New topics energy storage elements Capacitors Inductors EE

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New topics – energy storage elements Capacitors Inductors EE 42 and 100, Fall 2005

New topics – energy storage elements Capacitors Inductors EE 42 and 100, Fall 2005 Week 3 b 1

Books on Reserve for EE 42 and 100 in the Bechtel Engineering Library “The

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 “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. TK 5102. N 486 Shelved with dictionaries to right of entry gate. EE 42 and 100, Fall 2005 Week 3 b 2

The Capacitor Two conductors (a, b) separated by an insulator: difference in potential =

The Capacitor Two conductors (a, b) separated by an insulator: difference in potential = Vab => equal & opposite charges Q on conductors +Q -Q + - Vab Q = CVab (stored charge in terms of voltag where C is the capacitance of the structure, Ø 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 EE 42 and 100, Fall 2005 (F) F Week 3 b 3

Symbol: C or C C Electrolytic (polarized) capacitor Units: Farads (Coulombs/Volt) (typical range of

Symbol: C or C C Electrolytic (polarized) capacitor Units: Farads (Coulombs/Volt) (typical range of values: 1 p. F to 1 m. F; for “supercapacitors” up to a few F!) Current-Voltage relationship: ic If C (geometry) is unchanging, i. C = dv. C/dt + vc – Note: Q (vc) must be a continuous function of time EE 42 and 100, Fall 2005 Week 3 b 4

Voltage in Terms of Current; Capacitor Uses: Capacitors are used to store energy for

Voltage in Terms of Current; Capacitor 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 EE 42 and 100, Fall 2005 Week 3 b 5

EE 42 and 100, Fall 2005 Week 3 b 6

EE 42 and 100, Fall 2005 Week 3 b 6

Schematic Symbol and Water Model for a Capacitor EE 42 and 100, Fall 2005

Schematic Symbol and Water Model for a Capacitor EE 42 and 100, Fall 2005 Week 3 b 7

Stored Energy CAPACITORS STORE ELECTRIC ENERGY You might think the energy stored on a

Stored Energy CAPACITORS STORE ELECTRIC ENERGY 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 p. F capacitance charged to 5 Volts has ½(5 V)2 (1 p. F) = 12. 5 p. J (A 5 F supercapacitor charged to 5 volts stores 63 J; if it discharged at a constant rate in 1 ms energy is discharged at a 63 k. W rate!) EE 42 and 100, Fall 2005 Week 3 b 8

A more rigorous derivation ic EE 42 and 100, Fall 2005 Week 3 b

A more rigorous derivation ic EE 42 and 100, Fall 2005 Week 3 b + vc – 9

Example: Current, Power & Energy for a Capacitor i(t) v(t) 1 0 1 i

Example: Current, Power & Energy for a Capacitor i(t) v(t) 1 0 1 i (m. A) 0 1 2 3 4 5 – + v (V) 10 m. F t (ms) vc and q must be continuous functions of time; however, ic can be discontinuous. 2 EE 42 and 100, Fall 2005 3 4 5 Week 3 b t (ms) Note: In “steady state” (dc operation), time derivatives are zero C is an open circuit 10

p (W) i(t) 0 1 2 3 4 5 – + v(t) 10 m.

p (W) i(t) 0 1 2 3 4 5 – + v(t) 10 m. F t (ms) w (J) 0 1 2 EE 42 and 100, Fall 2005 3 4 5 Week 3 b t (ms) 11

Capacitors in Parallel i(t) i 1(t) i 2(t) + C 1 C 2 v(t)

Capacitors in Parallel i(t) i 1(t) i 2(t) + C 1 C 2 v(t) – + i(t) Ceq v(t) – Equivalent capacitance of capacitors in parallel is the sum EE 42 and 100, Fall 2005 Week 3 b 12

Capacitors in Series + v 1(t) – + v 2(t) – i(t) C 1

Capacitors in Series + v 1(t) – + v 2(t) – i(t) C 1 C 2 + i(t) Ceq v(t)=v 1(t)+v 2(t) – 1 1 1 = + Ceq C 1 C 2 EE 42 and 100, Fall 2005 Week 3 b 13

Capacitive Voltage Divider Q: Suppose the voltage applied across a series combination of capacitors

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? DQ 1=C 1 Dv 1 Q 1+DQ 1 C 1 v+Dv + – -Q 1 -DQ 1 Q 2+DQ 2 C 2 -Q 2 -DQ 2 + v 1+Dv 1 – Note that no net charge can be introduced to this node. Therefore, -DQ 1+DQ 2=0 + v 2(t)+Dv 2 – DQ 2=C 2 Dv 2 Note: Capacitors in series have the same incremental charge. EE 42 and 100, Fall 2005 Week 3 b 14

MEMS Airbag Deployment Accelerometer Chip about 1 cm 2 holding in the middle an

MEMS Airbag Deployment Accelerometer 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. EE 42 and 100, Fall 2005 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. Week 3 b 15

Application Example: MEMS Accelerometer to deploy the airbag in a vehicle collision • Capacitive

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 g 1 g 2 FIXED OUTER PLATES EE 42 and 100, Fall 2005 Week 3 b 16

Sensing the Differential Capacitance – Begin with capacitances electrically discharged – Fixed electrodes are

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 C 1 Vo C 2 –Vs EE 42 and 100, Fall 2005 Week 3 b 17

Application: Condenser Microphone EE 42 and 100, Fall 2005 Week 3 b 18

Application: Condenser Microphone EE 42 and 100, Fall 2005 Week 3 b 18

Practical Capacitors • A capacitor can be constructed by interleaving the plates with two

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 EE 42 and 100, Fall 2005 Week 3 b 19

The Inductor • An inductor is constructed by coiling a wire around some type

The Inductor • An inductor is constructed by coiling a wire around some type of form. + v. L(t) i. L _ • 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: Note: In “steady state” (dc operation), time derivatives are zero L is a short circuit EE 42 and 100, Fall 2005 Week 3 b 20

Symbol: L Units: Henrys (Volts • second / Ampere) (typical range of values: m.

Symbol: L Units: Henrys (Volts • second / Ampere) (typical range of values: m. H to 10 H) Current in terms of voltage: i. L + v. L – Note: i. L must be a continuous function of time EE 42 and 100, Fall 2005 Week 3 b 21

Schematic Symbol and Water Model of an Inductor EE 42 and 100, Fall 2005

Schematic Symbol and Water Model of an Inductor EE 42 and 100, Fall 2005 Week 3 b 22

Stored Energy INDUCTORS STORE MAGNETIC ENERGY Consider an inductor having an initial current i(t

Stored Energy INDUCTORS STORE MAGNETIC ENERGY Consider an inductor having an initial current i(t 0) = i 0 p(t ) = v(t )i(t ) = t w(t ) = ò p(t )dt = t 0 1 2 w(t ) = Li - Li 0 2 2 EE 42 and 100, Fall 2005 Week 3 b 23

Inductors in Series + v 1(t) – + v 2(t) – v(t) + –

Inductors in Series + v 1(t) – + v 2(t) – v(t) + – L 1 i(t) L 2 v(t) + – i(t) Leq + v(t)=v 1(t)+v 2(t) – Equivalent inductance of inductors in series is the sum EE 42 and 100, Fall 2005 Week 3 b 24

Inductors in Parallel i 1 i(t) L 1 + + i 2 v(t) L

Inductors in Parallel i 1 i(t) L 1 + + i 2 v(t) L 2 Leq v(t) – – EE 42 and 100, Fall 2005 i(t) Week 3 b 25

Summary Capacitor Inductor v cannot change instantaneously i can change instantaneously Do not short-circuit

Summary Capacitor Inductor 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: EE 42 and 100, Fall 2005 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: Week 3 b 26

Transformer – Two Coupled Inductors + + v 1 v 2 - - N

Transformer – Two Coupled Inductors + + v 1 v 2 - - N 1 turns N 2 turns |v 2|/|v 1| = N 2/N 1 EE 42 and 100, Fall 2005 Week 3 b 27

AC Power System EE 42 and 100, Fall 2005 Week 3 b 28

AC Power System EE 42 and 100, Fall 2005 Week 3 b 28

High-Voltage Direct-Current Power Transmission http: //www. worldbank. org/html/fpd/em/transmission/technology_abb. pdf Highest voltage +/- 600 k.

High-Voltage Direct-Current Power Transmission http: //www. worldbank. org/html/fpd/em/transmission/technology_abb. pdf Highest voltage +/- 600 k. V, in Brazil – brings 50 Hz power from 12, 600 MW Itaipu hydropower plant to 60 Hz network in Sao Paulo EE 42 and 100, Fall 2005 Week 3 b 29

Relative advantages of HVDC over HVAC power transmission • Asynchronous interconnections (e. g. ,

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. ) EE 42 and 100, Fall 2005 Week 3 b 30

Summary of Electrical Quantities Quantity Variable Unit Symbol Typical Values Defining Relations Charge Q

Summary of Electrical Quantities Quantity Variable Unit Symbol Typical Values Defining Relations Charge Q coulomb C 1 a. C to 1 C magnitude of 6. 242 × 1018 electron charges qe = 1. 602 x 10 -19 C Current I ampere A 1 m. A to 1 k. A 1 A = 1 C/s Voltage V volt V 1 m. V to 500 k. V 1 V = 1 N-m/C EE 42 and 100, Fall 2005 Week 3 b Important Equations Symbol i = dq/dt 31

Summary of Electrical Quantities (concluded) Power P watt W 1 m. W to 100

Summary of Electrical Quantities (concluded) Power P watt W 1 m. W to 100 MW 1 W = 1 J/s P = d. U/dt; P=IV Energy U joule J 1 f. J to 1 TJ 1 J = 1 N-m U = QV Force F newton N 1 N = 1 kgm/s 2 Time t second s Resistance R ohm W 1 W to 10 MW V = IR; P = V 2/R = I 2 R R Capacitanc e C farad F 1 f. F to 5 F Q = CV; i = C(dv/dt); U = (1/2)CV 2 C Inductance L henry H 1 m. H to 1 H v= L(di/dt); U = (1/2)LI 2 L EE 42 and 100, Fall 2005 Week 3 b 32