New topics – energy storage elements Capacitors Inductors

1. New topics – energy storage elements Capacitors Inductors

2. Books on Reserve for EECS 42 in Engineering Library “The Art of Electronics” by Horowitz and Hill (1st and 2nd editions) -- A terrific source book on electronics “Electrical Engineering Uncovered” by White and Doering (2nd edition) – Freshman intro to aspects of engineering and EE in particular ”Newton’s Telecom Dictionary: The authoritative resource for Telecommunications” by Newton (“18th edition – he updates it annually) – A place to find definitions of all terms and acronyms connected with telecommunications “Electrical Engineering: Principles and Applications” by Hambley (3rd edition) – Backup copy of text for EECS 42

3. Reader The EECS 42 Supplementary Reader is now available at Copy Central, 2483 Hearst Avenue (price: $12.99) It contains selections from two textbooks that we will use when studying semiconductor devices: Microelectronics: An Integrated Approach (by Roger Howe and Charles Sodini) Digital Integrated Circuits: A Design Perspective (by Jan Rabaey et al.)

4. The Capacitor Two conductors (a,b) separated by an insulator: difference in potential = Vab => equal & opposite charge Q on conductors Q = CVab where C is the capacitance of the structure, • positive (+) charge is on the conductor at higher potential (stored charge in terms of voltage) • Parallel-plate capacitor: • area of the plates = A (m2) • separation between plates = d (m) • dielectric permittivity of insulator =  (F/m) • => capacitance F (F)

5. C Symbol: Units: Farads (Coulombs/Volt) Current-Voltage relationship: C or C Electrolytic (polarized) capacitor (typical range of values: 1 pF to 1 mF; for “supercapa- citors” up to a few F!) ic + vc – If C (geometry) is unchanging, iC = dvC/dt Note: Q (vc) must be a continuous function of time

6. Voltage in Terms of Current; Capacitor Uses Uses: Capacitors are used to store energy for camera flashbulbs, in filters that separate various frequency signals, and they appear as undesired “parasitic” elements in circuits where they usually degrade circuit performance

8. Schematic Symbol and Water Model for a Capacitor

9. Thus, energy is . Stored Energy You might think the energy stored on a capacitor is QV= CV2, which has the dimension of Joules. But during charging, the average voltage across the capacitor was only half the final value of V for a linear capacitor. CAPACITORS STORE ELECTRIC ENERGY Example: A 1 pF capacitance charged to 5 Volts has ½(5V)2 (1pF) = 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!)

10. A more rigorous derivation ic + vc –

11. Example: Current, Power & Energy for a Capacitor i(t) v (V) v(t) – + 10 mF 1 t (ms) 0 1 2 3 4 5 vc and q must be continuous functions of time; however, ic can be discontinuous. i (mA) t (ms) 0 1 2 3 4 5 Note: In “steady state” (dc operation), time derivatives are zero  C is an open circuit

12. p (W) i(t) v(t) – + 10 mF t (ms) 0 1 2 3 4 5 w (J) t (ms) 0 1 2 3 4 5

13. Capacitors in Parallel i1(t) i2(t) + v(t) – i(t) C1 C2 + v(t) – i(t) Ceq Equivalent capacitance of capacitors in parallel is the sum

14. 1 1 1 = + C C C 1 2 eq Capacitors in Series + v1(t) – + v2(t) – + v(t)=v1(t)+v2(t) – C1 C2 i(t) i(t) Ceq

15. Capacitive Voltage Divider Q: Suppose the voltage applied across a series combination of capacitors is changed by Dv. How will this affect the voltage across each individual capacitor? DQ1=C1Dv1 Note that no net charge can can be introduced to this node. Therefore, -DQ1+DQ2=0 Q1+DQ1 + v1+Dv1 – C1 -Q1-DQ1 v+Dv + – Q2+DQ2 + v2(t)+Dv2 – C2 -Q2-DQ2 DQ2=C2Dv2 Note: Capacitors in series have the same incremental charge.

16. Application Example: MEMS Accelerometerto deploy the airbag in a vehicle collision • Capacitive MEMS position sensor used to measure acceleration (by measuring force on a proof mass) MEMS = micro- • electro-mechanical systems g1 g2 FIXED OUTER PLATES

17. Sensing the Differential Capacitance • Begin with capacitances electrically discharged • Fixed electrodes are then charged to +Vs and –Vs • Movable electrode (proof mass) is then charged to Vo Circuit model Vs C1 Vo C2 –Vs

18. Practical Capacitors • 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

19. 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: LiL • When the current changes, the magnetic flux changes  a voltage across the coil is induced: + vL(t) iL _ Note: In “steady state” (dc operation), time derivatives are zero  L is a short circuit

20. Symbol: Units: Henrys (Volts • second / Ampere) Current in terms of voltage: L (typical range of values: mH to 10 H) iL + vL – Note: iL must be a continuous function of time

21. Schematic Symbol and Water Model of an Inductor

22. Stored Energy INDUCTORS STORE MAGNETIC ENERGY Consider an inductor having an initial current i(t0) = i0 = = p ( t ) v ( t ) i ( t ) t ò = t t = w ( t ) p ( ) d t 0 1 1 2 = - 2 w ( t ) Li Li 0 2 2

23. Inductors in Series + v1(t) – + v2(t) – + v(t)=v1(t)+v2(t) – L1 L2 i(t) i(t) + – + – v(t) v(t) Leq Equivalent inductance of inductors in series is the sum

24. Inductors in Parallel + v(t) – + v(t) – i1 i2 i(t) i(t) Leq L1 L2

25. Summary • 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 (-> infinite voltage!) • n ind.’s in series: • n ind.’s in parallel: