Chapter 22: AC Circuits Figure 25-21. (a) Direct current. (b) Alternating current.

DC means direct current DC Circuit Summary DC Circuits DC means direct current The source of electrical energy is usually a battery If only resistors are in the circuit, the current is independent of time Ohm’s Law holds: I = V/R If the circuit contains capacitors & resistors, the current can vary with time but always approaches a constant value (I = V/R) a long enough time after closing the switch.

AC Circuit Introduction AC Circuits AC means alternating current The power source is a device that produces an electric potential that varies with time. There will be a frequency and peak voltage associated with the potential. Household electrical energy is supplied by an AC source: The standard frequency is f = 60 Hz AC power has numerous advantages over DC power

DC vs. AC Sources

Direct Current, or DC. or AC. Alternating Current, Current from a battery flows steadily in one direction. This is called Direct Current, or DC. By contrast, current from a power plant varies sinusoidally with time. This is called Alternating Current, or AC. Figure 25-21. (a) Direct current. (b) Alternating current.

Generating AC Voltages Most sources of AC voltage use a generator based on magnetic induction A shaft holds a coil with many loops of wire The coil is positioned between the poles of a permanent magnet The magnetic flux through the coil varies with time as the shaft turns This changing flux induces a voltage in the coil This induced voltage is the generator’s output

Generators Generators of electrical energy convert the mechanical energy of the rotating shaft into electrical energy. The principle of conservation of total energy still applies. The source of electrical energy in a circuit enables the transfer of electrical energy from a generator to an attached circuit.

Resistors in AC Circuits Assume a circuit with an AC generator & a resistor The voltage across the output of the AC source varies with time: V = Vmaxsin(2πƒt) V is the instantaneous potential difference Vmax is the amplitude of the AC voltage

Apply Ohm’s “Law” Since the voltage varies sinusoidally, so does the current

RMS Voltage To specify current & voltage values when they vary with time, rms values are adopted RMS means Root Mean Squared For the voltage:

RMS Current The root-mean-square value can be defined for any quantity that varies with time. For the current: The root-mean-square values of the voltage and current are typically used to specify the properties of an AC circuit

P = VmaxImaxsin2(2πƒt) P = I V Power The instantaneous power is the product of the instantaneous voltage & the instantaneous current P = I V Since both I & V vary with time, the power also varies with time: P = VmaxImaxsin2(2πƒt)

Average Power = (½)Maximum Power Pavg = ½(VmaxImax) = Vrms Irms The instantaneous power varies between Vmax Imax & 0 Average Power = (½)Maximum Power Pavg = ½(VmaxImax) = Vrms Irms This has the same mathematical form as the power in a DC circuit. Ohm’s Law can again be used to express the power in different ways:

AC Circuit Notation It is important to distinguish between instantaneous & average values of voltage, current & power. The amplitudes of the AC variations and the rms values are also important

Phasors The angle varies with time according to θ= 2πƒt AC circuits can be analyzed graphically with Phasors. An arrow has a length Vmax The arrow’s tail is tied to the origin. Its tip moves along a circle. The arrow makes an angle of θ with the horizontal. The angle varies with time according to θ= 2πƒt 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

The current in an AC circuit can also be represented by a phasor The 2 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

AC Circuits with Capacitors Assume an AC circuit containing a single capacitor The instantaneous charge on the plates is: q = CV = CVmaxsin(2πƒt) The capacitor’s voltage and charge are in phase with each other.

Current in Capacitors The current is the slope of the q-t plot. 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

I = Imax cos (2πƒt) I = Imax sin (2πƒt + ϕ) Current in Capacitors The current is a cosine function I = Imax cos (2πƒt) Equivalently, due to the relationship between sine & cosine functions I = Imax sin (2πƒt + ϕ) where ϕ = π/2

Capacitor Phasor Diagram The current is out of phase with the voltage The angle π/2 radians is called the phase angle, ϕ, between V & I For this circuit, the current & voltage are out of phase by 90°= π/2 radians

Current Value for a Capacitor The peak value of the current is ll Xc is called the reactance of the capacitor The SI unit of reactance is Ohms Reactance and resistance are different because the reactance of a capacitor depends on the frequency If the frequency is increased, the charge oscillates more rapidly & Δt is smaller, giving a larger current At high frequencies, the peak current is larger and the reactance is smaller

Vmax Imaxsin (2πƒt)cos (2πƒt) Power In A Capacitor For an AC circuit with a capacitor, P = VI = Vmax Imaxsin (2πƒt)cos (2πƒt) The average 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

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

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

I = Imaxsin (2πƒt – π/2) I = Imax sin (2πƒt + Φ) Current in Inductors The current equation can be rewritten as I = Imaxsin (2πƒt – π/2) Equivalently, I = Imax sin (2πƒt + Φ) where Φ = -π/2

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° Remember, for a capacitor, the phase difference was +90°

Current Value for an Inductor The peak value of the current is i l XL is called the reactance of the inductor SI unit of inductive reactance is Ohms As with the capacitor, inductive reactance depends on the frequency As the frequency is increased, the inductive reactance increases

-Vmax Imax sin (2πƒt) cos (2πƒt) Power in an Inductor For an AC circuit with an inductor P = VI = -Vmax Imax sin (2πƒt) cos (2πƒt) 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

Properties of AC Circuits