Week 3bEECS 42, Spring 2005 New topics – energy storage elements Capacitors Inductors

Week 3bEECS 42, Spring 2005 Books on Reserve for EECS 42 in Engineering Library “The Art of Electronics” by Horowitz and Hill (1 st and 2 nd editions) -- A terrific source book on electronics “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 “Electrical Engineering: Principles and Applications” by Hambley (3 rd edition) – Backup copy of text for EECS 42

Week 3bEECS 42, Spring 2005 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.) Reader

Week 3bEECS 42, Spring 2005 The Capacitor Two conductors (a,b) separated by an insulator: difference in potential = V ab => equal & opposite charge Q on conductors Q = CV ab where C is the 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 (stored charge in terms of voltage) F(F)

Week 3bEECS 42, Spring 2005 Symbol: Units: Farads (Coulombs/Volt) Current-Voltage relationship: or Note: Q (v c ) 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 C If C (geometry) is unchanging, i C = dv C /dt

Week 3bEECS 42, Spring 2005 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

Week 3bEECS 42, Spring 2005

Week 3bEECS 42, Spring 2005 Schematic Symbol and Water Model for a Capacitor

Week 3bEECS 42, Spring 2005 You might think the energy stored on a capacitor 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 for a linear capacitor. Thus, energy is. 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!) Stored Energy CAPACITORS STORE ELECTRIC ENERGY

Week 3bEECS 42, Spring vc–+vc– icic A more rigorous derivation

Week 3bEECS 42, Spring 2005 Example: Current, Power & Energy for a Capacitor –+–+ v(t)v(t) 10  F i(t)i(t) t (  s) v (V) t (  s) 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

Week 3bEECS 42, Spring w (J) –+–+ v(t)v(t) 10  F i(t)i(t) t (  s) p (W) t (  s)

Week 3bEECS 42, Spring 2005 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

Week 3bEECS 42, Spring 2005 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) – CCC eq 

Week 3bEECS 42, Spring 2005 Capacitive Voltage Divider Q: Suppose the voltage applied across a series combination of capacitors is changed by  v. How will this affect the voltage across each individual capacitor? v+vv+v C1C1 C2C2 + v 2 (t)+  v 2 – + v 1 +  v 1 – +–+– Note that no net charge can can be introduced to this node. Therefore,  Q 1 +  Q 2 =0 Q1+Q1Q1+Q1 - Q 1  Q 1 Q2+Q2Q2+Q2  Q 2  Q 2 Q1=C1v1Q1=C1v1 Q2=C2v2Q2=C2v2 Note: Capacitors in series have the same incremental charge.

Week 3bEECS 42, Spring 2005 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) MEMS = micro- electro-mechanical systems FIXED OUTER PLATES g1g1 g2g2

Week 3bEECS 42, Spring 2005 Sensing the Differential Capacitance –Begin with capacitances electrically discharged –Fixed electrodes are then charged to +V s and –V s –Movable electrode (proof mass) is then charged to V o C1C1 C2C2 VsVs –Vs–Vs VoVo Circuit model

Week 3bEECS 42, Spring 2005 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

Week 3bEECS 42, Spring 2005 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

Week 3bEECS 42, Spring 2005 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)

Week 3bEECS 42, Spring 2005 Schematic Symbol and Water Model of an Inductor

Week 3bEECS 42, Spring 2005 Stored Energy Consider an inductor having an initial current i(t 0 ) = i )( )()( )()()( 0 Li tw dptw titvtp t t      INDUCTORS STORE MAGNETIC ENERGY

Week 3bEECS 42, Spring 2005 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

Week 3bEECS 42, Spring 2005 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)–

Week 3bEECS 42, Spring 2005 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: Summary