The Ups and Downs of Circuits The End

The Ups and Downs of Circuits

The End is Near! • Quiz – Nov 18 th – Material since last quiz. (Induction) • Exam #3 – Nov 23 rd – WEDNESDAY • LAST CLASS – December 2 nd • FINAL EXAM – 12/5 10: 00 -12: 50 Room MAP 359 • Grades by end of week. Hopefully Maybe.

A circular region in the xy plane is penetrated by a uniform magnetic field in the positive direction of the z axis. The field's magnitude B (in teslas) increases with time t (in seconds) according to B = at, where a is a constant. The magnitude E of the electric field set up by that increase in the magnetic field is given in the Figure as a function of the distance r from the center of the region. Find a. [0. 030] T/s r VG

For the next problem, recall that i

R L

For the circuit of Figure 30 -19, assume that = 11. 0 V, R = 6. 00 W , and L = 5. 50 H. The battery is connected at time t = 0. 6 W 5. 5 H (a) How much energy is delivered by the battery during the first 2. 00 s? [23. 9] J (b) How much of this energy is stored in the magnetic field of the inductor? [7. 27] J (c) How much of this energy is dissipated in the resistor? [16. 7] J

Let’s put an inductor and a capacitor in the SAME circuit.

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

Current would begin to flow…. Energy Density in Capacitor Low High Low Energy Flows from Capacitor to the Inductor’s Magnetic Field

Energy Flow Energy

LC Circuit Low High Low

When t=0, i=0 so B=0 When t=0, voltage across the inductor = Q 0/C

The Math Solution:

Energy

Inductor

The Capacitor

Add ‘em Up …

Add Resistance

Actual RLC:

New Feature of Circuits with L and C n n These circuits can produce oscillations in the currents and voltages Without a resistance, the oscillations would continue in an un-driven circuit. With resistance, the current will eventually die out. q The frequency of the oscillator is shifted slightly from its “natural frequency” q The total energy sloshing around the circuit decreases exponentially There is ALWAYS resistance in a real circuit!

Types of Current n Direct Current q n Create New forms of life Alternating Current q Let there be light

Alternating emf DC Sinusoidal

Sinusoidal Stuff “Angle” Phase Angle

Same Frequency with PHASE SHIFT f

Different Frequencies

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

Producing AC Generator xxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxx

The Real World

A

The Flux:

OUTPUT WHAT IS AVERAGE VALUE OF THE EMF ? ?

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

Average Value For AC:

So … n n n 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

Average Value

RMS

Usually Written as:

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

Power

More Power - Details

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

Consider this circuit CURRENT AND VOLTAGE IN PHASE

Alternating Current Circuits An “AC” circuit is one in which the driving voltage and hence the current are sinusoidal in time. V(t) Vp fv p 2 p wt V = VP sin (wt - fv ) I = IP sin (wt - f. I ) -Vp w is the angular frequency (angular speed) [radians per second]. Sometimes instead of w we use the frequency f [cycles per second] Frequency f [cycles per second, or Hertz (Hz)] w = 2 p f

Phase Term V = VP sin (wt - fv ) V(t) Vp p fv -Vp 2 p wt

Alternating Current Circuits V = VP sin (wt - fv ) I = IP sin (wt - f. I ) I(t) Vp Vrms fv -Vp p 2 p wt Ip Irms f. I/w t -Ip Vp and Ip are the peak current and voltage. We also use the “root-mean-square” values: Vrms = Vp / and Irms=Ip / fv and f. I are called phase differences (these determine when V and I are zero). Usually we’re free to set fv=0 (but not f. I).

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.

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 Vp=Vrms = 170 V.

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 Vp=Vrms = 170 V. This 60 Hz is the frequency f: so w=2 p f=377 s -1.

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 Vp=Vrms = 170 V. This 60 Hz is the frequency f: so w=2 p f=377 s -1. So V(t) = 170 sin(377 t + fv). Choose fv=0 so that V(t)=0 at t=0: V(t) = 170 sin(377 t).

Resistors in AC Circuits R E ~ EMF (and also voltage across resistor): V = VP sin (wt) Hence by Ohm’s law, I=V/R: I = (VP /R) sin(wt) = IP sin(wt) (with IP=VP/R) V I p 2 p wt V and I “In-phase”

Capacitors in AC Circuits C Start from: q = C V [V=Vpsin(wt)] Take derivative: dq/dt = C d. V/dt So I = C d. V/dt = C VP w cos (wt) E ~ I = C w VP sin (wt + p/2) V I p 2 p wt This looks like IP=VP/R for a resistor (except for the phase change). So we call Xc = 1/(w. C) the Capacitive Reactance The reactance is sort of like resistance in that IP=VP/Xc. Also, the current leads the voltage by 90 o (phase difference). V and I “out of phase” by 90º. I leads V by 90º.

I Leads V? ? ? What the **(&@ does that mean? ? 2 V f 1 I I = C w VP sin (wt + p/2) Current reaches it’s maximum at an earlier time than the voltage!

Capacitor Example A 100 n. F capacitor is connected to an AC supply of peak voltage 170 V and frequency 60 Hz. C E ~ What is the peak current? What is the phase of the current? Also, the current leads the voltage by 90 o (phase difference).

Inductors in AC Circuits ~ L V = VP sin (wt) Loop law: V +VL= 0 where VL = -L d. I/dt Hence: d. I/dt = (VP/L) sin(wt). Integrate: I = - (VP / Lw) cos (wt) or V Again this looks like IP=VP/R for a resistor (except for the phase change). I p I = [VP /(w. L)] sin (wt - p/2) 2 p wt So we call the XL = w L Inductive Reactance Here the current lags the voltage by 90 o. V and I “out of phase” by 90º. I lags V by 90º.

Phasor Diagrams 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. Resistor Ip Vp wt

Phasor Diagrams 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. Resistor Ip Vp Capacitor Ip wt wt Vp

Phasor Diagrams 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. Resistor Ip Vp Capacitor Inductor Vp Ip wt wt Vp

i + + time i i LC Circuit i i + + +

Analyzing the L-C Circuit Total energy in the circuit: Differentiate : N o change in energy

Analyzing the L-C Circuit Total energy in the circuit: Differentiate : N o change in energy

Analyzing the L-C Circuit Total energy in the circuit: Differentiate : N o change in energy

Analyzing the L-C Circuit Total energy in the circuit: Differentiate : N o change in energy The charge sloshes back and forth with frequency w = (LC)-1/2