Energy-Storage Elements Capacitance and Inductance ELEC 308 Elements of Electrical Engineering Dr. Ron Hayne Images Courtesy of Allan Hambley and Prentice-Hall Energy-Storage Elements Remember Resistors convert electrical energy into heat Cannot store energy! Inductors and Capacitors can store energy and later return it to the circuit Do NOT generate energy! Passive elements, like resistors Capacitance is a circuit property that accounts for energy STORED in ELECTRIC fields Inductance is a circuit property that accounts for energy STORED in MAGNETIC fields ELEC 308 2 Inductance and Capacitance Uses Microphones Capacitance changes with sound pressure Linear variable differential transformer Position of moving iron core converted into voltage Conversion from DC-AC, AC-DC, AC-AC Electrical signal filters Combinations of inductances and capacitances in special circuits ELEC 308 3 Capacitors Constructed by separating two sheets of CONDUCTOR (usually metallic) by a thin layer of INSULATING material Insulating material called a DIELECTRIC Can be air, Mylar®, polyester, polypropylene, mica, etc. Parallel-plate Capacitor: ELEC 308 4 Fluid-Flow Analogy ELEC 308 5 Stored Charge in Terms of Voltage In an IDEAL capacitor Stored charge, q, is proportional to the voltage between the plates: q Cv Constant of proportionality is the capacitance, C Units are farads (F) Units equivalent to Coulombs per volt Farad is a VERY LARGE amount of capacitance Usually deal with capacitances from 1 pF to 0.01 F Occasionally, use femtofarads (in computer chips) ELEC 308 6 Current in Terms of Voltage Remember that current is the time rate of flow of charge In an IDEAL capacitor The relationship between current and voltage is dq d dv i Cv C dt dt dt ELEC 308 dv(t ) i (t ) C dt 7 Example 3.1 Plot the current vs. time ELEC 308 8 Stored Energy in a Capacitor Remember: pt v t it dv For an ideal capacitor: pt Cv dt For an ideal, uncharged capacitor (v(t0) = 0): wt 1 Cv 2 t 2 ELEC 308 9 Example 3.3 Plot current, power delivered and energy stored ELEC 308 10 Capacitances in Parallel ELEC 308 11 Capacitances in Series ELEC 308 12 Parallel-Plate Capacitors ELEC 308 13 Parallel-Plate Capacitors If d<<W and d<<L, the capacitance is approx. A WL C d d where ε is the dielectric constant of the material BETWEEN the plates For vacuum, the dielectric constant is 0 8.85 1012 F/m For other materials, r0 where εr is the relative dielectric constant See Table 3.1 on page 135 of textbook ELEC 308 14 Practical Capacitors Dimensions of 1μF parallel-plate capacitors are TOO LARGE for portable electronic devices Plates are rolled into smaller area Small-volume capacitors require very thin dielectrics (with HIGH dielectric constant) Dielectric materials break down when electric field intensity is TOO HIGH (become conductors) Real capacitors have MAXIMUM VOLTAGE RATING ELEC 308 15 Electrolytic Capacitors One plate is metallic aluminum or tantalum Dielectric is OXIDE layer on surface of the metal Other “plate” is ELECTROLYTIC SOLUTION Metal plate is immersed in the electrolytic solution Gives high capacitance per unit volume Requires that ONLY ONE polarity of voltage can be applied ELEC 308 16 Inductors Constructed by coiling a wire around some type of form ELEC 308 17 Voltage in Terms of Current In an IDEAL inductor Voltage across the coil is proportional to the time rate of change of the current Constant of proportionality is the inductance, L Units are henries (H) Units equivalent to volt-seconds per amperes Usually deal with inductances from 0.001μH to 100 H ELEC 308 18 Stored Energy in an Inductor Remember: pt v t it di For an ideal inductor: pt Lit dt For an ideal inductor with i(t0) = 0: 1 2 w t Li t 2 ELEC 308 19 Example 3.6 Plot voltage, power, and energy ELEC 308 20 Equivalent Inductance ELEC 308 21 Practical Inductors Cores (metallic iron forms) are made of thin sheets called laminations Voltages are induced in the core by the changing magnetic fields Cause eddy currents to flow in the core Dissipate energy Results in UNDESIRABLE core loss Can reduce eddy-current core loss Laminations Ferrite (iron oxide) cores Powdered iron with insulating binder ELEC 308 22 Electronic Photo Flash ELEC 308 23 Mutual Inductance Several coils wound on the same form Magnetic flux produced by one coil links the others Time-varying current flowing through one coil induces voltages on the other coils ELEC 308 24 Mutual Inductance Flux of one coil aids the flux produced by the other coil ELEC 308 25 Ideal Transformers v2 (t ) V2 rms N2 v1 (t ) N1 N2 V1rms N1 ELEC 308 26 Ideal Transformers ELEC 308 27 Power Transmission Losses Power Line Losses 2 Ploss RlineI rms Large Voltages and Small Currents Smaller Line Loss ELEC 308 28 Power Transmission Step-Up and Step-Down Transformers 99% Efficiency (vs. 50% with no transformers) ELEC 308 29 U.S. Power Grid ELEC 308 30 Summary Capacitance Inductance Voltage Current Power Energy Series Parallel ELEC 308 Voltage Current Power Energy Series Parallel Mutual Inductance Transformers 31