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Physics 272 April 8 Spring 2014 http://www.phys.hawaii.edu/~philipvd/pvd_14_spring_272_uhm.html Prof. Philip von Doetinchem philipvd@hawaii.edu Phys272 - Spring 14 - von Doetinchem - 218 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 - 219 L-C in parallel ● ● We assume that the system is oscillating with a special frequency Differential equation systems with the oscillations can be solved with an exponential approach Phys272 - Spring 14 - von Doetinchem - 220 L-C in parallel ● Now we have system of non-linear equations ● Let's write it down in matrix form: Phys272 - Spring 14 - von Doetinchem - 221 L-C in parallel ● No general solution exists, only for a special choice of when the determinant of the matrix is zero: Phys272 - Spring 14 - von Doetinchem - 222 L-C in parallel ● ● If the term under the square root is smaller than zero the system is oscillating This is driven by the choice of L,R,C Phys272 - Spring 14 - von Doetinchem - 223 L-C in parallel ● Right after closing the switch: Phys272 - Spring 14 - von Doetinchem - 224 L-C in parallel ● ● ● Use the following substitutions: Technically any linear combination of sine and cosine solutions to our problem are allowed before taking the initial conditions into account For charge in the capacitor we know that the charge is 0 after closing the switch (t=0): Phys272 - Spring 14 - von Doetinchem - 225 L-C in parallel ● ● Using our initial condition: Use this result to calculate the current in the inductor (Kirchhoff's loop rule): Phys272 - Spring 14 - von Doetinchem - 226 Review ● Mutual inductance ● Self-inductance ● Magnetic field energy ● RL circuit Phys272 - Spring 14 - von Doetinchem - 227 Review ● ● ● LC circuit LRC circuit is a realistic approach where inductors and wire have a non-zero resistance The resistance in this case is similar to friction in mechanics Phys272 - Spring 14 - von Doetinchem - 228 Alternating current ● ● ● Electric power distribution uses alternating current (AC) Transformer can easily be used to step voltage up and down High voltages with low currents are used for longdistance power transmission to keep i2R losses small Phys272 - Spring 14 - von Doetinchem - 230 Root-Mean-Square (rms) values ● ● ● Averaging a sinusoidal current is not very useful → average value is 0 Rectified average current is the average of the absolute current |I cost|: Another way of describing the alternating current is the root-mean-square value: Phys272 - Spring 14 - von Doetinchem - 231 Resistor in an AC circuit ● Current and voltage have both the same dependence on cosine: – When current is at maximum → voltage is at maximum – Current and voltage amplitudes are related in the same way as in a DC circuit Phys272 - Spring 14 - von Doetinchem - 232 Resistor in an AC circuit ● Current and voltage have both the same dependence on cosine: – When current is at maximum → voltage is at maximum – Current and voltage amplitudes are related in the same way as in a DC circuit Phys272 - Spring 14 - von Doetinchem - 233 Inductor in an AC circuit ● ● ● Ideal inductor with zero resistance Potential difference is not caused by dissipation of energy in wire, but by self-induced emf Voltage across the conductor is proportional to rate change Phys272 - Spring 14 - von Doetinchem - 234 Inductor in an AC circuit ● Voltage peaks occur a quarter cycle earlier → voltage leads the current by 90deg ● Inductive reactance: ● Be careful: current and voltage are out of phase Phys272 - Spring 14 - von Doetinchem - 235 The meaning of inductive reactance ● ● ● XL is description of the self-induced emf that opposes any change in current through a conductor More rapid variation in current increases inductive reactance High frequency voltages gives only small currents compared to lower-frequency voltages – This can be used to block high frequency noise Phys272 - Spring 14 - von Doetinchem - 236 An inductor in an AC circuit ● Current amplitude in a pure inductor in a radio receiver is 250A with voltage amplitude 3.6V at frequency 1.6MHz ● What inductance is needed: ● Change of current with different frequencies: Phys272 - Spring 14 - von Doetinchem - 237 Capacitor in an AC circuit ● Capacitor constantly charges and discharges in AC circuit – Current into one plate and equal current out of other plate – Equal displacement current between plates → effectively we can say that alternating current is going through the capacitor Phys272 - Spring 14 - von Doetinchem - 238 Capacitor in an AC circuit ● Current has greatest magnitude when the voltage is rising or falling most steeply ● Voltage lags the current by 90deg ● Capacitive reactance: ● Also here: voltage and current are out of phase Phys272 - Spring 14 - von Doetinchem - 239 The meaning of capacitive reactance ● ● With smaller frequency the capacitive reactance becomes higher Capacitors tend to pass high frequency current and to block low frequencies (opposite to inductors) Phys272 - Spring 14 - von Doetinchem - 240 A resistor and a capacitor in an AC circuit ● 200 resistor in series with a 5.0F capacitor ● Voltage across resistor is 1.2Vcos(2500Hz t) ● Current in circuit: ● Capacitive reactance: Phys272 - Spring 14 - von Doetinchem - 241 A resistor and a capacitor in an AC circuit ● 200W resistor in series with a 5.0mF capacitor ● Voltage across resistor is 1.2Vcos(2500Hz t) ● Voltage across capacitor ● Same current passes through resistor and capacitor, but voltages are different in amplitude and phase Phys272 - Spring 14 - von Doetinchem - 242 Comparing ac circuit elements ● ● ● ● Resistor shows no phase difference between voltage and current Inductors and capacitors have +/-90deg phase differences between voltage and current Resistance does not depend on the frequency Inductive and capacitive reactances depend on frequency Phys272 - Spring 14 - von Doetinchem - 243 Comparing ac circuit elements ● For →0: – alternating current case goes over into DC case: ● ● ● no current through capacitor no inductive effect For →∞: – current in inductor goes to zero – voltage across capacitor becomes zero (no charge build up) Phys272 - Spring 14 - von Doetinchem - 244 Crossover network for loudspeaker Phys272 - Spring 14 - von Doetinchem - 245 The L-R-C series circuit ● ● Instantaneous total voltage vad across all three components is equal to the source voltage Elements are connected in series → current at any instant is the same at every point in the circuit Phys272 - Spring 14 - von Doetinchem - 246 The L-R-C series circuit Phys272 - Spring 14 - von Doetinchem - 247 The L-R-C series circuit ● For any network of resistors, inductors, capacitors: impedance is defined as the ratio of: amplitude voltage/current amplitude Phys272 - Spring 14 - von Doetinchem - 248 The meaning of impedance and phase angle ● ● ● ● Impedance is similar to Ohm's law V=IR in DC circuits Impedance Z plays the role of R for AC circuit Alternating currents tend to follow path of least resistance Important for sinusoidally varying voltages: – Relations also work when replacing maximum voltages by RMS values Phys272 - Spring 14 - von Doetinchem - 249 The meaning of impedance and phase angle ● ● Impedance depends on R, L, C and In addition to impedance the phase angle between voltage and current is important C Phys272 - Spring 14 - von Doetinchem - 250 The meaning of impedance and phase angle ● L=0: ● C=0: ● R=0: ● L=0, C=0: Phys272 - Spring 14 - von Doetinchem - 251 Example for an L-R-C circuit ● LRC series circuit with R=300, L=60mH, C=0.5F sinusoidal voltage with amplitude voltage V=50V at =10,000rad/s ● Impedance: ● Amplitude current and phase angle: voltage leads current by 53deg Phys272 - Spring 14 - von Doetinchem - 252 Example for an L-R-C circuit ● ● ● ● LRC series circuit with R=300, L=60mH, C=0.5F sinusoidal voltage with amplitude voltage V=50V at =10,000rad/s Voltages across components: The total voltage of 50V is not equal to the scalar sum of the individual voltages! Vector sum has to be used Phys272 - Spring 14 - von Doetinchem - 253 Power in alternating-current circuits ● ● ● Alternating currents are very important for distributing and converting electric energy For a particular moment in time, the power delivered to a circuit element is: Resistor: – Voltage and current are in phase → energy is always supplied (product of V and I always positive) – Power is not constant, average power is: Phys272 - Spring 14 - von Doetinchem - 254 Power in alternating-current circuits ● Power in an inductor: – ● Voltage leads current by 90deg → average power supplied is zero (no energy transfer over one cycle) Power in a capacitor: – Voltage lags the current by 90deg → average power supplied is zero Phys272 - Spring 14 - von Doetinchem - 255 Power in a general AC circuit ● Power factor cos is important to determine how much current has to be drawn for a given voltage difference ● Drawing more current is undesirable: i2R losses increase ● Ideal inductors and capacitors do not absorb net power from the line ● Lagging current can be corrected for with capacitors in parallel to increase power factor Phys272 - Spring 14 - von Doetinchem - 256

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