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Physics 272 April 3 Spring 2014 http://www.phys.hawaii.edu/~philipvd/pvd_14_spring_272_uhm.html Prof. Philip von Doetinchem [email protected] Phys272 - Spring 14 - von Doetinchem - 192 Summary ● ● ● ● Flux through coil 2 is directly proportional to current in coil 1 if both coils are in vacuum Mutual inductance only depends on geometry of coils and the orientation of the coils to each other Mutual inductance is also the same for the case when coil 2 carries the current and induces a current in coil 1: Careful: – only time varying currents induce emf in second coil – Induced emf is directly proportional to rate of change of the current Phys272 - Spring 14 - von Doetinchem - 193 Summary ● Self-induction effect, elf-inductance L is defined as: ● current changes in circuit → magnetic flux changes self-induced emf after applying Faraday's law Phys272 - Spring 14 - von Doetinchem - 194 Summary ● ● Energy is stored in magnetic field in inductor like energy is stored in the electric field of a capacitor Magnetic energy density of a magnetic field: Phys272 - Spring 14 - von Doetinchem - 195 The R-L circuit ● ● ● Self-induced emf prevents rapid changes in circuit The greater the rate change of current → the greater the self-induced emf Inductors in circuits: – Use Kirchhoff's loop rule – When we go through an inductor in the same direction as the current → voltage drop (-Ldi/dt) – When we go through an inductor in the opposite direction to the current → voltage increase (Ldi/dt) Phys272 - Spring 14 - von Doetinchem - 196 The R-L circuit ● ● Closed S1: current growth based on inductor di/dt limits the rate change Closing S1 after some time time t: Phys272 - Spring 14 - von Doetinchem - 197 The R-L circuit ● ● At the instant when S1 is closed: – No current drop over resistor, i=0: – The greater L, the slower the increase – Current increases over time and approaches final current → di/dt goes to zero Final current does not depend on inductor Phys272 - Spring 14 - von Doetinchem - 198 The R-L circuit ● Solve differential equation Phys272 - Spring 14 - von Doetinchem - 199 The R-L circuit ● ● Solve differential equation For long times → current change gets slower Phys272 - Spring 14 - von Doetinchem - 200 Current decay in an R-L circuit ● ● ● If S1 is closed for a while → current reached final value → then close S2 and S1 at the same time Current goes to zero following an exponential decrease Energy stored in inductor slows down current decrease: Phys272 - Spring 14 - von Doetinchem - 201 R-L circuit ● Time constant for R-L circuit: small L current increases/decreases faster ● Compare to R-C circuit: ● Energy consideration: Phys272 - Spring 14 - von Doetinchem - 202 Example ● ● Sensitive electronics with R=175 operates at 36mA Critical: current should not rise above 4.9mA within the first 58s after turn on ● Use an inductor in series with component ● Required emf: ● Calculate required L from current increase formula in R-L circuit: Phys272 - Spring 14 - von Doetinchem - 203 The L-C circuit ● ● ● ● ● ● Start with a charged capacitor Capacitor discharges through inductor Capacitor potential equals induced emf Current increases Increasing current in inductor establishes magnetic field → energy stored in electric field of capacitor is transferred into magnetic field in inductor Once the capacitor is fully discharged the current in the circuit does not stop instantly Phys272 - Spring 14 - von Doetinchem - 207 The L-C circuit ● ● ● ● ● ● Instead it continues in the same direction for a little bit Capacitor is charging in opposite direction → current drops At the same time the magnetic field in the inductor becomes weaker Inductor provides emf in direction of current and supports charging of capacitor in opposite direction Capacitor is fully charged in opposite direction → no current flow, no magnetic field in inductor, no opposing emf Process starts the other way around → Oscillation Phys272 - Spring 14 - von Doetinchem - 208 Electrical osciallations in an L-C circuit ● ● Formal way: This is the same type of equation describing mechanical harmonic oscillations Phys272 - Spring 14 - von Doetinchem - 209 Electrical osciallations in an L-C circuit ● Solution: ● Current in LC circuit: Phys272 - Spring 14 - von Doetinchem - 210 Energy in an L-C circuit ● Energy conservation gives same result: Phys272 - Spring 14 - von Doetinchem - 211 Example ● ● A 300V has charged a 25F capacitor connected to a 10mH inductor Capacitor charge after 1.2ms: Phys272 - Spring 14 - von Doetinchem - 212 The L-R-C series circuit ● ● ● ● LC circuit is an idealization of the real world Every real circuit has a non-zero resistance value Resistance in a circuit can be regarded to in a similar way as friction in a mechanical setup In comparison to LC circuit: inductor stores less energy than initially stored in capacitor due to i2R losses in resistor Phys272 - Spring 14 - von Doetinchem - 215 Analyzing an L-R-C series circuit ● LRC circuit is oscillating (underdamped case): Phys272 - Spring 14 - von Doetinchem - 216 Analyzing an L-R-C series circuit ● LRC circuit is oscillating (underdamped case): exponential envelope ● When R becomes too large → system no longer oscillates → starts when value under the square root becomes negative Phys272 - Spring 14 - von Doetinchem - 217 L-C in parallel ● Follow a similar idea like in the easy case with L and C in series → use Kirchhoff's law Phys272 - Spring 14 - von Doetinchem - 218