginary Axis AB A + B Ch4 Sinusoidal Steady State Analysis

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4.2 Phasors Complex Numbers Arithmetic With Complex Numbers Subtraction : A = a + jb, B = c + jd, A - B = (a - c) + j(b - d) Real Axis Imaginary Axis AB A - B Ch4 Sinusoidal Steady State Analysis

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4.2 Phasors Complex Numbers Arithmetic With Complex Numbers Multiplication : A = A m B = B m A B = (A m B m ) ( ) Division: A = A m , B = B m A / B = (A m / B m ) ( ) Find ： P4.3, Ch4 Sinusoidal Steady State Analysis

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4.2 Phasors Phasors A phasor is a complex number that represents the magnitude and phase of a sinusoid: Phasor Diagrams A 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 Phasors Complex 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 C Key Words Key Words: I-V Relationship for R, L and C, Power conversion Ch4 Sinusoidal Steady State Analysis

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4.3 Phasor Relationships for R, L and C v~i relationship for a resistor Relationship between RMS : Wave and Phasor diagrams ： v、iv、i tt v i Resistor Suppose Ch4 Sinusoidal Steady State Analysis

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4.3 Phasor Relationships for R, L and C Time domain frequency domain Resistor With a resistor θ ﹦ φ, 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 Power Resistor p0p0 v、iv、i tt v i P=IV Average Power Transient Power Ch4 Sinusoidal Steady State Analysis

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4.3 Phasor Relationships for R, L and C Resistor P4.4 ，, R=10 ， Find i and P 。 Ch4 Sinusoidal Steady State Analysis

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4.3 Phasor Relationships for R, L and C v~i relationship Inductor Suppose Ch4 Sinusoidal Steady State Analysis

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4.3 Phasor Relationships for R, L and C v~i relationship Inductor Relationship between RMS: For DC ， f = 0 ， X L = 0. v(t) leads i(t) by 90º, or i(t) lags v(t) by 90º Ch4 Sinusoidal Steady State Analysis

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4.3 Phasor Relationships for R, L and C v ~ i relationship Inductor i(t) = I m e j t Represent 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 V and I. 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 relationship Inductor v、iv、i tt v i eLeL Wave and Phasor diagrams ： Ch4 Sinusoidal Steady State Analysis

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4.3 Phasor Relationships for R, L and C Power Inductor P tt v、iv、i tt v i ++ -- Energy stored: Average Power Reactive Power （ Var ） Ch4 Sinusoidal Steady State Analysis

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4.3 Phasor Relationships for R, L and C Inductor P4.5 ， L = 10mH ， v = 100sin t ， Find i L 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 relationship Capacitor Suppose: i(t) leads v(t) by 90º, or v(t) lags i(t) by 90º Relationship between RMS: For DC ， f = 0 ， X C Ch4 Sinusoidal Steady State Analysis

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v(t) = V m e j t Represent 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 V and I. 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 relationship Capacitor Ch4 Sinusoidal Steady State Analysis

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4.3 Phasor Relationships for R, L and C v ~ i relationship Capacitor v、iv、i tt v i Wave and Phasor diagrams ： Ch4 Sinusoidal Steady State Analysis

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4.3 Phasor Relationships for R, L and C Power Capacitor Average Power: P=0 Reactive Power （ Var ） P tt v、iv、i tt v i ++ -- Energy stored: Ch4 Sinusoidal Steady State Analysis

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4.3 Phasor Relationships for R, L and C Capacitor P4.7 ， Suppose C=20 F ， AC source v=100sin t ， Find X C and I for f = 50Hz, 50kHz 。 Ch4 Sinusoidal Steady State Analysis

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4.3 Phasor Relationships for R, L and C Review (v-I relationship) Time domainFrequency domain,,, v and i are in phase., v leads i by 90°., v lags i by 90°. R C L Ch4 Sinusoidal Steady State Analysis

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4.3 Phasor Relationships for R, L and C Summary R：R： L：L： C：C： 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 Impedance Key Words Key Words: complex currents and voltages. Impedance Phasor Diagrams Ch4 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 Ohm’s law: 4.4 Impedance Complex voltage ， Complex current ， Complex Impedance Z is called impedance. measured in ohms ( ) Ch4 Sinusoidal Steady State Analysis

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4.4 Impedance Complex Impedance Complex 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 frequency Ch4 Sinusoidal Steady State Analysis

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4.4 Impedance Complex Impedance Z R = R = 0 （ ＝ 0); or Z R = R 0 Resistor——The impedance is R =- /2 or Capacitor——The impedance is 1/j C = /2 or Inductor——The impedance is j L Ch4 Sinusoidal Steady State Analysis

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4.4 Impedance Complex Impedance Impedance in series/parallel can be combined as resistors. Current divider: Voltage divider: Ch4 Sinusoidal Steady State Analysis

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4.4 Impedance Complex Impedance P4.8, Ch4 Sinusoidal Steady State Analysis

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4.4 Impedance Complex Impedance Phasors and complex impedance allow us to use Ohm’s law with complex numbers to compute current from voltage and voltage from current 20k + - 1F1F10V 0 VCVC + - = 377 Find V C P4.9 How do we find V C ? First compute impedances for resistor and capacitor: Z R = 20k = 20k 0 Z C = 1/j (377 *1 F) = 2.65k -90 Ch4 Sinusoidal Steady State Analysis

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4.4 Impedance Complex Impedance 20k + - 1F1F10V 0 VCVC + - = 377 Find V C P4.9 20k 0 + - 2.65k -90 10V 0 VCVC + - Now use the voltage divider to find V C : 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 phasors –KCL & KVL –node analysis / loop analysis –superposition –Thevenin equivalents / Norton equivalents –source exchange The only difference is that now complex numbers are used. Complex Impedance Ch4 Sinusoidal Steady State Analysis

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4.4 Impedance Kirchhoff’s Laws KCL and KVL hold as well in phasor domain. KVL ： v k - Transient voltage of the # k branch KCL: i k - Transient current of the # k branch Ch4 Sinusoidal Steady State Analysis

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4.4 Impedance Admittance I = YV, Y is called admittance, the reciprocal of impedance, measured in siemens (S) Resistor: –The admittance is 1/R Inductor: –The admittance is 1/j L Capacitor: –The admittance is j C Ch4 Sinusoidal Steady State Analysis

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4.4 Impedance Phasor Diagrams A 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. 2mA 40 – 1F1F VCVC + – 1k VRVR + + – V I = 2mA 40 , V R = 2V 40 V C = 5.31V -50 , V = 5.67V -29.37 Real Axis Imaginary Axis VRVR VCVC V Ch4 Sinusoidal Steady State Analysis

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4.5 Parallel and Series Resonance Key Words Key Words: RLC Circuit, Series Resonance Parallel Resonance Ch4 Sinusoidal Steady State Analysis

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4.5 Parallel and Series Resonance v vRvR vLvL vCvC Phasor (2nd Order RLC Circuit ) Series RLC Circuit Ch4 Sinusoidal Steady State Analysis

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4.5 Parallel and Series Resonance (2nd Order RLC Circuit ) Z X = X L -X C R Phase difference: X L >X C >0 ， v leads i by ——Inductance Circuit X L <X C <0 ， v lags i by ——Capacitance Circuit X L =X C =0 ， v and i in phase——Resistors Circuit Series RLC Circuit Ch4 Sinusoidal Steady State Analysis

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4.5 Parallel and Series Resonance (2nd Order RLC Circuit ) v vRvR vLvL vCvC Series RLC Circuit Ch4 Sinusoidal Steady State Analysis

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4.5 Parallel and Series Resonance Series RLC Circuit (2nd Order RLC Circuit ) P4.9, R. L. C Series Circuit ， R = 30 ， L = 127mH ， C = 40 F ， Source, Find 1) X L 、 X C 、 Z ； 2) I and i ； 3) V R and v R ； V L and v L ； V C and v C ； 4) Phasor Diagrams P4.10 ， Computing by (complex numbers) Phasors v vRvR vLvL vCvC Ch4 Sinusoidal Steady State Analysis

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4.5 Parallel and Series Resonance Series Resonance (2nd Order RLC Circuit ) and —— Series Resonance Resonance condition f0f0 f X Resonance frequency Ch4 Sinusoidal Steady State Analysis

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4.5 Parallel and Series Resonance Series Resonance (2nd Order RLC Circuit ) Z min ； when V=constant, I=I max =I 0 。 When, Quality factor Q, Resonance condition: Ch4 Sinusoidal Steady State Analysis

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4.5 Parallel and Series Resonance Series Resonance (2nd Order RLC Circuit ) Ch4 Sinusoidal Steady State Analysis

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4.5 Parallel and Series Resonance Series Resonance (2nd Order RLC Circuit ) Ch4 Sinusoidal Steady State Analysis

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4.5 Parallel and Series Resonance Series Resonance (2nd Order RLC Circuit ) Ch4 Sinusoidal Steady State Analysis

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4.5 Parallel and Series Resonance Series Resonance (2nd Order RLC Circuit ) Ch4 Sinusoidal Steady State Analysis

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4.5 Parallel and Series Resonance Series Resonance (2nd Order RLC Circuit ) Ch4 Sinusoidal Steady State Analysis

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4.5 Parallel and Series Resonance Series Resonance (2nd Order RLC Circuit ) Ch4 Sinusoidal Steady State Analysis

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4.5 Parallel and Series Resonance Parallel RLC Circuit Parallel Resonance Parallel Resonance frequency In generally When, In phase with Z max I min : Ch4 Sinusoidal Steady State Analysis

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4.5 Parallel and Series Resonance Parallel RLC Circuit Z .Z . Quality factor Q, Ch4 Sinusoidal Steady State Analysis

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4.5 Parallel and Series Resonance Parallel RLC Circuit P4.10, v i i i 1 i2 i2 Find i 1 、 i 2 、 i Ch4 Sinusoidal Steady State Analysis

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4.5 Parallel and Series Resonance Parallel RLC Circuit Review For 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 Analysis Key Words Key Words: Bypass Capacitor RC Phase Difference Low-Pass and High-Pass Filter Ch4 Sinusoidal Steady State Analysis

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P4.11, Let 4.6 Examples for Sinusoidal Circuits Analysis Bypass Capacitor f ＝ 500Hz ， Determine V AB before the C is connected. And V AB after parallel C = 30 F v ii Before C is connected After C is connected Ch4 Sinusoidal Steady State Analysis

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P4.12, 4.6 Examples for Sinusoidal Circuits Analysis RC Phase Difference f = 300Hz, R = 100 。 If vo - vi = /4 ， C = ？ Ch4 Sinusoidal Steady State Analysis

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P4.13, The voltage sources are v i =240+100sin2 100t(V), R ＝ 200 ， C ＝ 50 F ， Determine V AC and V DC in output voltage v o. 4.6 Examples for Sinusoidal Circuits Analysis Low-Pass and High-Pass Filter V DC = 240V R C---- High-Pass Filter Ch4 Sinusoidal Steady State Analysis

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4.6 Examples for Sinusoidal Circuits Analysis Low-Pass and High-Pass Filter Ch4 Sinusoidal Steady State Analysis

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4.6 Examples for Sinusoidal Circuits Analysis Low-Pass and High-Pass Filter Ch4 Sinusoidal Steady State Analysis

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4.6 Examples for Sinusoidal Circuits Analysis Ch4 Sinusoidal Steady State Analysis

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4.6 Examples for Sinusoidal Circuits Analysis Low-Pass and High-Pass Filter Ch4 Sinusoidal Steady State Analysis

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4.6 Examples for Sinusoidal Circuits Analysis Ch4 Sinusoidal Steady State Analysis

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4.6 Examples for Sinusoidal Circuits Analysis Ch4 Sinusoidal Steady State Analysis

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4.6 Examples for Sinusoidal Circuits Analysis v 1 =120sin t v2v2 i3i3 i1 i1 i2 i2 P4.14, Find in the circuit of the fo ll owing fig. Complex Numbers Analysis Ch4 Sinusoidal Steady State Analysis

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4.6 Examples for Sinusoidal Circuits Analysis P4.15, Let V m ＝ 100V. Use Thevenin’s theorem to find v v Complex Numbers Analysis Ch4 Sinusoidal Steady State Analysis

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4.7 Magnetically Coupled Circuits Key Words Key Words: Self- inductance and Mutual inductance Magnetically Coupled Circuits and v ~ i relationship Dot convention Ideal transformer Ch4 Sinusoidal Steady State Analysis

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4.7 Magnetically Coupled Circuits Coupled Circuits and v~i relationship Magnetic flux: 1 ＝ f(i 1 ) ( 1 = N 1 1 ) The flux is proportional to the current in linear inductor: 1(t) ＝ L 1 i 1(t) L is a lumped element abstraction for the coil. i1i1 v1v1 v1v1 + - i1i1 Voltage be proportional to the time rate of change of the magnetic field. Ch4 Sinusoidal Steady State Analysis

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4.7 Magnetically Coupled Circuits Coupled Circuits and v~i relationship v1v1 + - i1i1 v2v2 + - i2i2 ——Ideal Coupled Circuits ’ v ~ i relationship L 1 、 L 2 、 M represent Ideal Coupled Inductor Mutual- inductance voltage Self- inductance voltage Ch4 Sinusoidal Steady State Analysis

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4.7 Magnetically Coupled Circuits v2v2 v1v1 + - i1i1 + - i2i2 v1v1 + - i1i1 v2v2 + - i2i2 Coupled Circuits and v ~ i relationship Ch4 Sinusoidal Steady State Analysis

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4.7 Magnetically Coupled Circuits Dot convention v1v1 + - i1i1 v2v2 + - i2i2 v1v1 + - i1i1 v2v2 + - i2i2 A current entering the dotted terminal of one coil produces an open circuit voltage with a positive voltage reference at the dotted terminal of the second coil. Inversely, current leaving of the dotted terminal of one coil produces a negative voltage reference at the dotted terminal of the other end. i1i1 i2i2 v1v1 v2v2 i1i1 i2i2 v1v1 v2v2 Ch4 Sinusoidal Steady State Analysis

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4.7 Magnetically Coupled Circuits Question ： The terminal is dotted ， how can we get v ~ i equations to coupled inductor ？ i1i1 i2i2 v1v1 v2v2 u2u2 i1i1 i2i2 v1v1 v2v2 Suppose direction of the i and is consistent with Dot convention! Steps to determine the coupled circuit voltage 2. For mutual inductance voltage1. For self inductance voltage + i / i - + v s - i / i + - v s - v m Ch4 Sinusoidal Steady State Analysis + v m

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4.7 Magnetically Coupled Circuits P4.16 ， For the circuit shown in following figures, determine v 1 and v 2. u2u2 i2i2 v1v1 v2v2 i1i1 L1L1 L2L2 M u2u2 i2i2 v1v1 v2v2 i1i1 L1L1 L2L2 M u2u2 i2i2 v1v1 v2v2 i1i1 L1L1 L2L2 M ＋ － Ch4 Sinusoidal Steady State Analysis

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4.7 Magnetically Coupled Circuits Coupled Circuits and v ~ i relationship v1v1 + - i1i1 v2v2 + - i2i2 ——Ideal Coupled Circuits ’ s v~i relationship For sinusoidal circuit, Ch4 Sinusoidal Steady State Analysis

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4.7 Magnetically Coupled Circuits Ideal transformer u2u2 i1i1 i2i2 v1v1 v2v2 N1N1 N2N2 (k = 1, L = , M = ) n=N 1 /N 2 For ideal transformer, p=0 Impedance Transformation ZLZL Z1Z1 Ch4 Sinusoidal Steady State Analysis

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