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 25
RLC Circuit (4) 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 26
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 25
Alternating Current (1) Now we consider a single loop circuit containing a capacitor, an inductor, a resistor, and a source of emf This source of emf is capable of producing a time varying voltage as opposed to the sources 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 V max is the amplitude or maximum value of the emf 28
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 45
Series RLC Circuit (4) 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 The voltage can be thought of as the projection of the vertical axis of the phasor V max representing the timevarying emf in the circuit as shown below In this figure we have replaced the sum of the two phasors V L and V C with the phasor V L - V C 46
Series RLC Circuit (5): Impedance The sum of the two phasors V L - V C and V R must equal V max 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 timevarying emf 47
Series RLC Circuit impedance Z = R 2 + active resistance ωl 1 Only active resistance determines losses! ωc 2 reactive resistance (reactance) Reactive resistance can be 0, at resonance 47
Resonant Behavior of RLC Circuit 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 V max = 7.5 V, L = 8.2 mh, C = 100 µf, and three resistances One can see that as the resistance is lowered, the maximum current at the resonant angular frequency increases and there is a more pronounced resonant peak 52
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Energy and Power in RLC Circuits (1) 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 since < sin 2 ωt >= 1/2 53
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 54
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 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 55
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 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 56
Applications 15
Frequency filters Response of RLC circuit strongly depends on frequency. Hi-Fi stereos have different speakers for low, middle and high frequencies Z = R 2 + ωl 1 2 ωc Impedance dominated by Xc Impedance dominated by XL 16
Frequency filters Z = R 2 + ωl 1 2 ωc 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) 17
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 = I 2 R The ability to raise and lower alternating voltages would be very useful in everyday life 57
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
Transformers 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. 20
Transformers (2) 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 N P turns connected to a source of emf We can assume that the primary windings act 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 (3) 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 Now consider the second coil with N S turns The time-varying emf in the primary coil induces a timevarying magnetic field in the iron core This core passes through the secondary coil 59
Transformers (4) Thus a time-varying voltage is induced in the secondary coil described by Faraday s Law Because both the primary and secondary coils experience the same changing magnetic field we can write "stepped up" Ns > Np, "stepped down" Ns < Np 60
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 P S = I S V S 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 I P 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 61
Increase in voltage equals decrease in current 25
Transformers (6) 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 V S = I S R We can then write the primary current as: With an effective primary resistance of 62
Transformers (7) Note that these equations assume no losses in the transformers and that the load is purely resistive Real transformers have small losses 63
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Let there be light! 29
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 30
Recall: electric induction E = t B E dl = S B t da 31
Magnetic induction Similarly, changing flux of electric field induces magnetic field But magnetic field is also generated by currents B dl = µ 0 I + µ 0 0 S B t da Displacement current µ 0 0 =1/c 2 32
Maxwell s equations E = ρ/ 0 electric charge produces E-field B =0 B = µ 0 J + µ 0 0 t E E = t B changing B-field also produce E-field = { x, y, z } no magnetic charge electric current and changing E-field produce B-field 33
E da = Q/ 0 B da =0 B E dl = S t da B dl = µ 0 I + µ 0 0 S B t da 34
Maxwell s equations in vacuum E = ρ/ 0 B =0 B = µ 0 J + µ 0 0 t E E = t B 35
Maxwell s equations in vacuum E = ρ/ 0 B =0 B = µ 0 J + µ 0 0 t E E = t B E =0 B =0 B = µ 0 0 t E E = t B 35
E =0 B =0 B = µ 0 0 t E E = t B 36
E =0 B =0 B = µ 0 0 t E E = t B 36
E =0 B =0 t B = µ 0 0 t E E = t B 36
E =0 B =0 t B = µ 0 0 t E E = t B B = µ 0 0 t E t E = 2 t B 36
Wave equation B = µ 0 0 t E t E = 2 t B B = ( B) B = B µ 0 0 = 1 c 2 B = 2 x + 2 y + 2 z B B 1 c 2 2 t B =0 B-field behaves like pendulum! 37
Wave equation B 1 c 2 2 t B =0 B = x 2 + y 2 z + 2 B B y (z,t) =B 0 sin(ωt kx) k = 2π λ k 2 ω2 c 2 =0 ω = kc wave length y z λ hump is moving with velocity c By c x 38
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) 39
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