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SECOND ORDER CIRCUITS

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SECOND ORDER CIRCUITS. Prepared by: Ertuğrul Eriş Reference book: Electric Circuits, Nielsson, Riedel Pearson, Prentence Hall,2007. Update 1: October 2011. RLC CIRCUITS. Second order circuits Two inductors, two capacitors, one inductor one capacitor DC Independent source /AC later

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  1. SECOND ORDER CIRCUITS Prepared by: Ertuğrul Eriş Reference book: Electric Circuits, Nielsson, Riedel Pearson, Prentence Hall,2007 Update 1: October 2011

  2. RLC CIRCUITS • Second order circuits • Two inductors, two capacitors, one inductor one capacitor • DC Independent source /AC later • Serial or parallel • Second order differential equation • At most second order derivative OR • Two first order differential equations, matrices Ertuğrul Eriş

  3. PARALLEL RLC CIRCUIT ANALYSIS • MATHEMATICAL • Second order single differential equation, or first order two differential equations • Homogenous solution: independent sources deactivated. • Particular solution: independent source type solution • General solution= Homogenous solution + particular solution • CIRCUIT BASED • Natural response • Independent sources are deactivated, initial conditions exsist • Forced solution • Independent sources active (step, basamak); initial conditions are(0) • General solution (Tam, genel çözüm)=Natural response+forced solution Ertuğrul Eriş

  4. SECOND ORDER CIRCUITS Paralel RLC Mathematical Homogenous+particular Circuit based Natural+forced + V0 _ I0 Seri RLC Mathematical Homogenous+particular Circuit based Natural+forced I0 + V0 _ Ertuğrul Eriş

  5. PARALLEL RLC CIRCUIT MATHEMATICAL ANALYSIS-1 + V0 _ I0 Second order single differential equation (capacitor voltage, inductor current): Two first order differential equations state equations (Durum Denklemleri ) Would it be enough to solve only one of the above equations? How could we find the other unknowns? Compare left and right hand equations Ertuğrul Eriş

  6. PARALLEL RLC CIRCUIT MATHEMATICAL ANALYSIS-2 + V0 _ I0 General (Tam) solution: = Homogenous (Homojen) solution + particular (Özel) solution independent sources deactivated for DC type souces it is a constant, For AC type sources, it is a sinusoidal function How do we know that the sum of homogenous and particular solution is general solution? Linearity!! Difference of homogenous and natural response? Ertuğrul Eriş

  7. PARALLEL RLC CIRCUIT MATHEMATICAL ANALYSIS-3 Tam çözüm (General)= = Homojen çözüm (Homogenous) + Özel çözüm (particular) The two unknown parameters in homogenous solution will be calculated by using the two initial conditions on the general solution = Transient state (Geçici hal,rejim) + Steady state (Sürekli hal,rejim) How do we know that the sum of homogenous and particular solution is general solution? Linearity!! Difference of homogenous and natural response? Ertuğrul Eriş

  8. PARALLEL RLC CIRCUIT MATHEMATICAL ANALYSIS-4 + V0 _ I0 I(t) independent current source If it is DC source then its derivative is (0) For Vc(t) general solution is equal to homogenous solution while particular solution is (0) A1 and A2 parameters will be calculated by vc(0), iL(0) initial conditions. dvc(0)/dt; diL(0)/dt can be calculated in terms of vc(0), iL(0) Ertuğrul Eriş

  9. PARALLEL RLC CIRCUIT MATHEMATICAL ANALYSIS-5 + V0 _ I0 If current source is a DC source then particular solution is estimated as a constant D and calculated. A1 and A2 parameters will be calculated by the known vc(0), iL(0) initial conditions. dvc(0)/dt; diL(0)/dt will be found by vc(0), iL(0) initial conditions Compare Vc(t) in previous slide and iL(t) (this slide)? Steady state (particular solution) is I, What does this mean? Difference of homogenous and natural response? Ertuğrul Eriş

  10. CHARACTERISTIC ROOTS • Four cases: • Reel roots α2>ω20, Overdamped • Complex conjugate roots α2<ω20, underdamped • Multiple roots α2=ω20, critically damped • Imaginary rootsα=0 Could the charecteristic roots be imaginar in a second order circuit? Why? Ertuğrul Eriş

  11. PARALLEL RLC CIRCUIT MATHEMATICAL ANALYSIS-6:REAL ROOTS + V0 _ I0 Ertuğrul Eriş

  12. PARALLEL RLC CIRCUIT MATHEMATICAL ANALYSIS-7: REAL ROOTS EXAMPLE VC(0+)=12V, iL(0+)=30 mA C=0,2 μF L=50 mH R=200Ω I=1 A + V0 _ I0 α = - (1/RC) = 12500 ω0 = 1/√LC = 10000 α > ω0 iL(0-) = iL(0) = iL(0+) = 30 mA iC = C (dv/dt) diL(0+)/dt = vc(0+)/L =240 A/s s1 = - 5000r/s s2 = - 20000r/s A1 = -1,277 A2 = 0,307 iL(t) = -1,277e-5000t + 0,307 e-20000t +1 A, t≥0 v(t) = 319 e-5000t - 307 e-20000t V, t≥0 iR(t) = 1,59 e-5000t - 1,53e-20000t A, t≥0 iC(t)= (-0,323e-5000t +1,223e-20000t) A, t≥0 Ertuğrul Eriş

  13. GENERAL SOLUTION FOR REAL ROOTS + V0 _ I0 VC(0+)=12V, iL(0+)=30 mA C=0,2 μF L=50 mH R=200Ω I=1 A s1 = - 5000r/s, s2 = - 20000r/s iL(t) = -1,277e-5000t + 0,307 e-20000t +1 A, t≥0 Transient and stady state solution, which elements will be short/open circuits, why? Ertuğrul Eriş

  14. PARALLEL RLC CIRCUIT MATHEMATICAL ANALYSIS-8: COMPLEX ROOTS + V _ I0 A1, A2 Will be complex conjugate!! Could B1, B2 be complex? Ertuğrul Eriş

  15. PARALLEL RLC CIRCUIT MATHEMATICAL ANALYSIS-9: COMPLEX ROOTS EXAMPLE VC(0+)=0 V, iL(0+)=-500 mA C=0,125 μF L=8mH R=20KΩ I=1 A + V0 _ I0 Ertuğrul Eriş

  16. GENERAL SOLUTION FOR COMPLEX ROOTS + V0 _ I0 VC(0+)=0 V, iL(0+)=-500 mA C=0,125 μF L=8mH R=20KΩ I=1 A Transient and stady state solution, which elements will be short/open circuits, why? Ertuğrul Eriş

  17. FIRST TWO MILISECONDS Ertuğrul Eriş

  18. PARALLEL RLC CIRCUIT MATHEMATICAL ANALYSIS-10: MULTIPLE ROOTS + V _ I0 Stability, (burn or not, kararlılık) ? Ertuğrul Eriş

  19. PARALLEL RLC CIRCUIT MATHEMATICAL ANALYSIS-11: MULTIPLE ROOTS EXAMPLE ω0 = 1/√LC = 1000 α = - (1/RC) = 1000 α = ω0 VC(0+)=0 V, iL(0+)=-500 mA C=0,125 μF L=8 H R=4KΩ I=1 A + V0 _ I0 Ertuğrul Eriş

  20. GENERAL SOLUTION FOR MULTIPLE ROOTS + V0 _ I0 VC(0+)=0 V, iL(0+)=-500 mA C=0,125 μF L=8 H R=4KΩ I=1 A Transient and stady state solution, which elements will be short/open circuits, why? Ertuğrul Eriş

  21. CIRCUIT BASED PARALLEL RLC CIRCUIT ANALYSIS + V0 _ General solution (Tam)= I0 = Natural solution (Öz) + Forced solution (Zorlanmış) independent spurces:deactivated, initial conditions iexsist independent sources: active, initial conditions (0) İnitial conditions (0) !! Difference of homogenous and natural response? Ertuğrul Eriş

  22. PARALLEL RLC CIRUITS: NATURAL RESPONSE-1 + V0 _ I0 Ertuğrul Eriş

  23. CHARACTERISTIC ROOTS • Four cases: • Reel roots α2>ω20, Overdamped • Complex conjugate roots α2<ω20, underdamped • Equal roots α2=ω20, critically damped • İmaginary t-rootsα=0 L=50mH, C=0.2μF ve a) R=200Ω, b) R=312.5Ω, c) R=250Ω Ertuğrul Eriş

  24. PARALLEL RLC CIRUITS: NATURAL RESPONSE, REAL ROOTS Ertuğrul Eriş

  25. PARALLEL RLC CIRUITS: NATURAL RESPONSE, REAL ROOTS EXAMPLE VC(0+)=12V, iL(0+)=30 mA α = - (1/RC) = 12500 ω0 = 1/√LC = 10000 α > ω0 iL(0-) = iL(0) = iL(0+) = 30 mA iC(0+) = -iL(0+) -iR(0+) = -90 mA iC = C (dv/dt) dvC(0+)/dt = - 450kV/s, s1 = - 5000r/s s2 = - 20000r/s A1 = -14V, A2 = 26V v(t)= (-14e-5000t+26e-20000t) v, t≥0 iR(t)=(-70e-5000t+130e-20000t) mA, t≥0 iC(t)= (14e-5000t-104e-20000t) mA, t≥0 iL(t)=(56e-5000t-26e-20000t) mA, t≥0 Ertuğrul Eriş

  26. PARALLEL RLC CIRUITS: NATURAL RESPONSE, REAL ROOTS EXAMPLE α = - (1/RC) = 12500 ω0 = 1/√LC = 10000 α > ω0 INDUCTORS CURRENT s1 = - 5000r/s s2 = - 20000r/s iL(t)=(56e-5000t-26e-20000t) mA, t≥0 VC(0+)=12V, iL(0+)=30 mA Ertuğrul Eriş

  27. PARALLEL RLC CIRUITS: NATURAL RESPONSE, COMPLEX ROOTS Ertuğrul Eriş

  28. PARALLEL RLC CIRUITS: NATURAL RESPONSE, COMPLEX ROOTS EXAMPLE 8mH V(0)=0, I(0)=-500mA Osilasyon, çınlama(ringing) Ertuğrul Eriş

  29. PARALLEL RLC CIRUITS: NATURAL RESPONSE, COMPLEX ROOTS EXAMPLE 8mH V(0)=0, I(0)=-500mA Ertuğrul Eriş

  30. PARALLEL RLC CIRUITS: NATURAL RESPONSE, MULTIPLE ROOTS Ertuğrul Eriş

  31. PARALLEL RLC CIRUITS: NATURAL RESPONSE, MULTIPLE ROOTS EXAMPLE 4kΩ V(0)= 0, iL(0)= -500 mA Ertuğrul Eriş

  32. PARALLEL RLC CIRUITS: NATURAL RESPONSE, MULTIPLE ROOTS EXAMPLE 4kΩ V(0)= 0, iL(0)= -500 mA V(0)=0, I(0)=-500mA Ertuğrul Eriş

  33. PARALLEL RLC CIRUITS: FORCED RESPONSE Initial energy 0 İnitial conditions0 VC(0)=0 IL(0) =0 + V0 _ I0 Ertuğrul Eriş

  34. PARALLEL RLC CIRUITS: FORCED RESPONSE, REAL ROOTS I(t)= 1 A İlk enerji 0 İlk koşullar 0 VC(0)=0 IL(0) =0 0,2 μF 50mH 200Ω Ertuğrul Eriş

  35. PARALLEL RLC CIRUITS: FORCED RESPONSE, REAL ROOTS I(t)= 1 A İlk enerji 0 İlk koşullar 0 VC(0)=0 IL(0) =0 Ertuğrul Eriş

  36. PARALLEL RLC CIRUITS: FORCED RESPONSE, COMPLEX ROOTS I(t)= 1 A İlk enerji 0 İlk koşullar 0 VC(0)=0 IL(0) =0 20 KΩ 0,125μF 8H Ertuğrul Eriş

  37. PARALLEL RLC CIRUITS: FORCED RESPONSE, COMPLEX ROOTS I(t)= 1 A İlk enerji 0 İlk koşullar 0 VC(0)=0 IL(0) =0 20 KΩ 0,125μF 8mH Ertuğrul Eriş

  38. PARALLEL RLC CIRUITS: FORCED RESPONSE, MULTIPLE ROOTS I(t)= 1 A İlk enerji 0 İlk koşullar 0 VC(0)=0 IL(0) =0 4 KΩ 0,125μF 8H Ertuğrul Eriş

  39. PARALLEL RLC CIRUITS: FORCED RESPONSE, MULTIPLE ROOTS 4 KΩ 0,125μF 8H I(t)= 1 A İlk enerji 0 İlk koşullar 0 VC(0)=0 IL(0) =0 Ertuğrul Eriş

  40. COMPARISION MATHEMATICAL AND CIRCUIT BASED SOLUTIONS FOR PARALEL RLCCIRCUITS Mathematical Circuit based Real root C=0,2μF L=50mH R=200Ω VC(0)=12V IL(0)=30mA Complex root C=0,125μF L=8mH R=20KΩ VC(0)=0V IL(0)=500mA Multiple roots C=0,125μF L=8H R=4KΩ VC(0)=0V IL(0)=500mA Ertuğrul Eriş

  41. EFFECT OF DIFFERENT RESISTANCES Ertuğrul Eriş

  42. SERIAL RLC CIRCUIT ANALYSIS Differential equation for Capacitor voltage? + V0 _ Tam çözüm (General)= = Natural response (Öz) + Forced response (Zorlanmış) Independent source: deactivated; Initial conditions: exsis Independent source: activated; Initial conditions: (0) I0 Difference of homogenous and natural response? İnitial con (0) !! Ertuğrul Eriş

  43. SERIAL RLC CIRCUIT ANALYSIS NATURAL RESPONSE Stabilty? Could there be imaginay roots? Why? Ertuğrul Eriş

  44. SERIAL RLC CIRCUIT ANALYSIS NATURAL RESPONSE EXAMPLE iL(0)=0 Vc(0)=100V What can we say about roots by looking the solutions? Stability? Aynı sonuçlara başka hangi diferansiyel denklemle ulaşılabilir? Ertuğrul Eriş

  45. SERIAL RLC CIRCUIT ANALYSIS FORCED RESPONSE iL(0)=0 Vc(0)=0 Stability? Ertuğrul Eriş

  46. SERIAL RLC CIRCUIT ANALYSIS FORCED RESPONSE EXAMPLE iL(0)=0 Vc(0)=0 100mH 560Ω 0.1 μF What can we say about roots by looking the solutions? Can we find the same solution by using different differential equation? Which one is it? Which part of the above solution is homogenous and particular? Which part of the above solution is transient and steady state? Ertuğrul Eriş

  47. SERIAL RLC CIRCUIT ANALYSIS FORCED RESPONSE EXAMPLE iL(0)=0 Vc(0)=0 100mH 560Ω 0.1 μF What can we say about roots by looking the solutions? Can we find the same solution by using different differential equation? Which one is it? Which part of the above solution is homogenous and particular? Which part of the above solution is transient and steady state? Ertuğrul Eriş

  48. SERIAL RLC CIRCUIT ANALYSIS GENERAL SOLUTION EXAMPLE Ertuğrul Eriş

  49. INTEGRATING CIRCUIT derivative? How? Ertuğrul Eriş

  50. INTEGRATING CIRCUIT EXAMPLE Vg=25mV Ertuğrul Eriş