Alternating-Current Circuits Chapter 22

Section 22.2 AC Circuit Notation

Phasors AC circuits can be analyzed graphically An arrow has a length V max The arrow’s tail is at the origin The arrow makes an angle of θ with the horizontal The angle varies with time according to θ = 2πƒt Section 22.2

Phasors, cont. The rotating arrow represents the voltage in an AC circuit The arrow is called a phasor A phasor is not a vector A phasor diagram provides a convenient way to illustrate and think about the time dependence in an AC circuit Section 22.2

Phasors, final The current in an AC circuit can also be represented by a phasor The two phasors always make the same angle with the horizontal axis as time passes The current and voltage are in phase For a circuit with only resistors Section 22.2

AC Circuits with Capacitors Assume an AC circuit containing a single capacitor The instantaneous charge is q = C V = C V max sin (2 πƒt) The capacitor’s voltage and charge are in phase with each other Section 22.3

Current in Capacitors The instantaneous current is the rate at which charge flows onto the capacitor plates in a short time interval The current is the slope of the q-t plot A plot of the current as a function of time can be obtained from these slopes Section 22.3

Current in Capacitors, cont. The current is a cosine function I = I max cos (2πƒt) Equivalently, due to the relationship between sine and cosine functions I = I max sin (2πƒt + Φ) where Φ = π/2 Section 22.3

Capacitor Phasor Diagram The current is out of phase with the voltage The angle π/2 is called the phase angle,Φ, between V and I For this circuit, the current and voltage are out of phase by 90 o Section 22.3

Current Value for a Capacitor The peak value of the current is The factor X c is called the reactance of the capacitor Units of reactance are Ohms Reactance and resistance are different because the reactance of a capacitor depends on the frequency If the frequency is increased, the charge oscillated more rapidly and Δt is smaller, giving a larger current At high frequencies, the peak current is larger and the reactance is smaller Section 22.3

Power In A Capacitor For an AC circuit with a capacitor, P = VI = V max I max sin (2 πƒt) cos (2 πƒt) The average value of the power over many oscillations is 0 Energy is transferred from the generator during part of the cycle and from the capacitor in other parts Energy is stored in the capacitor as electric potential energy and not dissipated by the circuit Section 22.3

AC Circuits with Inductors Assume an AC circuit containing a single inductor The voltage drop is V = L (ΔI / Δt) = V max sin (2 πƒt) The inductor’s voltage is proportional to the slope of the current-time relationship Section 22.4

Current in Inductors The instantaneous current oscillates in time according to a cosine function I = -I max cos (2πƒt) A plot of the current is shown Section 22.4

Current in Inductors, cont. The current equation can be rewritten as I = I max sin (2πƒt – π/2) Equivalently, I = I max sin (2πƒt + Φ) where Φ = -π/2 Section 22.4

Inductor Phasor Diagram The current is out of phase with the voltage For this circuit, the current and voltage are out of phase by -90 o Remember, for a capacitor, the phase difference was +90 o Section 22.4

Current Value for an Inductor The peak value of the current is The factor X L is called the reactance of the inductor Units of inductive reactance are Ohms As with the capacitor, inductive reactance depends on the frequency As the frequency is increased, the inductive reactance increases Section 22.4

Quiz! If the frequency in a circuit with a Capacitor is halved. The Reactance is? A) Double B) The Same C) Half D) Zero E) Disproportionate

Section 22.4 Properties of AC Circuits

Power In An Inductor For an AC circuit with an inductor, P = VI = -V max I max sin (2 πƒt) cos (2 πƒt) The average value of the power over many oscillations is 0 Energy is transferred from the generator during part of the cycle and from the inductor in other parts of the cycle Energy is stored in the inductor as magnetic potential energy Section 22.4

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 Section 22.5

LC Circuit, cont. 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 again 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 Section 22.5

LC Circuit, final 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 = q max cos (2πƒt) The current is I = I max sin (2πƒt) Section 22.5

Quiz! If the frequency in a circuit with an Inductor is doubled. The Reactance is? A) Double B) The Same C) Half D) Zero E) 42

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 Section 22.5

Energy Calculations 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 Section 22.5

Energy, final 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 Section 22.5

Frequency Oscillations – LC Circuit In an LC circuit, the instantaneous voltage across the capacitor and inductor are always equal Therefore, |V C | = |I X C | = |V L | = |I X L | Simplifying, X C = X L This assumed the current in the LC circuit is oscillating and hence applies only at the oscillation frequency This frequency is the resonance frequency Section 22.5

LRC Circuits Let the circuit contain a generator, resistor, inductor and capacitor in series LRC circuit From Kirchhoff’s Loop Rule, V AC = V L + V C + V R But the voltages are not all in phase, so the phase angles must also be taken into account Section 22.6

LRC Circuit – Phasor Diagram All the elements are in series, so the current is the same through each one All the current phasors are in the same direction Resistor: current and voltage are in phase Capacitor and inductor: current and voltage are 90 o out of phase, in opposite directions Section 22.6

Resonance The V C and V R 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 Section 22.6

Applications of Resonance 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 can be used to construct devices that are frequency dependent Section 22.6

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 V R = I R and V L = I X L 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 V total = 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 Section 22.7

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

Resonance in an LCR Circuit The current depends on the impedance, I max = V max /Z Since the impedance depends on the frequency, the current amplitude also varies with frequency For the maximum current, the impedance must be a minimum The minimum impedance occurs when Section 22.7

Resonance, cont. Solving for the frequency gives This is the same result as was found for the LC circuit The maximum current occurs at the resonant frequency This is the frequency at which the LCR circuit responds most strongly to an applied AC circuit Section 22.7

Behavior of Elements at Various Frequencies Section 22.8

Elements and Frequencies, cont. 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 reactances X C is largest at low frequencies, so the current through a capacitor is smallest at low frequencies X L is largest at high frequencies, so the current through an inductor is smallest at high frequencies Section 22.8

RL Circuit Example 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 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

RC Circuit Example 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

Quiz! A) 2.0 B) 0.5 C) 1.0 D) 0.1 Vac= 9V C1=10 uF C2= 2 uF F=2kHz How many Amps?

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 τ RL = L / R Define a corresponding frequency as ƒ RL = 1/ τ RL = R / L The high-frequency limit applies when the input frequency is much greater than ƒ RL A frequency higher than ~10 x ƒ RL falls into the high- frequency limit The low-frequency limit applies when the input frequency is much less than ƒ RL A frequency lower than ~ƒ RL / 10 falls into the low- frequency limit Section 22.8

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