Phasor Diagram for an Inductor

The phasors are at 90o with respect to each otherThis represents the phase difference between the current and voltageSpecifically, the current lags behind the voltage by 90o

Vector Addition of the Phasor Diagram

Vector addition is used to combine the voltage phasorsΔVL and ΔVC are in opposite directions, so they can be combinedTheir resultant is perpendicular to ΔVR

Total Voltage in RLC CircuitsFrom the vector diagram, ΔVmax can be calculated

( )( )

( )

22max

22max max max

22max max

( )I I I

I

R L C

L C

L C

V V V V

R X X

V R X X

Δ = Δ + Δ − Δ

= + −

Δ = + −

ImpedanceThe current in an RLC circuit is

Z is called the impedance of the circuit and it plays the role of resistance in the circuit, where

Impedance has units of ohmsAlso, ΔVmax = ImaxZ

( )max max

max 22I

L C

V VZR X X

Δ Δ= =

+ −

( )22L CZ R X X≡ + −

Impedance TriangleSince Imax is the same for each element, it can be removed from each term in the phasor diagramThe result is an impedance triangle

Impedance Triangle, cont.The impedance triangle confirms that

The impedance triangle can also be used to find the phase angle, φ

The phase angle can be positive or negative and determines the nature of the circuitAlso, cos φ =

( )22L CZ R X X≡ + −

1tan L CX XφR

− −⎛ ⎞= ⎜ ⎟⎝ ⎠

RZ

Power in an AC CircuitThe average power delivered by the generator is converted to internal energy in the resistor

Pav = ½ Imax ΔVmax cos φ = IrmsΔVrms cos φcos φ is called the power factor of the circuit

We can also find the average power in terms of R

Pav = I 2rmsR

*Note: “P” is used as the symbol for power here because the symbol used in the text is not available on all computer platforms.

Power in an AC Circuit, cont.The average power delivered by the source is converted to internal energy in the resistorNo power losses are associated with pure capacitors and pure inductors in an AC circuit

In a capacitor, during one-half of a cycle energy is stored and during the other half the energy is returned to the circuit and no power losses occur in the capacitorIn an inductor, the source does work against the back emf of the inductor and energy is stored in the inductor, but when the current begins to decrease in the circuit, the energy is returned to the circuit

Power and PhaseThe power delivered by an AC circuit depends on the phaseSome applications include using capacitors to shift the phase in heavy motors so that excessively high voltages are not needed

Resonance in an AC CircuitResonance occurs at the frequency ωowhere the current has its maximum value

To achieve maximum current, the impedance must have a minimum valueThis occurs when XL = XCSolving for the frequency gives

The resonance frequency also corresponds to the natural frequency of oscillation of an LC circuit

1oω LC=

Resonance, cont.Resonance occurs at the same frequency regardless of the value of RAs R decreases, the curve becomes narrower and tallerTheoretically, if R = 0 the current would be infinite at resonance

Real circuits always have some resistance

Power as a Function of Frequency Power can be expressed as a function of frequency in an RLC circuit

This shows that at resonance, the average power is a maximum

( )( )

2 2

22 2 2 2 2

rmsav

o

V RωP

R ω L ω ω

Δ=

+ −

Active Figure 33.19

(SLIDESHOW MODE ONLY)

Quality FactorThe sharpness of the resonance curve is usually described by a dimensionless parameter known as the quality factor, QQ = ωo / Δω = (ωoL) / R

Δω is the width of the curve, measured between the two values of ω for which Pavhas half its maximum value

These points are called the half-power points

Quality Factor, cont.A high-Q circuit responds only to a narrow range of frequencies

Narrow peak

A low-Q circuit can detect a much broader range of frequenciesTypical Q values in electronics range from 10 to 100

TransformersAn AC transformerconsists of two coils of wire wound around a core of soft ironThe side connected to the input AC voltage source is called the primary and has N1turns

Transformers, 2The other side, called the secondary, is connected to a resistor and has N2 turnsThe core is used to increase the magnetic flux and to provide a medium for the flux to pass from one coil to the other

Eddy current losses are minimized by using a laminated coreIron is used as the core material because it is a soft ferromagnetic substance and reduces hysteresis losses

Transformers, 3Assume an ideal transformer

One in which the energy losses in the windings and the core are zero

Typical transformers have power efficiencies of 90% to 99%

In the primary, The rate of change of the flux is the same for both coils

1 1BdV N

dtΦ

Δ = −

Transformers, 4The voltage across the secondary is

The voltages are related by

When N2 > N1, the transformer is referred to as a step-up transformerWhen N2 < N1, the transformer is referred to as a step-down transformer

22 1

1

NV VN

Δ = Δ

2 2BdV N

dtΦ

Δ = −

Transformers, 5The power input into the primary equals the power output at the secondary

I1ΔV1 = I2ΔV2

The equivalent resistance of the load resistance when viewed from the primary is

2

1eq

2L

NR RN

⎛ ⎞= ⎜ ⎟⎝ ⎠

Transformers, finalA transformer may be used to match resistances between the primary circuit and the loadThis way, maximum power transfer can be achieved between a given power source and the load resistance

In stereo terminology, this technique is called impedance matching

RectifierThe process of converting alternating current to direct current is called rectificationA rectifier is the converting deviceThe most important element in a rectifier circuit is the diode

A diode is a circuit element that conducts current in one direction but not the other

Rectifier Circuit

The arrow on the diode ( ) indicates the direction of the current in the diodeBecause of the diode, the alternating current in the load resistor is reduced to the positive portion of the cycle

Half-Wave Rectifier

The solid line in the graph is the result through the resistorIt is called a half-wave rectifier because current is present in the circuit during only half of each cycle

Half-Wave Rectifier, Modification

A capacitor can be added to the circuitThe circuit is now a simple DC power supplyThe time variation in the circuit is close to zero

It is determined by the RC time constant of the circuitThis is represented by the dotted lines in the previous graph

Filter Circuit, ExampleA filter circuit is one used to smooth out or eliminate a time-varying signalAfter rectification, a signal may still contain a small AC component

This component is often called a rippleBy filtering, the ripple can be reducedFilters can also be built to respond differently to different frequencies

High-Pass FilterThe circuit shown is one example of a high-pass filterA high-pass filter is designed to preferentially pass signals of higher frequency and block lower frequency signals

Active Figure 33.25

(SLIDESHOW MODE ONLY)

High-Pass Filter, contAt low frequencies, ΔVoutis much smaller than ΔVin

At low frequencies, the capacitor has high reactance and much of the applied voltage appears across the capacitor

At high frequencies, the two voltages are equal

At high frequencies, the capacitive reactance is small and the voltage appears across the resistor

Low-Pass Filter

At low frequencies, the reactance and voltage across the capacitor are highAs the frequency increases, the reactance and voltage decreaseThis is an example of a low-pass filter

Quick Quiz 33.1

Consider the voltage phasor in the figure below, shown at three instants of time. Choose the part of the figure that represents the instant of time at which the instantaneous value of the voltage has the largest magnitude.

Answer: (a). The phasor in part (a) has the largest projection onto the vertical axis.

Quick Quiz 33.1

Quick Quiz 33.2

For the voltage phasor in the figure below, choose the part of the figure that represents the instant of time at which the instantaneous value of the voltage has the smallest magnitude.

Answer: (b). The phasor in part (b) has the smallest-magnitude projection onto the vertical axis.

Quick Quiz 33.2

Quick Quiz 33.3

Which of the following statements might be true for a resistor connected to a sinusoidal AC source? (a) av = 0 and iav = 0 (b) av = 0 and iav > 0 (c) av > 0 and iav = 0 (d) av > 0 and iav > 0

Answer: (c). The average power is proportional to the rms current, which, as Figure 33.5 shows, is nonzero even though the average current is zero. Condition (a) is valid only for an open circuit, and conditions (b) and (d) can never be true because iav = 0 for AC circuits.

Quick Quiz 33.3

Quick Quiz 33.4

Consider the AC circuit in the figure below. The frequency of the AC source is adjusted while its voltage amplitude is held constant. The lightbulb will glow the brightest at (a) high frequencies (b) low frequencies (c) The brightness will be the same at all frequencies.

Answer: (b). For low frequencies, the reactance of the inductor is small so that the current is large. Most of the voltage from the source is across the bulb, so the power delivered to it is large.

Quick Quiz 33.4

Quick Quiz 33.5

Consider the AC circuit in the figure below. The frequency of the AC source is adjusted while its voltage amplitude is held constant. The lightbulb will glow the brightest at (a) high frequencies (b) low frequencies (c) The brightness will be same at all frequencies.

Answer: (a). For high frequencies, the reactance of the capacitor is small so that the current is large. Most of the voltage from the source is across the bulb, so the power delivered to it is large.

Quick Quiz 33.5

Quick Quiz 33.6

Consider the AC circuit in this figure. The frequency of the AC source is adjusted while its voltage amplitude is held constant. The lightbulb will glow the brightest at (a) high frequencies (b) low frequencies (c) The brightness will be same at all frequencies.

Answer: (b). For low frequencies, the reactance of the capacitor is large so that very little current exists in the capacitor branch. The reactance of the inductor is small so that current exists in the inductor branch and the lightbulb glows. As the frequency increases, the inductive reactance increases and the capacitive reactance decreases. At high frequencies, more current exists in the capacitor branch than the inductor branch and the lightbulb glows more dimly.

Quick Quiz 33.6

Quick Quiz 33.7

Label each part of the figure below as being XL > XC, XL = XC, or XL < XC.

Answer: (a) XL < XC. (b) XL = XC. (c) XL > XC.

Quick Quiz 33.7

Quick Quiz 33.9

The impedance of a series RLC circuit at resonance is (a) larger than R (b) less than R (c) equal to R (d) impossible to determine

Answer: (c). At resonance, XL = XC. According to Equation 33.25, this gives us Z = R.

Quick Quiz 33.9

Quick Quiz 33.10

An airport metal detector (see page 1003) is essentially a resonant circuit. The portal you step through is an inductor (a large loop of conducting wire) within the circuit. The frequency of the circuit is tuned to its resonance frequency when there is no metal in the inductor. Any metal on your body increases the effective inductance of the loop and changes the current in it. If you want the detector to detect a small metallic object, the circuit should have (a) a high quality factor(b) a low quality factor

Answer: (a). The higher the quality factor, the more sensitive the detector. As you can see from Figure 33.19, when Q = ω0/Δω is high, a slight change in the resonance frequency (as might happen when a small piece of metal passes through the portal) causes a large change in current that can be detected easily.

Quick Quiz 33.10

Quick Quiz 33.11a

Suppose you are designing a high-fidelity system containing both large loudspeakers (woofers) and small loudspeakers (tweeters). If you wish to deliver low-frequency signals to a woofer, what device would you place in series with it? (a) an inductor (b) a capacitor (c) a resistor

Answer: (a). The current in an inductive circuit decreases with increasing frequency (see Eq. 33.9). Thus, an inductor connected in series with a woofer blocks high-frequency signals and passes low-frequency signals.

Quick Quiz 33.11a

Quick Quiz 33.11b

Remember, you are designing a high-fidelity system containing both large loudspeakers (woofers) and small loudspeakers (tweeters). If you wish to deliver high-frequency signals to a tweeter, what device would you place in series with it? (a) an inductor (b) a capacitor (c) a resistor

Answer: (b). The current in a capacitive circuit increases with increasing frequency (see Eq. 33.17). When a capacitor is connected in series with a tweeter, the capacitor blocks low-frequency signals and passes high-frequency signals.

Quick Quiz 33.11b