RLC Circuit (3)

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RLC Circuit (3)

We can then write the differential equation for charge on the capacitor

The solution of this differential equation is

(damped harmonic oscillation!), where

Monday, March 31, 2014

If we charge the capacitor then hook it up to the circuit, we will observe a charge in the circuit that varies sinusoidally with time and while at the same time decreasing in amplitude

This behavior with time is illustrated below

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RLC Circuit (4)


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RLC Circuit (3)

We can then write the differential equation for charge on the capacitor

The solution of this differential equation is

(damped harmonic oscillation!), where


Now we consider a single loop circuitcontaining a capacitor, an inductor,a resistor, and a source of emf

This source of emf is capable of producinga time varying voltage as opposed to thesources of emf we have studied in previous chapters

We will assume that this source of emf provides a sinusoidal voltage as a function of time given by

where ω is the angular frequency of the emf and Vmax is the amplitude or maximum value of the emf

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Alternating Current (1)


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Series RLC Circuit (3)

The voltage phasors for an RLC circuit are shown below

The instantaneous voltages across each of the components are represented by the projections of the respective phasors on the vertical axis


Kirchhoff’s loop rules tells that the voltage drops across all the devices at any given time in the circuit must sum to zero, which gives us

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Series RLC Circuit (4)

The voltage can be thought of as the projection of the vertical axis of the phasor Vmax representing the time-varying emf in the circuit as shown below

In this figure we have replaced the sum of the two phasors VL and VC with the phasor VL - VC


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Series RLC Circuit (5): Impedance

The sum of the two phasors VL - VC and VR must equal Vmax so

Now we can put in our expression for the voltage across the components in terms of the current and resistance or reactance

We can then solve for the current in the circuit

The denominator in the equation is called the impedance

The impedance of a circuit depends on the frequency of the time-varying emf


Z =

�

R2 +

�ωL− 1

ωC

�2

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Series RLC Circuit

active resistance

reactive resistance (reactance)

Only active resistance determines losses!

impedance

Reactive resistance can be 0, at resonance


The resonant behavior of an RLC circuit resembles the response of a damped oscillator

Here we show the calculated maximum current as a function of the ratio of the angular frequency of the time varying emf divided by the resonant angular frequency, for a circuit with Vmax = 7.5 V, L = 8.2 mH, C = 100 µF, and three resistances

One can see that as theresistance is lowered, themaximum current at theresonant angular frequencyincreases and there is a morepronounced resonant peak

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Resonant Behavior of RLC Circuit


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When an RLC circuit is in operation, some of the energy in the circuit is stored in the electric field of the capacitor, some of the energy is stored in the magnetic field of the inductor, and some energy is dissipated in the form of heat in the resistor

The energy stored in the capacitor and inductor do not change in steady state operation

Therefore the energy transferred from the source of emf to the circuit is transferred to the resistor

The rate at which energy is dissipated in the resistor is the power P given by

The average power is given by

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Energy and Power in RLC Circuits (1)

since < sin2 ωt >= 1/2


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We define the root-mean-square (rms) current to be

So we can write the average power as

We can make similar definitions for other time-varying quantities• rms voltage:

• rms time-varying emf:

The currents and voltages measured by an alternating current ammeter or voltmeter are rms values


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For example, we normally say that the voltage in the wall socket is 110 V This rms voltage would correspond to a maximum voltage of

We can then re-write our formula for the current as

Which allows us to express the average power dissipated as


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For example, we normally say that the voltage in the wall socket is 110 V This rms voltage would correspond to a maximum voltage of

We can then re-write our formula for the current as

Which allows us to express the average power dissipated as


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We can relate the phase constant to the ratio of the maximum value of the voltage across the resistor divided by the maximum value of the time-varying emf

We can see that the maximum power is dissipated when φ = 0

We call cos(φ) the power factor


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We can relate the phase constant to the ratio of the maximum value of the voltage across the resistor divided by the maximum value of the time-varying emf

We can see that the maximum power is dissipated when φ = 0

We call cos(φ) the power factor

If both L =0 and C =0, or at resonance, R/Z=1. Maximal power


Applications

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Frequency filters

Response of RLC circuit strongly depends on frequency. Hi-Fi stereos have different speakers for low, middle and

high frequencies

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Impedance dominated by Xc

Impedance dominated by XL

Z =

�

R2 +

�ωL− 1

ωC

�2


Frequency filters

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C

High frequency pass filter (at low

frequencies capacitance is like a broken circuit)

Low frequency pass filter (at high

frequencies inductance is like a broken circuit)

Z =

�

R2 +

�ωL− 1

ωC

�2


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Transformers (1)

When using or generating electrical power, high currents and low voltages are desirable for convenience and safety

When transmitting electric power, high voltages and low currents are desirable• The power loss in the transmission wires goes as P = I2R

The ability to raise and lower alternating voltages would be very useful in everyday life


Transformers

Same idea as with the CD player and disconnected speakers, but with coils.

Mutual inductance depends on “linked” magnetic flux

Use metal bar to contain the magnetic flux

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Changing current in loop 1 induces varying B-field and magnetic flux through loop 1 and 2

Varying magnetic flux through loop 2 induces EMF in the loop 2.

Transformers

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To transform alternating currents and voltages from high to low one uses a transformer

A transformer consists of two sets of coils wrapped around an iron core. Huge inductor.

Consider the primary windings with NPturns connected to a source of emf

We can assume that the primary windingsact as an inductor

The current is out of phase with the voltage (for R=0)and no power is delivered to the transformer

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Transformers (2)


A transformer that takes voltages from lower to higher is called a step-up transformer and a transformer that takes voltages from higher to lower is called a step-down transformer

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Transformers (3)

Now consider the second coil with NS turns

The time-varying emf in the primary coil induces a time-varying magnetic field in the iron core

This core passes through the secondary coil


Thus a time-varying voltage is induced in the secondary coil described by Faraday’s Law

Because both the primary and secondarycoils experience the same changing magneticfield we can write

"stepped up" Ns > Np, "stepped down" Ns < Np

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Transformers (4)


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Transformers (5) If it’s just the inductor, power is 0.

If we now connect a resistor R across the secondary windings, a current will begin to flow through the secondary coil

The power in the secondary circuit is then PS = ISVS This current will induce a time-varying magnetic field that

will induce an emf in the primary coil

The emf source then will produce enough current IP to maintain the original emf

This induced current will not be 90° out of phase with the emf, thus power can be transmitted to the transformer

Energy conservation tells that the power produced by the emf source in the primary coil will be transferred to the secondary coil so we can write

R


Increase in voltage equals decrease in current

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When the secondary circuit begins to draw current, then current must be supplied to the primary circuit

We can define the current in the secondary circuit as VS = ISR

We can then write the primary current as:

With an effective primary resistance of

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Transformers (6)


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Transformers (7)

Note that these equations assume no losses in the transformers and that the load is purely resistive• Real transformers have small losses


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Let there be light!

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EM field likes to oscillate

Changing magnetic field produces electric field (Faraday’s law of induction).

Changing electric field also produces magnetic field (e.g. LC circuit).

Analogue with a swing: potential energy changes into kinetic energy, kinetic energy changes into potential energy

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Recall: electric induction

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∇×E = −∂tB

�E · dl = −

�

S

∂B

∂t· dA


Magnetic induction

Similarly, changing flux of electric field induces magnetic field

But magnetic field is also generated by currents

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�B · dl = µ0I + µ0�0

�

S

∂B

∂t· dA

Displacement current

µ0�0 = 1/c2


∇ = {∂x, ∂y, ∂z}

Maxwell’s equations

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∇ ·E = ρ/�0

∇ ·B = 0

∇×B = µ0J+ µ0�0∂tE

∇×E = −∂tB

electric charge produces E-field

no magnetic charge

electric current and changing E-field produce B-field

changing B-field also produce E-field


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�E · dA = Q/�0

�B · dA = 0

�E · dl = −

�

S

∂B

∂t· dA

�B · dl = µ0I + µ0�0

�

S

∂B

∂t· dA


Maxwell’s equations in vacuum

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∇ ·E = ρ/�0

∇ ·B = 0

∇×B = µ0J+ µ0�0∂tE

∇×E = −∂tB


Maxwell’s equations in vacuum

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∇ ·E = ρ/�0

∇ ·B = 0

∇×B = µ0J+ µ0�0∂tE

∇×E = −∂tB∇ ·E = 0

∇ ·B = 0

∇×B = µ0�0∂tE

∇×E = −∂tB


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∇ ·E = 0

∇ ·B = 0

∇×B = µ0�0∂tE

∇×E = −∂tB


∇×

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∇ ·E = 0

∇ ·B = 0

∇×B = µ0�0∂tE

∇×E = −∂tB


∇×∂t

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∇ ·E = 0

∇ ·B = 0

∇×B = µ0�0∂tE

∇×E = −∂tB


∇×∂t

∇×∇×B = µ0�0∇× ∂tE

∂t∇×E = −∂2tB

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∇ ·E = 0

∇ ·B = 0

∇×B = µ0�0∂tE

∇×E = −∂tB


µ0�0 =1

c2

∆B− 1

c2∂2tB = 0

Wave equation

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∇×∇×B = µ0�0∇× ∂tE

∂t∇×E = −∂2tB

∇×∇×B = ∇(∇ ·B)−∆B = −∆B

B-field behaves like pendulum!

∆B =�∂2x + ∂2

y + ∂2z

�B


∆B =�∂2x + ∂2

y + ∂2z

�B

By(z, t) = B0 sin(ωt− kx)

k =2π

λ

ω = kc

k2 − ω2

c2= 0

Wave equation

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∆B− 1

c2∂2tB = 0

wave length

hump is moving with velocity c

c

λx

y

z By


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B

E

- E & B oscillate (in phase)- Changing E-field generates B-field- Changing B-field generates E-field- Fixed phase (humps or troughs) propagate with speed of light- All in vacuum, no carrier (no aether)


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RLC Circuit (3)