Induction - Spring Time Varying Circuits April 10, 2006

Induction - Spring What is going on? There are only 7 more classes and the final is 3 weeks away. There are only 7 more classes and the final is 3 weeks away. Scotty, beam me somewhere else! Scotty, beam me somewhere else! Exam Issues Exam Issues Look Ashamed! Look Ashamed! Inductor Circuits Inductor Circuits Quiz Friday Quiz Friday AC Next week & Following Monday AC Next week & Following Monday

Induction - Spring Second Problem a h Both Currents are going into the page. B   Other Wire

Induction - Spring First Problem VV

Induction - Spring The Last two Problems were similar to WebAssigns that were also reviewed in class. Circular Arc – Easy Biot-Savart Moving Rod

Induction - Spring Question What about these problems was “unfair”? Why so many blank or completely wrong pages?

Induction - Spring And Now ….. From the past

Induction - Spring Max Current Rate of increase = max emf V R =iR ~current

Induction - Spring Solve the loop equation.

Induction - Spring We also showed that

Induction - Spring LR Circuit i Steady Source

Induction - Spring Time Dependent Result:

Induction - Spring RLRL

Induction - Spring At t=0, the charged capacitor is now connected to the inductor. What would you expect to happen??

Induction - Spring The math … For an RLC circuit with no driving potential (AC or DC source):

Induction - Spring The Graph of that LR (no emf) circuit..

Induction - Spring

Induction - Spring Mass on a Spring Result Energy will swap back and forth. Add friction  Oscillation will slow down  Not a perfect analogy

Induction - Spring

Induction - Spring LC Circuit High Q/C Low High

Induction - Spring The Math Solution (R=0):

Induction - Spring New Feature of Circuits with L and C These circuits produce oscillations in the currents and voltages Without a resistance, the oscillations would continue in an un-driven circuit. With resistance, the current would eventually die out.

Induction - Spring Variable Emf Applied emf Sinusoidal DC

Induction - Spring Sinusoidal Stuff “Angle” Phase Angle

Induction - Spring Same Frequency with PHASE SHIFT 

Induction - Spring Different Frequencies

Induction - Spring Note – Power is delivered to our homes as an oscillating source (AC)

Induction - Spring Producing AC Generator x x x x x x x x x x x x x x x x x x x x x x x

Induction - Spring The Real World

Induction - Spring A

Induction - Spring

Induction - Spring The Flux:

Induction - Spring April 12, 2006

Induction - Spring Schedule Today Today Finish Inductors Finish Inductors Friday Friday Quiz on this weeks material Quiz on this weeks material Some problems and then AC circuits Some problems and then AC circuits Monday Monday Last FULL week of classes Last FULL week of classes Following Monday is last day of class Following Monday is last day of class FINAL IS LOOMING! FINAL IS LOOMING!

Induction - Spring Some Problems

Induction - Spring Calculate the resistance in an RL circuit in which L = 2.50 H and the current increases to 90.0% of its final value in 3.00 s.

Induction - Spring Show that I = I 0 e – t/τ is a solution of the differential equation where τ = L/R and I 0 is the current at t = 0.

Induction - Spring Consider the circuit in Figure P32.17, taking ε = 6.00 V, L = 8.00 mH, and R = 4.00 Ω. (a) What is the inductive time constant of the circuit? (b) Calculate the current in the circuit 250 μs after the switch is closed. (c) What is the value of the final steady-state current? (d) How long does it take the current to reach 80.0% of its maximum value?

Induction - Spring In the circuit shown in Figure P32.17, let L = 7.00 H, R = 9.00 Ω, and ε = 120 V. What is the self-induced emf s after the switch is closed?

Induction - Spring A 140-mH inductor and a 4.90-Ω resistor are connected with a switch to a 6.00-V battery as shown in Figure P (a) If the switch is thrown to the left (connecting the battery), how much time elapses before the current reaches 220 mA? (b) What is the current in the inductor 10.0 s after the switch is closed? (c) Now the switch is quickly thrown from a to b. How much time elapses before the current falls to 160 mA?

Induction - Spring At t = 0, an emf of 500 V is applied to a coil that has an inductance of H and a resistance of 30.0 Ω. (a) Find the energy stored in the magnetic field when the current reaches half its maximum value. (b) After the emf is connected, how long does it take the current to reach this value?

Induction - Spring The switch in Figure P32.52 is connected to point a for a long time. After the switch is thrown to point b, what are (a) the frequency of oscillation of the LC circuit, (b) the maximum charge that appears on the capacitor, (c) the maximum current in the inductor, and (d) the total energy the circuit possesses at t = 3.00 s?

Induction - Spring Back to Variable Sources

Induction - Spring Source Voltage:

Induction - Spring Average value of anything: Area under the curve = area under in the average box T h

Induction - Spring Average Value For AC:

Induction - Spring So … Average value of current will be zero. Power is proportional to i 2 R and is ONLY dissipated in the resistor, The average value of i 2 is NOT zero because it is always POSITIVE

Induction - Spring Average Value

Induction - Spring RMS

Induction - Spring Usually Written as:

Induction - Spring Example: What Is the RMS AVERAGE of the power delivered to the resistor in the circuit: E R ~

Induction - Spring Power

Induction - Spring More Power - Details

Induction - Spring AC Circuits April 17, 2006

Induction - Spring Last Days … If you need to, file your taxes TODAY! If you need to, file your taxes TODAY! Due at midnight. Due at midnight. Note: File on Web has been updated. Note: File on Web has been updated. This week This week Monday & Wednesday – AC Circuits followed by problem based review Monday & Wednesday – AC Circuits followed by problem based review Friday – Review problems Next Week Friday – Review problems Next Week Monday – Complete Problem review. Monday – Complete Problem review.

Induction - Spring Final Examination Will contain 8-10 problems. One will probably be a collection of multiple choice questions. Will contain 8-10 problems. One will probably be a collection of multiple choice questions. Problems will be similar to WebAssign problems. Class problems may also be a source. Problems will be similar to WebAssign problems. Class problems may also be a source. You have 3 hours for the examination. You have 3 hours for the examination. SCHEDULE: MONDAY, MAY 10:00 AM SCHEDULE: MONDAY, MAY 10:00 AM ar/exam/ ar/exam/ ar/exam/ ar/exam/

Induction - Spring Back to AC

Induction - Spring Resistive Circuit We apply an AC voltage to the circuit. Ohm’s Law Applies

Induction - Spring Consider this circuit CURRENT AND VOLTAGE IN PHASE

Induction - Spring

Induction - Spring Alternating Current Circuits  is the angular frequency (angular speed) [radians per second]. Sometimes instead of  we use the frequency f [cycles per second] Frequency  f [cycles per second, or Hertz (Hz)]  f V = V P sin (  t -  v ) I = I P sin (  t -  I ) An “AC” circuit is one in which the driving voltage and hence the current are sinusoidal in time.  vv  V(t) tt VpVp -V p

Induction - Spring  vv  V(t) tt VpVp -V p V = VP sin (wt -  v ) Phase Term

Induction - Spring V p and I p are the peak current and voltage. We also use the “root-mean-square” values: V rms = V p / and I rms =I p /  v and  I are called phase differences (these determine when V and I are zero). Usually we’re free to set  v =0 (but not  I ). Alternating Current Circuits V = V P sin (  t -  v ) I = I P sin (  t -  I )  vv  V(t) tt VpVp -V p V rms I/I/ I(t) t IpIp -Ip-Ip I rms

Induction - Spring Example: household voltage In the U.S., standard wiring supplies 120 V at 60 Hz. Write this in sinusoidal form, assuming V(t)=0 at t=0.

Induction - Spring Example: household voltage In the U.S., standard wiring supplies 120 V at 60 Hz. Write this in sinusoidal form, assuming V(t)=0 at t=0. This 120 V is the RMS amplitude: so V p =V rms = 170 V.

Induction - Spring Example: household voltage In the U.S., standard wiring supplies 120 V at 60 Hz. Write this in sinusoidal form, assuming V(t)=0 at t=0. This 120 V is the RMS amplitude: so V p =V rms = 170 V. This 60 Hz is the frequency f: so  =2  f=377 s -1.

Induction - Spring Example: household voltage In the U.S., standard wiring supplies 120 V at 60 Hz. Write this in sinusoidal form, assuming V(t)=0 at t=0. This 120 V is the RMS amplitude: so V p =V rms = 170 V. This 60 Hz is the frequency f: so  =2  f=377 s -1. So V(t) = 170 sin(377t +  v ). Choose  v =0 so that V(t)=0 at t=0: V(t) = 170 sin(377t).

Induction - Spring Review: Resistors in AC Circuits E R ~ EMF (and also voltage across resistor): V = V P sin (  t) Hence by Ohm’s law, I=V/R: I = (V P /R) sin(  t) = I P sin(  t) (with I P =V P /R) V and I “In-phase” V tt  I 

Induction - Spring This looks like I P =V P /R for a resistor (except for the phase change). So we call X c = 1/(  C) the Capacitive Reactance Capacitors in AC Circuits E ~ C Start from: q = C V [V=V p sin(  t)] Take derivative: dq/dt = C dV/dt So I = C dV/dt = C V P  cos (  t) I = C  V P sin (  t +  /2) The reactance is sort of like resistance in that I P =V P /X c. Also, the current leads the voltage by 90 o (phase difference). V tt   I V and I “out of phase” by 90º. I leads V by 90º.

Induction - Spring I Leads V??? What the does that mean?? I V Current reaches it’s maximum at an earlier time than the voltage! 1 2 I = C  V P sin (  t +  /2)  Phase= - (  /2)

Induction - Spring Capacitor Example E ~ C A 100 nF capacitor is connected to an AC supply of peak voltage 170V and frequency 60 Hz. What is the peak current? What is the phase of the current? Also, the current leads the voltage by 90o (phase difference). I=V/X C

Induction - Spring Again this looks like I P =V P /R for a resistor (except for the phase change). So we call X L =  L the Inductive Reactance Inductors in AC Circuits L V = V P sin (  t) Loop law: V +V L = 0 where V L = -L dI/dt Hence: dI/dt = (V P /L) sin(  t). Integrate: I = - (V P / L  cos (  t) or I = [V P /(  L)] sin (  t -  /2) ~ Here the current lags the voltage by 90 o. V tt   I V and I “out of phase” by 90º. I lags V by 90º.

Induction - Spring

Induction - Spring Phasor Diagrams VpVp IpIp  t Resistor A phasor is an arrow whose length represents the amplitude of an AC voltage or current. The phasor rotates counterclockwise about the origin with the angular frequency of the AC quantity. Phasor diagrams are useful in solving complex AC circuits. The “y component” is the actual voltage or current. A phasor is an arrow whose length represents the amplitude of an AC voltage or current. The phasor rotates counterclockwise about the origin with the angular frequency of the AC quantity. Phasor diagrams are useful in solving complex AC circuits. The “y component” is the actual voltage or current.

Induction - Spring Phasor Diagrams VpVp IpIp  t VpVp IpIp ResistorCapacitor A phasor is an arrow whose length represents the amplitude of an AC voltage or current. The phasor rotates counterclockwise about the origin with the angular frequency of the AC quantity. Phasor diagrams are useful in solving complex AC circuits. The “y component” is the actual voltage or current. A phasor is an arrow whose length represents the amplitude of an AC voltage or current. The phasor rotates counterclockwise about the origin with the angular frequency of the AC quantity. Phasor diagrams are useful in solving complex AC circuits. The “y component” is the actual voltage or current.

Induction - Spring Phasor Diagrams VpVp IpIp  t VpVp IpIp VpVp IpIp ResistorCapacitor Inductor A phasor is an arrow whose length represents the amplitude of an AC voltage or current. The phasor rotates counterclockwise about the origin with the angular frequency of the AC quantity. Phasor diagrams are useful in solving complex AC circuits. The “y component” is the actual voltage or current. A phasor is an arrow whose length represents the amplitude of an AC voltage or current. The phasor rotates counterclockwise about the origin with the angular frequency of the AC quantity. Phasor diagrams are useful in solving complex AC circuits. The “y component” is the actual voltage or current.

Induction - Spring Steady State Solution for AC Current (2) Expand sin & cos expressions Collect sin  d t & cos  d t terms separately These equations can be solved for I m and  (next slide) High school trig! cos  d t terms sin  d t terms

Induction - Spring Steady State Solution for AC Current (2) Expand sin & cos expressions Collect sin  d t & cos  d t terms separately These equations can be solved for I m and  (next slide) High school trig! cos  d t terms sin  d t terms

Induction - Spring Solve for  and I m in terms of R, X L, X C and Z have dimensions of resistance Let’s try to understand this solution using “phasors” Steady State Solution for AC Current (3) Inductive “reactance” Capacitive “reactance” Total “impedance”

Induction - Spring REMEMBER Phasor Diagrams? VpVp IpIp  t VpVp IpIp VpVp IpIp ResistorCapacitor Inductor A phasor is an arrow whose length represents the amplitude of an AC voltage or current. The phasor rotates counterclockwise about the origin with the angular frequency of the AC quantity. Phasor diagrams are useful in solving complex AC circuits. A phasor is an arrow whose length represents the amplitude of an AC voltage or current. The phasor rotates counterclockwise about the origin with the angular frequency of the AC quantity. Phasor diagrams are useful in solving complex AC circuits.

Induction - Spring Reactance - Phasor Diagrams VpVp IpIp  t VpVp IpIp VpVp IpIp ResistorCapacitor Inductor

Induction - Spring “Impedance” of an AC Circuit R L C ~ The impedance, Z, of a circuit relates peak current to peak voltage: (Units: OHMS)

Induction - Spring “Impedance” of an AC Circuit R L C ~ The impedance, Z, of a circuit relates peak current to peak voltage: (Units: OHMS) (This is the AC equivalent of Ohm’s law.)

Induction - Spring Impedance of an RLC Circuit R L C ~ E As in DC circuits, we can use the loop method: E - V R - V C - V L = 0 I is same through all components.

Induction - Spring Impedance of an RLC Circuit R L C ~ E As in DC circuits, we can use the loop method: E - V R - V C - V L = 0 I is same through all components. BUT: Voltages have different PHASES  they add as PHASORS.

Induction - Spring Phasors for a Series RLC Circuit IpIp V Rp (V Cp - V Lp ) VPVP  V Cp V Lp

Induction - Spring Phasors for a Series RLC Circuit By Pythagoras’ theorem: (V P ) 2 = [ (V Rp ) 2 + (V Cp - V Lp ) 2 ] IpIp V Rp (V Cp - V Lp ) VPVP  V Cp V Lp

Induction - Spring Phasors for a Series RLC Circuit By Pythagoras’ theorem: (V P ) 2 = [ (V Rp ) 2 + (V Cp - V Lp ) 2 ] = I p 2 R 2 + (I p X C - I p X L ) 2 IpIp V Rp (V Cp - V Lp ) VPVP  V Cp V Lp

Induction - Spring Impedance of an RLC Circuit Solve for the current: R L C ~

Induction - Spring Impedance of an RLC Circuit Solve for the current: Impedance: R L C ~

Induction - Spring The circuit hits resonance when 1/  C-  L=0:  r =1/ When this happens the capacitor and inductor cancel each other and the circuit behaves purely resistively: I P =V P /R. Impedance of an RLC Circuit The current’s magnitude depends on the driving frequency. When Z is a minimum, the current is a maximum. This happens at a resonance frequency:  The current dies away at both low and high frequencies. rr L=1mH C=10  F

Induction - Spring Phase in an RLC Circuit IpIp V Rp (V Cp - V Lp ) VPVP  V Cp V Lp We can also find the phase: tan  = (V Cp - V Lp )/ V Rp or; tan  = (X C -X L )/R. or tan  = (1/  C -  L) / R

Induction - Spring Phase in an RLC Circuit At resonance the phase goes to zero (when the circuit becomes purely resistive, the current and voltage are in phase). IpIp V Rp (V Cp - V Lp ) VPVP  V Cp V Lp We can also find the phase: tan  = (V Cp - V Lp )/ V Rp or; tan  = (X C -X L )/R. or tan  = (1/  C -  L) / R More generally, in terms of impedance: cos  R/Z

Induction - Spring Power in an AC Circuit V(t) = V P sin (  t) I(t) = I P sin (  t) P(t) = IV = I P V P sin 2 (  t) Note this oscillates twice as fast. V tt   I tt  P  = 0 (This is for a purely resistive circuit.)

Induction - Spring The power is P=IV. Since both I and V vary in time, so does the power: P is a function of time. Power in an AC Circuit Use, V = V P sin (  t) and I = I P sin (  t+  ) : P(t) = I p V p sin(  t) sin (  t+  ) This wiggles in time, usually very fast. What we usually care about is the time average of this: (T=1/f )

Induction - Spring Power in an AC Circuit Now:

Induction - Spring Power in an AC Circuit Now:

Induction - Spring Power in an AC Circuit Use: and: So Now:

Induction - Spring Power in an AC Circuit Use: and: So Now: which we usually write as

Induction - Spring Power in an AC Circuit  goes from to 90 0, so the average power is positive) cos(  is called the power factor. For a purely resistive circuit the power factor is 1. When R=0, cos(  )=0 (energy is traded but not dissipated). Usually the power factor depends on frequency.