1. ECEN 301Discussion #10 – Energy Storage1 DateDayClass No. TitleChaptersHW Due date Lab Due date Exam 6 OctMon10Energy Storage3.7, 4.1NO LAB 7 OctTue NO LAB 8 OctWed11Dynamic Circuits4.2 – 4.4 9 OctThu 10 OctFri Recitation HW 4 11 OctSat 12 OctSun 13 OctMon12Exam 1 Review LAB 4EXAM 1 14 OctTue Schedule…

2. ECEN 301Discussion #10 – Energy Storage2 Energy Moro. 7: 48 48 Wherefore, my beloved brethren, pray unto the Father with all the energy of heart, that ye may be filled with this love, which he hath bestowed upon all who are true followers of his Son, Jesus Christ; that ye may become the sons of God; that when he shall appear we shall be like him, for we shall see him as he is; that we may have this hope; that we may be purified even as he is pure. Amen.

3. ECEN 301Discussion #10 – Energy Storage3 Lecture 10 – Energy Storage Maximum Power Transfer Capacitors Inductors

4. ECEN 301Discussion #10 – Energy Storage4 Maximum Power Transfer uEquivalent circuit representations are important in power transfer analysis vTvT +–+– RTRT Load +v–+v– i uIdeally all power from source is absorbed by the load ÙBut some power will be absorbed by internal circuits (represented by R T )

5. ECEN 301Discussion #10 – Energy Storage5 Maximum Power Transfer uEfficiently transferring power from source to load means that internal resistances (R T ) must be minimized Ùi.e. for a given R L we want R T as small as possible! uFor a given R T there is a specific R L that will maximize power transfer to the load vTvT +–+– RTRT RLRL iLiL

6. ECEN 301Discussion #10 – Energy Storage6 Maximum Power Transfer uConsider the power (P L ) absorbed by the load vTvT +–+– RTRT RLRL iLiL

7. ECEN 301Discussion #10 – Energy Storage7 Maximum Power Transfer uConsider the power (P L ) absorbed by the load vTvT +–+– RTRT RLRL iLiL BUT Therefore

8. ECEN 301Discussion #10 – Energy Storage8 Maximum Power Transfer uThe maximum power transfer can be found by calculating dP L /dR L = 0 vTvT +–+– RTRT RLRL iLiL OR NB: assumes R T is fixed and R L is variable

9. ECEN 301Discussion #10 – Energy Storage9 Maximum Power Transfer To transfer maximum power to the load, the source and load resistors must be matched (i.e. R T = R L ). Power is attenuated as R L departs from R T When R L < R T : P L is lowered since v L goes down When R L > R T : P L is lowered since less current (i L ) flows

10. ECEN 301Discussion #10 – Energy Storage10 Maximum Power Transfer Maximum power theorem: maximum power is delivered by a source (represented by its Thévenin equivalent circuit) is attained when the load (R L ) is equal to the Thévenin resistance R T vTvT +–+– RTRT RLRL iLiL NB: Substitute (R T = R L ) to get P LMax

11. ECEN 301Discussion #10 – Energy Storage11 Maximum Power Transfer Efficiency of power transfer (η): the ratio of the power delivered to the load (P out ), to the power supplied by the source (P in ) NB: P out = P L

12. ECEN 301Discussion #10 – Energy Storage12 Maximum Power Transfer Efficiency of power transfer (η): the ratio of the power delivered to the load (P out ), to the power supplied by the source (P in ) vTvT +–+– RTRT RLRL iLiL

13. ECEN 301Discussion #10 – Energy Storage13 Maximum Power Transfer Efficiency of power transfer (η): the ratio of the power delivered to the load (P out ), to the power supplied by the source (P in ) vTvT +–+– RTRT RLRL iLiL NB: Substitute (R T = R L ) to get P inMax

14. ECEN 301Discussion #10 – Energy Storage14 Maximum Power Transfer Efficiency of power transfer (η): the ratio of the power delivered to the load (P out ), to the power supplied by the source (P in ) vTvT +–+– RTRT RLRL iLiL In other words, at best, only 50% efficiency can be achieved

15. ECEN 301Discussion #10 – Energy Storage15 Maximum Power Transfer uExample1: Find the maximum power delivered to R L Ùv s = 18V, R 1 = 3Ω, R 2 = 6Ω, R 3 = 2Ω R2R2 R 1 RLRL R 3 vsvs +–+–

16. ECEN 301Discussion #10 – Energy Storage16 Maximum Power Transfer uExample1: Find the maximum power delivered to R L Ùv s = 18V, R 1 = 3Ω, R 2 = 6Ω, R 3 = 2Ω 1.Find Thévenin equivalent circuit vTvT +–+– RTRT RLRL

17. ECEN 301Discussion #10 – Energy Storage17 Maximum Power Transfer uExample1: Find the maximum power delivered to R L Ùv s = 18V, R 1 = 3Ω, R 2 = 6Ω, R 3 = 2Ω 1.Find Thévenin equivalent circuit 2.Set R L = R T 3.Calculate P LMax vTvT +–+– RTRT RLRL

18. ECEN 301Discussion #10 – Energy Storage18 Dynamic Energy Storage Elements Capacitor and Inductor

19. ECEN 301Discussion #10 – Energy Storage19 Storage Elements uTo this point, circuits have consisted of either: ÙDC Sources (active element, provide energy) ÙResistors (passive element, absorb energy) uSome passive (non-source) elements can store energy ÙCapacitors – store voltage ÙInductors – store current +C–+C– +C–+C– +L–+L– + L – + C – Capacitor symbols Inductor symbols

20. ECEN 301Discussion #10 – Energy Storage20 Ideal Capacitor uA capacitor (C) is composed of: Ù2 parallel conducting plates cross-sectional area A ÙSeparated distance d by either: Air Dielectric (insulating) material d A Dielectric (or air) ε – relative permittivity (dielectric constant) NB: the parallel plates do not touch – thus under DC current a capacitor acts like an open circuit + – C – Capacitance

21. ECEN 301Discussion #10 – Energy Storage21 Ideal Capacitor Capacitance: a measure of the ability to store energy in the form of separated charge (or an electric field) + + + + + + + + + + + – – – – – – – – – – – – – – – The charge q+ on one plate is equal to the charge q– on the other plate Farad (F): unit of capacitance. 1 farad = 1 coulomb/volt (C/V)

22. ECEN 301Discussion #10 – Energy Storage22 Ideal Capacitor uNo current can flow through a capacitor if the voltage across it is constant (i.e. connected to a DC source) uBUT when a DC source is applied to a capacitor, there is an initial circuit current as the capacitor plates are charged (‘charging’ the capacitor) + + + + – – – – – – vsvs +–+– Charging the capacitor: a current flows as electrons leave one plate to accumulate on the other. This occurs until the voltage across the capacitor = source voltage i

23. ECEN 301Discussion #10 – Energy Storage23 Ideal Capacitor uCapacitor with an AC source: since AC sources periodically reverse voltage directions, the capacitor plates will change charge polarity accordingly ÙWith each polarity change, the capacitor goes through a ‘charging’ state, thus current continuously flows ÙJust as with a DC source, no electrons cross between the plates ÙJust as with a DC source, voltage across a capacitor cannot change instantaneously + + + + – – – – – – v s (t) +–+– i – – – – + + + + + –+–+ i

24. ECEN 301Discussion #10 – Energy Storage24 Ideal Capacitor ui – v characteristic for an ideal capacitor Differentiate both sides BUT OR

25. ECEN 301Discussion #10 – Energy Storage25 Ideal Capacitor ui – v characteristic for an ideal capacitor NB: Assumes we know the value of the capacitor from time (τ = -∞) until time (τ = t) Instead Initial time (often t 0 = 0) Initial condition

26. ECEN 301Discussion #10 – Energy Storage26 Ideal Capacitor uExample2: calculate the current through the capacitor C = 0.1μF with the voltage as shown: v(t) = 5(1-e -t/10 -6 )

27. ECEN 301Discussion #10 – Energy Storage27 Ideal Capacitor uExample2: calculate the current through the capacitor C = 0.1μF with the voltage as shown: v(t) = 5(1-e -t/10 -6 )

28. ECEN 301Discussion #10 – Energy Storage28 Ideal Capacitor uExample2: calculate the current through the capacitor C = 0.1μF with the voltage as shown: v(t) = 5(1-e -t/10 -6 ) NB: the capacitor’s current jumps ‘instantaneously’ to 0.5A. The ability of a capacitor’s current to change instantaneously is an important property of capacitors

29. ECEN 301Discussion #10 – Energy Storage29 Ideal Capacitor uExample3: find the voltage v(t) for a capacitor C = 0.5F with the current as shown and v(0) = 0

30. ECEN 301Discussion #10 – Energy Storage30 Ideal Capacitor uExample3: find the voltage v(t) for a capacitor C = 0.5F with the current as shown and v(0) = 0

31. ECEN 301Discussion #10 – Energy Storage31 Ideal Capacitor uExample3: find the voltage v(t) for a capacitor C = 0.5F with the current as shown and v(0) = 0

32. ECEN 301Discussion #10 – Energy Storage32 Ideal Capacitor uExample3: find the voltage v(t) for a capacitor C = 0.5F with the current as shown and v(0) = 0

33. ECEN 301Discussion #10 – Energy Storage33 Ideal Capacitor uExample3: find the voltage v(t) for a capacitor C = 0.5F with the current as shown and v(0) = 0 NB: The final value of the capacitor voltage after the current source has stopped charging the capacitor depends on two things: 1.The initial capacitor voltage 2.The history of the capacitor current

34. ECEN 301Discussion #10 – Energy Storage34 Series Capacitors uCapacitor Series Rule: two or more circuit elements are said to be in series if the current from one element exclusively flows into the next element. ÙCapacitors in series add the same way resistors in parallel add C EQ C1C1 ∙ ∙ ∙ C2C2 C3C3 CnCn CNCN

35. ECEN 301Discussion #10 – Energy Storage35 Parallel Capacitors uParallel Rule: two or more circuit elements are said to be in parallel if the elements share the same terminals ÙCapacitors in parallel add the same way resistors in series add C1C1 C2C2 C3C3 CnCn CNCN C EQ

36. ECEN 301Discussion #10 – Energy Storage36 Parallel and Series Capacitors uExample4: determine the equivalent capacitance C EQ ÙC 1 = 2mF, C 2 = 2mF, C 3 = 1mF, C 4 = 1/3mF, C 5 = 1/3mF, C 6 = 1/3mF C 1 C 2 C 3 C 4 C 5 C 6

37. ECEN 301Discussion #10 – Energy Storage37 Parallel and Series Capacitors uExample4: determine the equivalent capacitance C EQ ÙC 1 = 2mF, C 2 = 2mF, C 3 = 1mF, C 4 = 1/3mF, C 5 = 1/3mF, C 6 = 1/3mF C 1 C 2 C 3 C 4 C 5 C 6

38. ECEN 301Discussion #10 – Energy Storage38 Parallel and Series Capacitors uExample4: determine the equivalent capacitance C EQ ÙC 1 = 2mF, C 2 = 2mF, C 3 = 1mF, C 4 = 1/3mF, C 5 = 1/3mF, C 6 = 1/3mF C 1 C 2 C 3 C EQ1

39. ECEN 301Discussion #10 – Energy Storage39 Parallel and Series Capacitors uExample4: determine the equivalent capacitance C EQ ÙC 1 = 2mF, C 2 = 2mF, C 3 = 1mF, C 4 = 1/3mF, C 5 = 1/3mF, C 6 = 1/3mF C 1 C 2 C EQ2

40. ECEN 301Discussion #10 – Energy Storage40 Parallel and Series Capacitors uExample4: determine the equivalent capacitance C EQ ÙC 1 = 2mF, C 2 = 2mF, C 3 = 1mF, C 4 = 1/3mF, C 5 = 1/3mF, C 6 = 1/3mF C EQ

41. ECEN 301Discussion #10 – Energy Storage41 Energy Storage in Capacitors Capacitor energy W C (t): can be found by taking the integral of power ÙInstantaneous power P C = iv Since the capacitor is uncharged at t = -∞, v(- ∞) = 0, thus:

42. ECEN 301Discussion #10 – Energy Storage42 Ideal Inductor uInductors are made by winding a coil of wire around a core ÙThe core can be an insulator or ferromagnetic material l N turns Cross-sectional area A Diameter d μ – relative permeability + – L – Inductance

43. ECEN 301Discussion #10 – Energy Storage43 Ideal Inductor Inductance: a measure of the ability of a device to store energy in the form of a magnetic field Henry (H): unit of inductance. 1 henry = 1 volt-second/ampere (V-s/A) Ideally the resistance through an inductor is zero (i.e. no voltage drop), thus an inductor acts like a short circuit in the presence of a DC source. BUT there is an initial voltage across the inductor as the current builds up (much like ‘charging’ with capacitors) i

44. ECEN 301Discussion #10 – Energy Storage44 Ideal Inductor uInductor with an AC source: since AC sources periodically reverse current directions, the current flow through the inductor also changes ÙWith each current direction change, the current through the inductor must ‘build up’, thus there is a continual voltage drop across the inductor ÙJust as with a DC source, current across an inductor cannot change instantaneously v s (t) +–+– i –+–+ i

45. ECEN 301Discussion #10 – Energy Storage45 Ideal Inductor uI – v characteristic for an ideal inductor NB: note the duality between inductors and capacitors

46. ECEN 301Discussion #10 – Energy Storage46 Ideal Inductor ui – v characteristic for an ideal inductor NB: Assumes we know the value of the inductor from time (τ = -∞) until time (τ = t) Instead Initial time (often t 0 = 0) Initial condition

47. ECEN 301Discussion #10 – Energy Storage47 Ideal Inductor uExample5: find the voltage across an inductor L = 0.1H when the current is: i(t) = 20 t e -2t

48. ECEN 301Discussion #10 – Energy Storage48 Ideal Inductor uExample5: find the voltage across an inductor L = 0.1H when the current is: i(t) = 20 t e -2t

49. ECEN 301Discussion #10 – Energy Storage49 Ideal Inductor uExample5: find the voltage across an inductor L = 0.1H when the current is: i(t) = 20 t e -2t NB: the inductor’s voltage jumps ‘instantaneously’ to 2V. The ability of an inductor’s voltage to change instantaneously is an important property of inductors

50. ECEN 301Discussion #10 – Energy Storage50 Series Inductors uSeries Rule: two or more circuit elements are said to be in series if the current from one element exclusively flows into the next element. ÙInductors in series add the same way resistors in series add L EQ L1L1 ∙ ∙ ∙ L2L2 L3L3 LnLn LNLN

51. ECEN 301Discussion #10 – Energy Storage51 Parallel Inductors uParallel Rule: two or more circuit elements are said to be in parallel if the elements share the same terminals ÙInductors in parallel add the same way resistors in parallel add L1L1 L2L2 L3L3 LnLn LNLN L EQ

52. ECEN 301Discussion #10 – Energy Storage52 Energy Storage in Capacitors Capacitor energy W L (t): can be found by taking the integral of power ÙInstantaneous power P L = iv Since the inductor is uncharged at t = -∞, i(- ∞) = 0, thus:

53. ECEN 301Discussion #10 – Energy Storage53 Energy Storage in Capacitors uExample6: calculate the power and energy stored in a 0.1-H inductor when: Ùi = 20 t e -2t A (i = 0 for t < 0) ÙV = 2 e -2t (1- 2t)V (for t >= 0)

54. ECEN 301Discussion #10 – Energy Storage54 Energy Storage in Capacitors uExample6: calculate the power and energy stored in a 0.1-H inductor when: Ùi = 20 t e -2t A (i = 0 for t < 0) ÙV = 2 e -2t (1- 2t)V (for t >= 0)

55. ECEN 301Discussion #10 – Energy Storage55 Energy Storage in Capacitors uExample6: calculate the power and energy stored in a 0.1-H inductor when: Ùi = 20 t e -2t A (i = 0 for t < 0) ÙV = 2 e -2t (1- 2t)V (for t >= 0)

56. ECEN 301Discussion #10 – Energy Storage56 Ideal Capacitors and Inductors InductorsCapacitors Passive sign convention Voltage Current Power + L – i + C – i

57. ECEN 301Discussion #10 – Energy Storage57 Ideal Capacitors and Inductors InductorsCapacitors Energy An instantaneous change is not permitted in: CurrentVoltage Will permit an instantaneous change in: VoltageCurrent With DC source element acts as a: Short CircuitOpen Circuit