LCR

Lecture 22 1

Electromagnetic Oscillations

C q U_E

2

2

 2

2 1

Li U_B _C

q U_E

2

2

 2

2 1

Li U_B

December 5, 2007

Oscillating Quantities

• _{We will write oscillating quantities with a lower-case symbol, and the}

corresponding amplitude of the oscillation with upper case.

• _Examples:

Oscillating Quantity Amplitude

Voltage v V

Current i I

Charge q Q

) cos( 

 Q t

q

) (

cos 2

2

2 2

2

  

 t

C Q C

q

dt t d

I dt

di cos( )

Derivation of Oscillation Frequency

• _{We have shown qualitatively that LC circuits act like an oscillator.}

• _{We can discover the frequency of oscillation by looking at the equations}

governing the total energy.

• _{Since the total energy is constant, the time derivative should be zero:}

• _{But and , so making these substitutions:}

• _{This is a second-order, homogeneous differential equation, whose solution}

is

• _{i.e. the charge varies according to a cosine wave with amplitude Q and}

frequency . Check by taking two time derivatives of charge:

• _{Plug into original equation:}

2 2

2 1 2C Li

q U

U

U  _E  _B  

0    dt di Li dt dq C q dt dU 2 2 dt q d dt

di _{ 2} 0

2   C q dt q d L ) cos( 

 Q t

q

) sin( 

  

 Q t

dt dq ) cos( 2 2 2     

 Q t

dt q d 0 ) cos( ) cos( 2 2 2      

    t 

C Q t LQ C q dt q d

L _ 2 _ 1 _ 0

C L LC 1   dt dq i 

December 5, 2007

1. The expressions below give the charge on a capacitor

in an LC circuit. Choose the one that will have the

greatest maximum current?

A.

q

= 2 cos 4

t

B.

q

= 2 cos(4

t+

p

/2)

C. q

= 2 sin

t

D. q

= 4 cos 4

t

E.

q

= 2 sin 5

t

2. The three circuits below have identical inductors and

capacitors. Rank the circuits according to the time

taken to fully discharge the capacitor during an

oscillation, greatest first.

A. I, II, III.

B. II, I, III.

C. III, I, II.

D. III, II, I.

E.

II, III, I.

Time to Discharge Capacitor

Charge, Current & Energy Oscillations

• _{The solution to the equation is , which gives the}

charge oscillation.

• _{From this, we can determine the corresponding oscillation of current:}

• _{And energy}

• _{But recall that , so .} • _{That is why our graph for the energy oscillation}

had the same amplitude for both U_E and U_B.

• Note that

) cos( 

 Q t

q 0 2 2   C q dt q d L ) sin( 

 

 

 Q t

dt dq i ) ( cos 2 2 2 2 2      t C Q C q

U_E sin ( )

2 1 2

1 2 _ 2_2 2 _{ }

 Li LQ t

U_B

LC

1 

 _{sin ( )}

2 2 2     t C Q U_B C Q t t C Q U

U_{E B}

2 )] ( sin ) ( [cos 2 2 2 2 2          

Constant

Damped Oscillations



Recall that all circuits have at least a little bit of

resistance.



_{In this general case, we really have an RLC}

circuit, where the oscillations get smaller with

time. They are said to be “damped oscillations.”

Damped Oscillations



Then the power equation becomes

R i dt di Li dt dq C q dt

dU _{    2}

Power lost due to resistive heating



As before, substituting and

gives the differential equation for

q

dt dq

i  ₂

2 dt q d dt di _ 0 2 2    C q dt dq R dt q d L 2 2 _(R/2L)

    ) cos( 2 / _ _

_Qe _t

q Rt L

December 5, 2007

3. How does the resonant frequency



for an ideal LC

circuit (no resistance) compare with



’

for a

non-ideal one where resistance cannot be ignored?

A. The resonant frequency for the non-ideal circuit is

higher

than for the ideal one (



’ >



).

B. The resonant frequency for the non-ideal circuit is

lower

than

for the ideal one (



’ <



).

C. The resistance in the circuit does not affect the resonant

frequency—they are the same (



’ =



).

Alternating Current

•

_{The electric power out of a home or office power socket is in the form of}

alternating current (AC), as opposed to the direct current (DC) of a battery.

•

_{Alternating current is used because it is easier to transport, and easier to}

“transform” from one voltage to another using a transformer.

•

_{In the U.S., the frequency of oscillation of AC is 60 Hz. In most other countries}

it is 50 Hz.



The figure at right shows one way to make an

alternating current by rotating a coil of wire in

a magnetic field. The slip rings and brushes

allow the coil to rotate without twisting the

connecting wires. Such a device is called a

generator

.



It takes power to rotate the coil, but that power

can come from moving water (a water turbine),

or air (windmill), or a gasoline motor (as in

your car), or steam (as in a nuclear power

plant).

t

d

m 



December 5, 2007

RLC Circuits with AC Power

• _{When an RLC circuit is driven with an AC}

power source, the “driving” frequency is the frequency of the power source, while the

circuit can have a different “resonant” frequency .

• _{Let’s look at three different circuits driven by}

an AC EMF. The device connected to the EMF is called the “load.”

• _{What we are interested in is how the voltage}

oscillations across the load relate to the current oscillations.

• _{We will find that the “phase” relationships}

change, depending on the type of load (resistive, capacitive, or inductive).

d



2

) 2 / ( /

1 LC R L

 

A Resistive Load

• _{Phasor Diagram: shows the}

instantaneous phase of either voltage or current.

• _{For a resistor, the current follows}

the voltage, so the voltage and current are in phase (  0).

• _If • _Then



t R

V t I

i R _d

d R

R  sin  sin

t V

December 5, 2007

4.

The plot below shows the current and voltage oscillations in a

purely resistive circuit. Below that are four curves. Which

color curve best represents the power dissipated in the

resistor?

A. The green curve (straight line).

B. The blue curve.

C. The black curve.

D. The red curve.

E.

None are correct.

Power in a Resistive Circuit

P

• _{For a capacitive load, the voltage across the capacitor is}

proportional to the charge

• _{But the current is the time derivative of the charge}

• _{In analogy to the resistance, which is the proportionality}

constant between current and voltage, we define the “capacitive reactance” as

• _{So that .}

• _{The phase relationship is that}   90_{º, and current leads}

voltage.

A Capacitive Load

t X

V

i _d

C C

C  cos

C X

d

C _

1



t C

Q C

q

v_C   sin_d

t CV

dt dq

December 5, 2007

An Inductive Load

• _{For an inductive load, the voltage across the inductor is}

proportional to the time derivative of the current

• _{But the current is the time derivative of the charge}

• _{Again in analogy to the resistance, which is the}

proportionality constant between current and voltage, we define the “inductive reactance” as

• _{So that .}

• _{The phase relationship is that}   90_{º, and current lags}

voltage.

t X

V

i _d

L L

L   cos

L X_L _d dt

di L

v L

L 

t L

V dt

t L

V

i _d

d L d

L

L sin _ cos

  

    



5.

We just learned that capacitive reactance is and

inductive reactance is . What are the units of

reactance?

A. Seconds per coulomb.

B. Henry-seconds.

C. Ohms.

D. Volts per Amp.

E.

The two reactances have different units.

Units of Reactance

L X_L _d

C X

d

C _

1

December 5, 2007

Summary Table

Circuit

Element Symbol Resistance or Reactance Phase of Current ConstantPhase Amplitude Relation

Resistor R R In phase

with v_R 0º (0 rad) VR=IRR

Capacitor C _XC=1/_dC Leads vR

by 90º

90º (p/2) VC=ICXC

Inductor L X_L=_dL Lags v_R by

90º

December 5, 2007

Summary

• _{Energy in inductor:}

• _{LC circuits: total electric + magnetic energy is conserved} • _{LC circuit:}

• _{LRC circuit:}

• _{Resistive, capacitive, inductive}

2

2 1

Li

U_B _{Energy in magnetic field}

2 2

2 1 2C Li

q U

U

U  _E  _B  

) cos( 

 Q t

q

LC

1





Charge equationCurrent equationOscillation frequency

) sin( 

  

 Q t

i

Charge equation

Oscillation frequency

2 2 _(R/2L)

    ) cos( 2 / _{ }

_Qe _t

q Rt L

t R

V t I

i R _d

d R

R  sin  sin _X t

V

i _d

C C

C  cos t

X V

i _d

L L

L   cos

C X d C _ 1  L X_L _d R

X_R 

Lecture 22 19

Resonant Circuits

•

_{Resonant frequency}

_{: the frequency at which the}

impedance of a series RLC circuit or the admittance

of a parallel RLC circuit is purely real, i.e., the

imaginary term is zero (ωL=1/ωC)

•

_{For both series and parallel RLC circuits, the}

resonance frequency is

•

_{At resonance the voltage and current are in phase,}

(i.e., zero phase angle) and the power factor is unity

C

L

1

0





Quality Factor (Q)

•

_{An energy analysis of a RLC circuit provides a basic}

definition of the

quality factor

(Q) that is used across

engineering disciplines, specifically:

•

_{The quality factor is a measure of the sharpness of}

the resonance peak; the larger the Q value, the

sharper the peak

where

BW

=bandwidth

Cycle

per

Dissipated

Energy

at

Stored

Energy

Max

W

W

Q

D

S

₂

2

p



p





BW

Q





Lecture 22 21

Series RLC Circuit

•

_{For a series RLC circuit the quality factor is}

C

L

R

CR

R

L

Q

BW

Q

1

1

0

0















Parallel RLC Circuit

•

_{For a parallel RLC circuit, the quality factor is}

L

C

R

CR

L

R

Q

BW

Q











0

0





