1. Week 3b. Prof. WhiteEE 42 and 100, Spring 20061 Books on Reserve for EE 42 and 100 in the Bechtel Engineering Library “The Art of Electronics” by Horowitz and Hill (2 nd edition) -- A terrific source book on practical electronics (also a copy in 140 Cory lab bookcase) “Electrical Engineering Uncovered” by White and Doering (2 nd edition) – Freshman intro to aspects of engineering and EE in particular ”Newton’s Telecom Dictionary: The authoritative resource for Telecommunications” by Newton (18 th edition – he updates it annually) – A place to find definitions of all terms and acronyms connected with telecommunications. TK5102.N486 Shelved with dictionaries to right of entry gate. Announcements

2. Week 3b. Prof. WhiteEE 42 and 100, Spring 20062 New topics – energy storage elements Capacitors Inductors

3. Week 3b. Prof. WhiteEE 42 and 100, Spring 20063 The Capacitor Two conductors (a,b) separated by an insulator: difference in potential = V ab => equal & opposite charges Q on conductors Q = CV ab where C is the so-called capacitance of the structure,  positive (+) charge is on the conductor at higher potential Parallel-plate capacitor: area of the plates = A (m 2 ) separation between plates = d (m) dielectric permittivity of insulator =  (F/m) => capacitance Q = Magnitude of charge stored on each conductor F V ab + - +Q -Q (F)

4. Week 3b. Prof. WhiteEE 42 and 100, Spring 20064

5. Week 3b. Prof. WhiteEE 42 and 100, Spring 20065 Symbol: Units: Farads (Coulombs/Volt) Current-Voltage relationship: or Note: Q(t) must be a continuous function of time +vc–+vc– icic C C (typical range of values: 1 pF to 1  F; for “supercapa- citors” up to a few F!) Electrolytic (polarized) capacitor has + sign on one plate If C (geometry) is unchanging, i C = dv C /dt

6. Week 3b. Prof. WhiteEE 42 and 100, Spring 20066

7. Week 3b. Prof. WhiteEE 42 and 100, Spring 20067 A capacitor can be constructed by interleaving the plates with two dielectric layers and rolling them up, to achieve a compact size. To achieve a small volume, a very thin dielectric with a high dielectric constant is desirable. However, dielectric materials break down and become conductors when the electric field (units: V/cm) is too high. –Real capacitors have maximum voltage ratings –An engineering trade-off exists between compact size and high voltage rating Practical Capacitors

8. Week 3b. Prof. WhiteEE 42 and 100, Spring 20068 Schematic Symbol and Water Model for a Capacitor

9. Week 3b. Prof. WhiteEE 42 and 100, Spring 20069 Capacitor Uses Capacitors are used to: store energy for camera flashbulbs; in filters that separate signals having different fre- quencies in resonant circuits to tune a radio and oscillators that generate a time-varying voltage at a desired frequency; Capacitors also appear as undesired “parasitic” elements in circuits where they usually degrade circuit perfor- mance (example, conductors on printed circuit boards

10. Week 3b. Prof. WhiteEE 42 and 100, Spring 200610 Capacitors used in MEMS Airbag Deployment Accelerometer (MEMS = MicroElectroMechanical Systems) Chip about 1 cm 2 holding in the middle an electromechanical accelerometer around which are electronic test and calibration circuits (Analog Devices, Inc.) Hundreds of millions have been sold. Airbag of car that crashed into the back of a stopped Mercedes. Within 0.3 seconds after deceleration the bag is supposed to be empty. Driver was not hurt in any way; chassis distortion meant that this car was written off.

11. Week 3b. Prof. WhiteEE 42 and 100, Spring 200611 Application Example: MEMS Accelerometer to deploy the airbag in a vehicle collision Capacitive MEMS position sensor used to measure acceleration (by measuring force on a proof mass) FIXED OUTER PLATES g1g1 g2g2

12. Week 3b. Prof. WhiteEE 42 and 100, Spring 200612 Application: Condenser Microphone V out =  x E const

13. Week 3b. Prof. WhiteEE 42 and 100, Spring 200613 Capacitor Voltage in Terms of Current Charge is integral of current through capacitor and also equals capacitance C time capacitor voltage:

14. Week 3b. Prof. WhiteEE 42 and 100, Spring 200614 You might think the energy stored on a capacitor charged to voltage V is QV = CV 2, which has the dimension of Joules. But during charging, the average voltage across the capacitor was only half the final value of V Thus, energy is. Example: The energy stored in a 1 pF capacitance charged to 5 Volts equals ½ (1pF) (5V) 2 = 12.5 pJ (A 5F supercapacitor charged to 5 volts stores 63 J; if it discharged at a constant rate in 1 ms, energy is discharged at a 63 kW rate!) Stored Energy CAPACITORS STORE ELECTRIC ENERGY

15. Week 3b. Prof. WhiteEE 42 and 100, Spring 200615 +vc–+vc– icic A more rigorous derivation

16. Week 3b. Prof. WhiteEE 42 and 100, Spring 200616 Example: Current, Power & Energy for a Capacitor –+–+ v(t)v(t) 10  F i(t)i(t) t (s) v (V) 023451 t (s) 0 23451 1 i (  A) v c and q must be continuous functions of time; however, i c can be discontinuous. Note: In “steady state” (dc operation), time derivatives are zero  C is an open circuit

17. Week 3b. Prof. WhiteEE 42 and 100, Spring 200617 023451 w (J) –+–+ v(t)v(t) 10  F i(t)i(t) t (s) 0 23451 p (W) t (s)

18. Week 3b. Prof. WhiteEE 42 and 100, Spring 200618 Capacitors in Parallel i(t)i(t) +v(t)–+v(t)– C1C1 C2C2 i1(t)i1(t)i2(t)i2(t) i(t)i(t) +v(t)–+v(t)– C eq Equivalent capacitance of capacitors in parallel is the sum

19. Week 3b. Prof. WhiteEE 42 and 100, Spring 200619 Capacitors in Series i(t)i(t) C1C1 + v 1 (t) – i(t)i(t) + v(t)=v 1 (t)+v 2 (t) – C eq C2C2 + v 2 (t) – 21 111 CCC eq 

20. Week 3b. Prof. WhiteEE 42 and 100, Spring 200620 The Inductor An inductor is constructed by coiling a wire around some type of form. Current flowing through the coil creates a magnetic field and a magnetic flux that links the coil: Li L When the current changes, the magnetic flux changes  a voltage across the coil is induced: iLiL vL(t)vL(t) + _ Note: In “steady state” (dc operation), time derivatives are zero  L is a short circuit

21. Week 3b. Prof. WhiteEE 42 and 100, Spring 200621 Symbol: Units: Henrys (Volts second / Ampere) Current in terms of voltage: Note: i L must be a continuous function of time +vL–+vL– iLiL L (typical range of values:  H to 10 H)

22. Week 3b. Prof. WhiteEE 42 and 100, Spring 200622 Schematic Symbol and Water Model of an Inductor

23. Week 3b. Prof. WhiteEE 42 and 100, Spring 200623 Stored Energy Consider an inductor having an initial current i(t 0 ) = i 0 2 0 2 2 1 2 1 )( )()( )()()( 0 Li tw dptw titvtp t t      INDUCTORS STORE MAGNETIC ENERGY

24. Week 3b. Prof. WhiteEE 42 and 100, Spring 200624 Inductors in Series v(t)v(t) L1L1 + v 1 (t) – v(t)v(t) + v(t)=v 1 (t)+v 2 (t) – L eq L2L2 + v 2 (t) – +–+– +–+– i(t)i(t) i(t)i(t) Equivalent inductance of inductors in series is the sum

25. Week 3b. Prof. WhiteEE 42 and 100, Spring 200625 L1L1 i(t)i(t) i2i2 i1i1 Inductors in Parallel L2L2 +v(t)–+v(t)– L eq i(t)i(t) +v(t)–+v(t)–

26. Week 3b. Prof. WhiteEE 42 and 100, Spring 200626 Capacitor v cannot change instantaneously i can change instantaneously Do not short-circuit a charged capacitor (-> infinite current!) n cap.’s in series: n cap.’s in parallel: Inductor i cannot change instantaneously v can change instantaneously Do not open-circuit an inductor with current flowing (-> infinite voltage!) n ind.’s in series: n ind.’s in parallel: Summary q = Cv C

27. Week 3b. Prof. WhiteEE 42 and 100, Spring 200627 Transformer – Two Coupled Inductors N 1 turns N 2 turns + - v1v1 + v2v2 - |v 2 |/|v 1 | = N 2 /N 1 See Hambley pp. 712-4 on Ideal Transformer

28. Week 3b. Prof. WhiteEE 42 and 100, Spring 200628 AC Power System

29. Week 3b. Prof. WhiteEE 42 and 100, Spring 200629 Relative advantages of HVDC over HVAC power transmission Asynchronous interconnections (e.g., 50 Hz to 60 Hz system) Environmental – smaller footprint, can put in underground cables more economically,... Economical -- cheapest solution for long distances, smaller loss on same size of conductor (skin effect), terminal equipment cheaper Power flow control (bi-directional on same set of lines) Added benefits to the transmission (system stability, power quality, etc.)

30. Week 3b. Prof. WhiteEE 42 and 100, Spring 200630 * Highest DC voltage system: +/- 600 kV, in Brazil – brings 50 Hz power from 12,600 MW Itaipu hydropower plant to 60 Hz network in Sao Paulo High-voltage DC power systems

31. Week 3b. Prof. WhiteEE 42 and 100, Spring 200631 QuantityVariableUnitUnit Symbol Typical Values Defining Relations Important Equations Symbol ChargeQcoulombC1aC to 1Cmagnitude of 6.242 × 10 18 electron charges q e = - 1.602x10 -19 C i = dq/dt CurrentIampereA 1  A to 1kA 1A = 1C/s VoltageVvoltV 1  V to 500kV 1V = 1N-m/C Summary of Electrical Quantities

32. Week 3b. Prof. WhiteEE 42 and 100, Spring 200632 PowerPwattW 1  W to 100MW 1W = 1J/sP = dU/dt; P=IV EnergyUjouleJ1fJ to 1TJ1J = 1N-mU = QV ForceFnewtonN 1N = 1kg- m/s 2 Timetseconds ResistanceRohm  1  to 10M  V = IR; P = V 2 /R = I 2 R R CapacitanceCfaradF1fF to 5FQ = CV; i = C(dv/dt);U = (1/2)CV 2 C InductanceLhenryH 1  H to 1H v = L(di/dt); U = (1/2)LI 2 L Summary of Electrical Quantities (concluded)