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25
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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25
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
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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45
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
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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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47
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
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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
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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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Energy and Power in RLC Circuits (2)
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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55
Energy and Power in RLC Circuits (3)
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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55
Energy and Power in RLC Circuits (3)
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
Monday, March 31, 2014
56
Energy and Power in RLC Circuits (4)
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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56
Energy and Power in RLC Circuits (4)
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
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Applications
15
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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
16
Impedance dominated by Xc
Impedance dominated by XL
Z =
�
R2 +
�ωL− 1
ωC
�2
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Frequency filters
17
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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57
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
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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
19
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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
20
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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
58
Transformers (2)
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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
59
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
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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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61
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
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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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63
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
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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
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∇ = {∂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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34
�E · dA = Q/�0
�B · dA = 0
�E · dl = −
�
S
∂B
∂t· dA
�B · dl = µ0I + µ0�0
�
S
∂B
∂t· dA
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Maxwell’s equations in vacuum
35
∇ ·E = ρ/�0
∇ ·B = 0
∇×B = µ0J+ µ0�0∂tE
∇×E = −∂tB
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Maxwell’s equations in vacuum
35
∇ ·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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36
∇ ·E = 0
∇ ·B = 0
∇×B = µ0�0∂tE
∇×E = −∂tB
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∇×
36
∇ ·E = 0
∇ ·B = 0
∇×B = µ0�0∂tE
∇×E = −∂tB
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∇×∂t
36
∇ ·E = 0
∇ ·B = 0
∇×B = µ0�0∂tE
∇×E = −∂tB
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∇×∂t
∇×∇×B = µ0�0∇× ∂tE
∂t∇×E = −∂2tB
36
∇ ·E = 0
∇ ·B = 0
∇×B = µ0�0∂tE
∇×E = −∂tB
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µ0�0 =1
c2
∆B− 1
c2∂2tB = 0
Wave equation
37
∇×∇×B = µ0�0∇× ∂tE
∂t∇×E = −∂2tB
∇×∇×B = ∇(∇ ·B)−∆B = −∆B
B-field behaves like pendulum!
∆B =�∂2x + ∂2
y + ∂2z
�B
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∆B =�∂2x + ∂2
y + ∂2z
�B
By(z, t) = B0 sin(ωt− kx)
k =2π
λ
ω = kc
k2 − ω2
c2= 0
Wave equation
38
∆B− 1
c2∂2tB = 0
wave length
hump is moving with velocity c
c
λx
y
z By
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39
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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