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SINUSOIDAL STEADY-STATE ANALYSIS

SINUSOIDAL STEADY-STATE ANALYSIS. Prepared by Ertuğrul Eriş Reference Book: Electric Circuits Nilsson, Riedel. Güncelleme: 2 : Feb 2013. SINUSOIDAL STEADY-STATE ANALYSIS (SİNÜSOİDAL SÜREKLİ HAL). The Sinusoidal Source/Signal The Sinusoidal Response The Phasor ( Fazör )

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SINUSOIDAL STEADY-STATE ANALYSIS

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  1. SINUSOIDAL STEADY-STATE ANALYSIS Prepared by Ertuğrul Eriş Reference Book: Electric Circuits Nilsson, Riedel Güncelleme: 2 : Feb 2013 Ertuğrul Eriş

  2. SINUSOIDAL STEADY-STATE ANALYSIS(SİNÜSOİDAL SÜREKLİ HAL) • The Sinusoidal Source/Signal • The Sinusoidal Response • The Phasor (Fazör) • The Passive Circuit Elements in The Frequency Domain • Kirchhoff ‘s Law in The Frequency Domain • Series/parellel, Delta-to-Wye simplifications • Source transformations, Thevenin-Norton equivalent circuits • The Node-voltage Method • The Mesh-current Method • The Transformer (Transformatör (M elemanı)) • The Ideal Transformer Ertuğrul Eriş

  3. ELECTRONIC/ELECTRICAL ENNGINEERING • Information transfer • Audio (Ses), text, Image, video (görüntü hareketsiz, hareketli) • Why electrical (Electronic engineering): • Information process, storage, transmit/recieve • Signal transfors required (microphone, speakers;...) • Other fields (Electrical engineering) • Energy source (alternatives) • Heat • Light Ertuğrul Eriş

  4. POWER/SIGNAL SOURCES • Upto now DC power sources included circuits • System/circuit inputs • Power inputs • DC (batteries, transistor included circuits) • AC sources, power sources • Sıgnal inputs • Electrical signals «audio, image, video» representing voice, still and moving pictures • Sum of sinusoidal signals (fourier sseries) • System/circuit outputs • Sinusoidal signals Ertuğrul Eriş

  5. WHY SINUSOIDAL SIGNALS • Analog Electricals signals • v(t), i(t) • Mathematical • continious • Periodical • Fourier transforms • Sum of iferent frequeny and amplitıde sinusoidal signals • Linear circuits Ertuğrul Eriş

  6. NON SINUSOIDAL SIGNALS • Digital Electrical signals • V(t), i(t) • Advantages:Noise elimination,Strorage, accuracy • Disadvantages • Analog-Digital-Analog converters • Digital sistemler • Boolean Algebra/ Logic design • Digital control, digital communicatin Ertuğrul Eriş

  7. LINEAR CIRCUITS/ORDINARY DIFFERENTIAL EQUATIONS • Linear Circuit • Homogenity: sources multiplied by k→solutions multiplied by k • Super position: all sources included soluion → sum of each sources alone solutions • Inputs • DC Power, first semestr RC,RL,RLC • AC power or signal (Circuit Analysis, this course) • Linear circuit analysis/ Ordinary differeential equation solution • Homogenous solution= transient response (Geçici çözüm ): (zamanla kaybolan çözüm) • Particular solution=steady-state response (Sürekli çözüm) (devamlı gözlenen) Ertuğrul Eriş

  8. SINUSOIDAL STEADY-STATE SOLUTION / PARTICULAR SOLUTION OF A ORDINARY DIFFERENTIAL EQUATION • General solutıon • Homogenous solutıon(transient): • Depending on the chracteristic roots «under, over, critically damped» exponentially decreasing funtion in a few mınutes it becomes zero. • Particular solutıon: • The same type (DC, sınusoidal) as source functions • Şimdiye kadar kaynaklar DC idi, şimdiden sonra AC • General solutıon= Homogenous+ Particular Since homogenous solution disapears in a few seconds then general solution becomes particular solution which is called: SINUSOIDAL STEADY-STATE SOLUTION (Yani Sinüsoidal Sürekli Hal Çözümü) Ertuğrul Eriş

  9. SINUSOIDAL STEADY-STATE SOLUTION / PARTICULAR SOLUTION OF A ORDINARY DIFFERENTIAL EQUATION • Particular solution • Independent Source type • Sınusoidal sources, • Elements’ current and voltages are also sinusoidal • Frequency: same as source frequency • But «Phase and amplitude»will change Ertuğrul Eriş

  10. LINEAR CIRCUIT ANALYSIS SUMMARY

  11. SINUSOIDAL SIGNAL • Parameters • Periodic: Frequency/period • Amplitute(genlik) • Phase(Faz) Ertuğrul Eriş

  12. PHASE OF SINUSOIDAL SIGNAL (Faz) Degree= (180/ π )radian VmCos(ωt+Φ) ile VmCos(ωt) phase difference: Vm Cos(ωt+Φ) → Vm ; ωt+Φ = 0 → t = (- Φ/ ω) Vm Cos(ωt) (time) Vm Cos(ωt+Φ) Phase chage→shift : Φ positve→shift left; Φ negative →shift right Ertuğrul Eriş

  13. MEASUREMENT RMS=Root of the Mean value of the Squared function Why rms? edaquate? How to measure frequency, phase? Ertuğrul Eriş

  14. GENERAL SOLUTION IN t-DOMAIN Transient solution (geçici hal) Sinusoidal steady-state solution (sürekli sinüsoidal hal) θ=arctg ωL/R • OBSERVATION: • Independent source: sinusoidal. • Homogenous solution becomes zero. • Particular solution: • source type, sınusoidal • Frequency same as source frequency • Phase and amplitude change, not the same as source Proteusta simulation available on course web-site. Ertuğrul Eriş

  15. TRANSIENT SOLUTION COULD BE IMPORTANT Transient solution (geçici hal) Sinusoidal steady-state solution (sürekli sinüsoidal hal) θ=arctg ωL/R Asynchronous electric motor Vs=220*1,41sin (2*pi*50*t) R=0,1Ω, L=1H (Asynchronous electric motor) Above solution Φ=-π/2 (Above cos becomes sin) «Inrush current» For t=T/2=10ms → i(t)=2 Amps For t=30s →i(t)=1 Amp Engineering conclusion: Delayed 2-3Amp fuse Proteusta simulation available on course web-site. Ertuğrul Eriş

  16. EXAMPLE • R=1KΩ, L=10mH, • V=sin(ω t), Vm=1V, f=100khz, Φ=0 • Θ=arctg(ωL/R)=810 • τ=(L/R)=10μs • (Vm/√(R2+ω2))=157 μA • i(t)=-157 μA sin(810)e(-t/10 μs) +157 μA sin(ωt-810) • t=0→i(0)=0 • t= L/R=time costant=10 μs • 10μs→ i(10 μs)=97,61 μA • «proteus» solution is available on course web-side Ertuğrul Eriş

  17. PHASOR (FAZÖR) :FREQUENCY DOMAIN ω-domain t-domain vs ω-domain Ertuğrul Eriş

  18. COMPLEX NUMBERS: POLAR/RECTANGULAR TRANSFORM • Polar form • De jθ = D(cos θ+ j sin θ) • Rectangular form • A+jB • A= Dcos θ B= Dsin θ Ertuğrul Eriş

  19. COMPLEX NUMBERS: RECTANGULAR/POLARTRANSFORM • A+jB 1. kadran • D= √A2+B2 θ= arctn (B/A) • -A-jB 3. kadran • D= √A2+B2 ,θ= arctn (B/A)+180 • -A+jB 2. kadran • D= √A2+B2 ,θ= 180-arctn (B/A) • A-jB 4. kadran • D= √A2+B2 ,θ= -arctn (B/A) Nilsson Appendix B Ertuğrul Eriş

  20. FREQUENCY DOMAIN: RESISTANCE V=R I Phasor «bold» Ertuğrul Eriş

  21. FREQUENCY DOMAIN: INDUCTANCE V = jωL I Voltage leading current π / 2 Current lagging voltage - π / 2 j = e jπ/2 = cos(π/2) + j sin(π/2) Ertuğrul Eriş

  22. FREQUENCY DOMAIN: CONDUCTANCE V = (1/jωc) I Voltage lagging current -π/ 2 Current leading voltage +π/ 2 j = e jπ/2 = cos(π/2) + j sin(π/2) Ertuğrul Eriş

  23. IMPEDANCE: REACTANCE ADMITTANCE: CONDUCTANCE SUSCEPTANCE (empedans,reakyans,admitans,iletkenlik,süseptans) • IMPEDANCE (Empedans): V=Z I • ADMITTANCE (Admitance) : I=Y V • Current and voltage phasors are complex numbers • IMPEDANCE and ADMITTANCE complex numbers, not phasor, • Reactance: Imaginary part of ımpedance • Conductance: Real part of admittance • suseptance: Imaginary part of admittance Ertuğrul Eriş

  24. KIRCHOFF’S LAWS IN FREQUENCY DOMAIN • Element voltages and currents become Element voltage phasors and current phasors, • Advantages of frequency domain? • Disadvantages of frequency domain? Σvi = 0; Σii = 0 Ertuğrul Eriş

  25. COMPARISON t-DOAIN AND ω-DOMAIN SOLUTIONS transient Stady-state solution θ=arctg ωL/R Which solution has not been calculated? What are the results? Compare source and output signals? Ertuğrul Eriş

  26. TRANSIENT AND STEADY SOLUTIONS Ertuğrul Eriş

  27. A SINUSOIDAL RELATION • Sum of two sinusoidal signals with the same frequency equals to a single sinusoidal signal with the same frequency Cos(ωtŦΦ)=cos ωt cos Φ±sin ωt sin Φ • sin(ωt±Φ)=cos ωt sin Φ ± sin ωt cos Φ Y1=20cos(ωt-30), Y2=40cos(ωt+60) Y1+Y2= 44.72 cos(ωt+33.430) Y1=20e-j30Y2=40ej60 Y1+Y2=44.72 ej33,43 Ertuğrul Eriş

  28. COMBINING IMPEDANCES IN SERIAL Zab= Z1+Z2+…….+Zn Ertuğrul Eriş

  29. EXAMPLE Which solution in t-domain? benefits? Ertuğrul Eriş

  30. COMBINING IMPEDANCES IN PARALLEL Yab= Y1+Y2+…….+Yn Ertuğrul Eriş

  31. EXAMPLE Ertuğrul Eriş

  32. NODE VOLTAGE METHOD IN ω-DOMAIN What about C and L values? t-domain solutions? How? C=10μF, L=(2/5)mH, ω=104 r/s unique? Ertuğrul Eriş

  33. MESH-CURRENTMETHOD IN ω-DOMAIN I1 = -58e j63.4 =58e j244 =58e -j116 I2 = -64.2e j67.5 Ix = 6.32e j(180-71,5) ω=1000r/s L1=2mH L2=3mH C=62,5μF t- domain solutions? Ertuğrul Eriş

  34. DELTA-WYE TRANSFORMS (YILDIZ-ÜÇGEN DÖNÜŞÜMLERİ) Ertuğrul Eriş

  35. EXAMPLE Ertuğrul Eriş

  36. SOURCE TRANSFORMS IN ω-DOMAIN Other interpratation: Thévenin Norton equvalance circuits transforms Ertuğrul Eriş

  37. EXAMPLE Ertuğrul Eriş

  38. THÉVENIN EQUIVALANCE IN ω-DOMAIN Thévenin equivalent circuit with respect to a-b terminals Is it possible to fınd Thévenin equivalent circuit of a circuit which include ınductors and capacitors?Why? Ertuğrul Eriş

  39. NORTON EQUIVALANCE IN ω-DOMAIN Norton equivalent circuit with respect to a-b terminals Norton /Thevenin equivalance transform: IN= V0 / Zth ; ZN=Zth Ertuğrul Eriş

  40. EXAMPLE a b Norton current=I0=(430/51)+j(20/51) Ertuğrul Eriş

  41. MUTUAL INDUCTANCE(Linear Transformer) Ertuğrul Eriş

  42. MUTUAL INDUCTANCE EXAMPLE c d (cd) Terminal Thevenin Equivalent: Vth = Vcdopen circuit voltage Zth = ımpedance seen looking into the terminals (cd) Ertuğrul Eriş

  43. IDEAL TRANSFORMER + I2 I1 + V2 V1 M2=L1L2 Ertuğrul Eriş

  44. IMPEDANCE MATCH BY USING IDEAL TRAFO n=(n1/n2) Z1=V1/I1=n2*ZL Speaker and amplifier impedance match!, Impedance seen looking into the right of V1 Ertuğrul Eriş

  45. SSS SUMMARY • CONSTRAINTS • Linear circuits • Independent sources are sinusoidal (AC) • Particular solution only! • BENEFITS • No differential equations but • Linear algebraic equations • LOSTS/ADDITIONALS • Transient solution • time→Phasor →time transforms • Stability cannot be observed Ertuğrul Eriş

  46. SSS: AN EVALUATION • Sinusoidal independent sources • Asin(ωt+φ) • Sinusoidal elements’ current and voltages • B(ω) sin[ωt+θ(ω)] • No changes: sinusoidal functions, frequency • Chages: Amplitude and Phase depending on the elements’ values and the frequency Ertuğrul Eriş

  47. TIME VS FREQUENCY DOMAINS Ertuğrul Eriş

  48. PROGRAM DESIGN DEPT, PROGRAM G R A D U A T E S T U D E N T STUDENT P R OG R A M O U T C O M E S PROGRAM OUTCOMES P R OG R A M O U T C O M E S STATE, ENTREPRENEUR FIELD QALIFICATIONS EU/NATIONAL QUALIFICATIONS KNOWLEDGE SKILLS COMPETENCES NEWCOMERSTUDENT ORIENTIATION GOVERNANCE Std. questionnaire ALUMNI, PARENTS ORIENTIATION STUDENT PROFILE Std. questionnaire FACULTY NGO STUDENT, ??? CIRCICULUM ??? INTRERNAL CONSTITUENT Std. questionnaire EXTRERNAL CONSTITUENT EXTRERNAL CONSTITUENT REQUIREMENTS EU/NATIONAL FIELD QUALIFICATIONS PROGRAM OUTCOMES QUESTIONNAIRES QUALITY IMP. TOOLS GOAL: NATIONAL/INTERNATIONAL ACCREDITION

  49. BLOOM’S TAXONOMYANDERSON AND KRATHWOHL (2001) !!Listening !! Doesn’t exits in the original!!! http://www.learningandteaching.info/learning/bloomtax.htm Ertuğrul Eriş

  50. ULUSAL LİSANS YETERLİLİKLER ÇERÇEVESİ BLOOMS TAXONOMY Ertuğrul Eriş

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