Ch. 22: More on AC Circuits

LC Circuit Most useful circuits contain multiple circuit elements Will start with an LC circuit, containing just an inductor and a capacitor No AC generator is included, but some excess charge is placed on the capacitor at t = 0

After t = 0, the charge moves from one capacitor plate to the other and current passes through the inductor Eventually, the charge on each capacitor plate falls to zero The inductor opposes change in the current, so the induced emf now acts to maintain the current at a nonzero value This current continues to transport charge from one capacitor plate to the other, causing the capacitor’s charge and voltage to reverse sign Eventually the charge on the capacitor returns to its original value

The charge is q = qmax cos (2πƒt) The current is I = Imax sin (2πƒt) LC Circuit The voltage and current in the circuit oscillate between positive and negative values The circuit behaves as a simple harmonic oscillator The charge is q = qmax cos (2πƒt) The current is I = Imax sin (2πƒt)

Energy in an LC Circuit Capacitors and inductors store energy A capacitor stores energy in its electric field and depends on the charge An inductor stores energy in its magnetic field and depends on the current As the charge and current oscillate, the energies stored also oscillate

Energy in AC Circuits (LC) For the capacitor, For the inductor, The energy oscillates back and forth between the capacitor and its electric field and the inductor and its magnetic field The total energy must remain constant

The maximum energy in the capacitor must equal the maximum energy in the inductor From energy considerations, the maximum value of the current can be calculated This shows how the amplitudes of the current and charge oscillations in the LC circuits are related

Frequency Oscillations – LC Circuit In an LC circuit, the instantaneous voltage across the capacitor and inductor are always equal Therefore, |VC| = |I XC| = |VL| = |I XL| Simplifying, XC = XL This assumed the current in the LC circuit is oscillating and hence applies only at the oscillation frequency This frequency is the resonant frequency

LRC Circuits VAC = VL + VC + VR LRC circuit Let the circuit contain a generator, resistor, inductor and capacitor in series LRC circuit From Kirchhoff’s Loop Rule, VAC = VL + VC + VR But the voltages are not all in phase, so the phase angles must also be taken into account

LRC Circuit – Phasor Diagram All the elements are in series, so the current is the same through each one All the current phasors have the same orientation Resistor: current and voltage are in phase Capacitor and inductor: current and voltage are 90° out of phase, in opposite directions

Resonance The VC and VR values are the same at the resonance frequency Only the resistor is left to “resist” the flow of the current This cancellation between the voltages occurs only at the resonance frequency The resonance frequency corresponds to the highest current

Applications of Resonance Resonance is used in radios, cell phones and other similar applications Tuning a radio Changes the value of the capacitance in the LCR circuit so the resonance frequency matches the frequency of the station you want to listen to LCR circuits are used to construct devices that are frequency dependent

Real Inductors in AC Circuits A typical inductor includes a nonzero resistance Due to the wire itself The inductor can be modeled as an ideal inductor in series with a resistor The current can be calculated using phasors Section 22.7

Real Inductor, cont. The elements are in series, so the current is the same through both elements Voltages are VR = I R and VL = I XL The voltages must be added as phasors The phase differences must be included The total voltage has an amplitude of Section 22.7

Impedance The impedance, Z, is a measure of how strongly a circuit “impedes” current in a circuit The impedance is defined as Vtotal = I Z where This is the impedance for an RL circuit only The impedance for a circuit containing other elements can also be calculated using phasors The angle between the current and the impedance can also be calculated

Impedance, LCR Circuit The current phasor is on the horizontal axis The total voltage is The impedance is Section 22.7

Elements and Frequencies in AC Circuits Resistor Resistors in an AC circuit behave very much like resistors in a DC circuit The current is always in phase with the voltage Capacitor or inductor Both are frequency dependent Due to the frequency dependence of the reactants XC is largest at low frequencies, so the current through a capacitor is smallest at low frequencies XL is largest at high frequencies, so the current through an inductor is smallest at high frequencies Section 22.8

Elements at Various Frequencies – Summary Section 22.8

High-Pass Filter (LR Circuit) When the input frequency is very low, the reactance of the inductor is small The inductor acts as a wire Voltage drop will be 0 At very high frequencies, the inductor acts as an open circuit No current is passed The output voltage is equal to the input voltage This circuit acts as a high-pass filter Section 22.8

Low-Pass Filter (RC Circuit) When the input frequency is very low, the reactance of the capacitor is large The current is very small The capacitor acts as an open circuit The output voltage is equal to the input voltage At high frequencies, the capacitor acts as a short circuit The inductor acts as a wire The output voltage is 0 This circuit acts as a low-pass filter Section 22.8

Application of a Low-Pass Filter A low-pass filter is used in radios and MP3 players A music signal often contains static Static comes from unwanted high-frequency components in the music These high frequencies can be filtered out by using a low-pass filter Section 22.8

Frequency Limits, RL Circuit For an RL circuit, the input frequency is compared to the RL time constant The time constant is τRL = L / R Define a corresponding frequency as ƒRL = 1/ τRL = R / L The high-frequency limit applies when the input frequency is much greater than ƒRL A frequency higher than ~10 x ƒRL falls into the high-frequency limit The low-frequency limit applies when the input frequency is much less than ƒRL A frequency lower than ~ƒRL / 10 falls into the low-frequency limit Section 22.8

Frequency Limits, RC Circuit For an RC circuit, the input frequency is compared to the RC time constant The time constant is τRC = R C Define a corresponding frequency as ƒRC = 1/ τRC = 1 / RC The high-frequency limit applies when the input frequency is much greater than ƒRC A frequency higher than ~10 x ƒRC falls into the high-frequency limit The low-frequency limit applies when the input frequency is much less than ƒRC A frequency lower than ~ƒRC / 10 falls into the low-frequency limit Section 22.8

Frequency Limits, LC Circuit The resonant frequency determines the boundary between high- and low-frequency limits Remember, Section 22.8

Filter Application – Stereo Speakers Many stereo speakers actually contain two separate speakers A tweeter is designed to perform well at high frequencies A woofer is designed to perform well at low frequencies The AC signal passes through a crossover network A combination of low-pass and high-pass filters The outputs of the filter are sent to the speaker which is most efficient at that frequency Section 22.8

Transformers Transformers are devices that can increase or decrease the amplitude of an applied AC voltage A simple transformer consists of two solenoid coils with the loops arranged so that all or most of the magnetic field lines and flux generated by one coil pass through the other coil Section 22.9

Transformers, cont. The wires are covered with a non-conducting layer so that current cannot flow directly from one coil to the other An AC current in one coil will induce an AC voltage across the other coil An AC voltage source is typically attached to one of the coils called the input coil The other coil is called the output coil

Transformers, Equations Faraday’s Law applies to both coils If the input coil has Nin coils and the output coil has Nout turns, the flux in the coils is related by The voltages are related by Section 22.9

Transformers, final The ratio of the turns can be greater than or less than one Therefore, the input voltage can be transformed to a different value Transformers cannot change DC voltages Since they are based on Faraday’s Law Section 22.9

Practical Transformers Most practical transformers have central regions filled with a magnetic material This produces a larger flux, resulting in a larger voltage at both the input and output coils The ratio Vout / Vin is not affected by the presence of the magnetic material Section 22.9

Applications of Transformers Transformers are used in the transmission of electric power over long distances Many household appliances use transformers to convert the AC voltage at a wall socket to the smaller voltages needed in many devices Two steps are needed – converting 120 V to 9 V then AC to DC Section 22.9

Transformers and Power The output voltage of a transformer can be made much larger by arranging the number of coils According to the principle of conservation of energy, the energy delivered through the input coil must either be stored in the transformer’s magnetic field or transferred to the output circuit Over many cycles, the stored energy is constant The power delivered to the input coil must equal the output power Section 22.9

Power, cont. Since P = V I, if Vout is greater than Vin, then Iout must be smaller than Iin Pin = Pout only in an ideal transformer In real transformers, the coils always have a small electrical resistance This causes some power dissipation For a real transformer, the output power is always less than the input power Usually by only a small amount Section 22.9

Motors An AC voltage source can be use to power a motor The AC source is connected to a coil wound around a horseshoe magnet Called the input coil The input coil induces a magnetic field that circulates through the horseshoe magnet Section 22.10

Motors, cont. A second coil is mounted between the poles of the horseshoe magnet and attached to a rotating shaft The forces acting on the second coil produce a torque on the coil This causes the shaft to rotate As the AC current in the input coil changes direction, so do the forces The torques continue to produce a rotation that is always in the same direction The oscillations of the AC current and field make the shaft rotate Section 22.10

Advantages of AC vs. DC Biggest advantage is in the systems that distribute electric power across long distances The power generated at a power plant must be distributed to distance places The power plant acts as an AC generator Section 22.11

Advantages, cont. There is power dissipated in the power lines Pave = (Irms )2 Rline The power company wants to minimize these power losses, so they want to make Irms as small as possible The voltage is increased by using a transformer The increase in voltage is done in order to decrease the current A transformer is used to drop the high voltages in the power lines to the lower voltages at the house Section 22.11

Advantages, final The power lines have typical voltages of 500,000 V or higher The transformer reduces the voltage to a maximum voltage of 170 V Typically 5% to 10% of the energy that leaves the power plant is dissipated in the resistance of the power lines Section 22.11