electrical meters

Electrical Meters

Galvanometer

• A galvanometer is the main component in analog meters for measuring current and voltage

• Digital meters are in common use– Digital meters operate

under different principles

Galvanometer

• A galvanometer consists of a coil of wire mounted so that it is free to rotate on a pivot in a magnetic field

• The field is provided by permanent magnets

• A torque acts on a current in the presence of a magnetic field

Galvanometer

• The torque is proportional to the current

– The larger the current, the greater the torque

– The greater the torque, the larger the rotation of the coil before the spring resists enough to stop the rotation

• The deflection of a needle attached to the coil is proportional to the current

• Once calibrated, it can be used to measure currents or voltages

Ammeter

• An ammeter is a device that measures current

• The ammeter must be connected in series with the elements being measured

– The current must pass directly through the ammeter

Ammeter in a Circuit

• The ammeter is connected in series with the elements in which the current is to be measured

• Ideally, the ammeter should have zero resistance so the current being measured is not altered

Ammeter from Galvanometer

• The galvanometer typically has a resistance of 60 W

• To minimize the resistance, a shunt resistance, Rp, is placed in parallel with the galvanometer

Ammeter

• The value of the shunt resistor must be much less than the resistance of the galvanometer– Remember, the equivalent resistance of resistors in parallel

will be less than the smallest resistance

• Most of the current will go through the shunt resistance, this is necessary since the full scale deflection of the galvanometer is on the order of 1 mA

Voltmeter

• A voltmeter is a device that measures potential difference

• The voltmeter must be connected in parallel with the elements being measured

– The voltage is the same in parallel

Voltmeter in a Circuit

• The voltmeter is connected in parallel with the element in which the potential difference is to be measured– Polarity must be observed

• Ideally, the voltmeter should have infinite resistance so that no current would pass through it– Corrections can be made to

account for the known, non-infinite, resistance of the voltmeter

Voltmeter

• The value of the added resistor must be much greater than the resistance of the galvanometer– Remember, the equivalent resistance of resistors in series

will be greater than the largest resistance

• Most of the current will go through the element being measured, and the galvanometer will not alter the voltage being measured

Inductance

Some Terminology

• Use emf and current when they are caused by batteries or other sources

• Use induced emf and induced current when they are caused by changing magnetic fields

• When dealing with problems in electromagnetism, it is important to distinguish between the two situations

Self-Inductance

• When the switch is closed, the current does not immediately reach its maximum value

• Faraday’s law can be used to describe the effect

Self-Inductance

• As the current increases with time, the magnetic flux through the circuit loop due to this current also increases with time

• This increasing flux creates an induced emf in the circuit

Self-Inductance

• The direction of the induced emf is such that it would cause an induced current in the loop which would establish a magnetic field opposing the change in the original magnetic field

• The direction of the induced emf is opposite the direction of the emf of the battery

• This results in a gradual increase in the current to its final equilibrium value

Self-Inductance

• This effect is called self-inductance

– Because the changing flux through the circuit and the resultant induced emf arise from the circuit itself

• The emf εL is called a self-induced emf

Self-Inductance, Equations

• An induced emf is always proportional to the time rate of change of the current

– The emf is proportional to the flux, which is proportional to the field and the field is proportional to the current

• L is a constant of proportionality called the inductance of the coil and it depends on the geometry of the coil and other physical characteristics

L

d Iε L

dt

Inductance of a Coil

• A closely spaced coil of N turns carrying current I has an inductance of

• The inductance is a measure of the opposition to a change in current

B LN εL

I d I dt

Inductance of a Solenoid

• Assume a uniformly wound solenoid having Nturns and length ℓ

– Assume ℓ is much greater than the radius of the solenoid

• The flux through each turn of area A is

B o o

NBA μ nI A μ I A

Inductance of a Solenoid

• The inductance is

• This shows that L depends on the geometry of the object

2

oB μ N ANL

I

RL Circuit, Introduction

• A circuit element that has a large self-inductance is called an inductor

• The circuit symbol is

• We assume the self-inductance of the rest of the circuit is negligible compared to the inductor

– However, even without a coil, a circuit will have some self-inductance

Effect of an Inductor in a Circuit

• The inductance results in a back emf

• Therefore, the inductor in a circuit opposes changes in current in that circuit

– The inductor attempts to keep the current the same way it was before the change occurred

– The inductor can cause the circuit to be “sluggish” as it reacts to changes in the voltage

RL Circuit, Analysis

• An RL circuit contains an inductor and a resistor

• Assume S2 is connected to a

• When switch S1 is closed (at time t = 0), the current begins to increase

• At the same time, a back emf is induced in the inductor that opposes the original increasing current

RL Circuit, Analysis

• Applying Kirchhoff’s loop rule to the previous circuit in the clockwise direction gives

• Looking at the current, we find

0d I

ε I R Ldt

1 Rt LεI e

R

RL Circuit, Analysis

• The inductor affects the current exponentially

• The current does not instantly increase to its final equilibrium value

• If there is no inductor, the exponential term goes to zero and the current would instantaneously reach its maximum value as expected

RL Circuit, Time Constant

• The expression for the current can also be expressed in terms of the time constant, t, of the circuit

– where t = L / R

• Physically, t is the time required for the current to reach 63.2% of its maximum value

1 t τεI e

R

RL Circuit, Current-Time Graph

• The equilibrium value of the current is e /R and is reached as tapproaches infinity

• The current initially increases very rapidly

• The current then gradually approaches the equilibrium value

RL Circuit, Current-Time Graph

• The time rate of change of the current is a maximum at t = 0

• It falls off exponentially as t approaches infinity

• In general,

t τd I εe

dt L

RL Circuit Without A Battery

• Now set S2 to position b

• The circuit now contains just the right hand loop

• The battery has been eliminated

• The expression for the current becomes

t tτ τ

i

εI e I e

R

Energy in a Magnetic Field

• In a circuit with an inductor, the battery must supply more energy than in a circuit without an inductor

• Part of the energy supplied by the battery appears as internal energy in the resistor

• The remaining energy is stored in the magnetic field of the inductor

Energy in a Magnetic Field

• Looking at this energy (in terms of rate)

– Ie is the rate at which energy is being supplied by the battery

– I2R is the rate at which the energy is being delivered to the resistor

– Therefore, LI (dI/dt) must be the rate at which the energy is being stored in the magnetic field

2 d II ε I R LI

dt

Energy in a Magnetic Field

• Let U denote the energy stored in the inductor at any time

• The rate at which the energy is stored is

• To find the total energy, integrate and

dU d ILI

dt dt

2

0

1

2

I

U L I d I LI

Energy Density of a Magnetic Field

• Given U = ½ L I2 and assume (for simplicity) a solenoid with L = mo n2 V

• Since V is the volume of the solenoid, the magnetic energy density, uB is

• This applies to any region in which a magnetic field exists (not just the solenoid)

22

21

2 2o

o o

B BU μ n V V

μ n μ

2

2B

o

U Bu

V μ

Energy Storage Summary

• A resistor, inductor and capacitor all store energy through different mechanisms

– Charged capacitor

• Stores energy as electric potential energy

– Inductor

• When it carries a current, stores energy as magnetic potential energy

– Resistor

• Energy delivered is transformed into internal energy

Example: The Coaxial Cable

• Calculate L for the cable

• The total flux is

• Therefore, L is

2

ln2

bo

Ba

o

μ IB dA dr

πr

μ I b

π a

ln2

oB μ bL

I π a

Mutual Inductance

• The magnetic flux through the area enclosed by a circuit often varies with time because of time-varying currents in nearby circuits

• This process is known as mutual inductionbecause it depends on the interaction of two circuits

Mutual Inductance

• The current in coil 1 sets up a magnetic field

• Some of the magnetic field lines pass through coil 2

• Coil 1 has a current I1 and N1 turns

• Coil 2 has N2 turns

Mutual Inductance

• The mutual inductance M12 of coil 2 with respect to coil 1 is

• Mutual inductance depends on the geometry of both circuits and on their orientation with respect to each other

2 1212

1

NM

I

Induced emf in Mutual Inductance

• If current I1 varies with time, the emf induced by coil 1 in coil 2 is

• If the current is in coil 2, there is a mutual inductance M21

• If current 2 varies with time, the emf induced by coil 2 in coil 1 is

12 12 2 12

d d Iε N M

dt dt

21 21

d Iε M

dt

Mutual Inductance

• In mutual induction, the emf induced in one coil is always proportional to the rate at which the current in the other coil is changing

• The mutual inductance in one coil is equal to the mutual inductance in the other coil

– M12 = M21 = M

• The induced emf’s can be expressed as2 1

1 2andd I d I

ε M ε Mdt dt

LC Circuits

• A capacitor is connected to an inductor in an LC circuit

• Assume the capacitor is initially charged and then the switch is closed

• Assume no resistance and no energy losses to radiation

Oscillations in an LC Circuit

• Under the previous conditions, the current in the circuit and the charge on the capacitor oscillate between maximum positive and negative values

• With zero resistance, no energy is transformed into internal energy

• Ideally, the oscillations in the circuit persist indefinitely– The idealizations are no resistance and no

radiation

Oscillations in an LC Circuit

• The capacitor is fully charged

– The energy U in the circuit is stored in the electric field of the capacitor

– The energy is equal to Q2max / 2C

– The current in the circuit is zero

– No energy is stored in the inductor

• The switch is closed

Oscillations in an LC Circuit

• The current is equal to the rate at which the charge changes on the capacitor

– As the capacitor discharges, the energy stored in the electric field decreases

– Since there is now a current, some energy is stored in the magnetic field of the inductor

– Energy is transferred from the electric field to the magnetic field

Oscillations in an LC Circuit

• Eventually, the capacitor becomes fully discharged– It stores no energy

– All of the energy is stored in the magnetic field of the inductor

– The current reaches its maximum value

• The current now decreases in magnitude, recharging the capacitor with its plates having opposite their initial polarity

Oscillations in an LC Circuit

• The capacitor becomes fully charged and the cycle repeats

• The energy continues to oscillate between the inductor and the capacitor

• The total energy stored in the LC circuit remains constant in time and equals

221

2 2IC L

QU U U L

C

LC Circuit Analogy to Spring-Mass System

• The potential energy ½kx2 stored in the spring is analogous to the electric potential energy (Qmax)

2/(2C) stored in the capacitor

• All the energy is stored in the capacitor at t = 0

• This is analogous to the spring stretched to its amplitude

LC Circuit Analogy to Spring-Mass System

• The kinetic energy (½ mv2) of the spring is analogous to the magnetic energy (½ L I2) stored in the inductor

• At t = ¼ T, all the energy is stored as magnetic energy in the inductor

• The maximum current occurs in the circuit

• This is analogous to the mass at equilibrium

LC Circuit Analogy to Spring-Mass System

• At t = ½ T, the energy in the circuit is completely stored in the capacitor

• The polarity of the capacitor is reversed

• This is analogous to the spring stretched to -A

LC Circuit Analogy to Spring-Mass System

• At t = ¾ T, the energy is again stored in the magnetic field of the inductor

• This is analogous to the mass again reaching the equilibrium position

LC Circuit Analogy to Spring-Mass System

• At t = T, the cycle is completed

• The conditions return to those identical to the initial conditions

• At other points in the cycle, energy is shared between the electric and magnetic fields

Time Functions of an LC Circuit

• In an LC circuit, charge can be expressed as a function of time– Q = Qmax cos (ωt + φ)

– This is for an ideal LC circuit

• The angular frequency, ω, of the circuit depends on the inductance and the capacitance– It is the natural frequency of oscillation of the

circuit1ω

LC

Time Functions of an LC Circuit

• The current can be expressed as a function of time

• The total energy can be expressed as a function of time

max

dQI ωQ sin(ωt φ)

dt

22 2 21

2 2max

C L max

QU U U cos ωt LI sin ωt

c

Charge and Current in an LC Circuit

• The charge on the capacitor oscillates between Qmax and

-Qmax

• The current in the inductor oscillates between Imax and -Imax

• Q and I are 90o out of phase with each other

– So when Q is a maximum, I is zero, etc.

Energy in an LC Circuit – Graphs

• The energy continually oscillates between the energy stored in the electric and magnetic fields

• When the total energy is stored in one field, the energy stored in the other field is zero

The RLC Circuit

• A circuit containing a resistor, an inductor and a capacitor is called an RLCCircuit

• Assume the resistor represents the total resistance of the circuit

RLC Circuit, Analysis

• The total energy is not constant, since there is a transformation to internal energy in the resistor at the rate of dU/dt = -I2R

– Radiation losses are still ignored

• The circuit’s operation can be expressed as2

20

d Q dQ QL R

dt dt C

RLC Circuit Compared to Damped Oscillators

• The RLC circuit is analogous to a damped harmonic oscillator

• When R = 0

– The circuit reduces to an LC circuit and is equivalent to no damping in a mechanical oscillator

RLC Circuit Compared to Damped Oscillators

• When R is small:

– The RLC circuit is analogous to light damping in a mechanical oscillator

– Q = Qmax e-Rt/2L cos ωdt

– ωd is the angular frequency of oscillation for the circuit and

12 2

1

2d

Rω

LC L

RLC Circuit Compared to Damped Oscillators

• When R is very large, the oscillations damp out very rapidly

• There is a critical value of R above which no oscillations occur

• If R = RC, the circuit is said to be critically damped

• When R > RC, the circuit is said to be overdamped

4 /CR L C

Damped RLC Circuit, Graph

• The maximum value of Qdecreases after each oscillation

– R < RC

• This is analogous to the amplitude of a damped spring-mass system

Alternating Current Circuit

AC Circuits

• An AC circuit consists of a combination of circuit elements and a power source

• The power source provides an alternative voltage, Dv

• Notation Note

– Lower case symbols will indicate instantaneous values

– Capital letters will indicate fixed values

AC Voltage

• The output of an AC power source is sinusoidal and varies with time according to the following equation:

– Δv = ΔVmax sin ωt

• Δv is the instantaneous voltage

• ΔVmax is the maximum output voltage of the source– Also called the voltage amplitude

• ω is the angular frequency of the AC voltage

AC Voltage

• The angular frequency is

– ƒ is the frequency of the source

– T is the period of the source

• The voltage is positive during one half of the cycle and negative during the other half

22 ƒ

πω π

T

Resistors in an AC Circuit

• Consider a circuit consisting of an AC source and a resistor

• The AC source is symbolized by

• ΔvR = DVmax= Vmax sin wt

• ΔvR is the instantaneous voltage across the resistor

Resistors in an AC Circuit

• The instantaneous current in the resistor is

• The instantaneous voltage across the resistor is also given as

ΔvR = Imax R sin ωt

sin sin maxmaxIR

R

v Vi ωt ωt

R R

D D

Resistors in an AC Circuit

• The graph shows the current through and the voltage across the resistor

• The current and the voltage reach their maximum values at the same time

• The current and the voltage are said to be in phase

Resistors in an AC Circuit

• For a sinusoidal applied voltage, the current in a resistor is always in phase with the voltage across the resistor

• The direction of the current has no effect on the behavior of the resistor

• Resistors behave essentially the same way in both DC and AC circuits

Phasor Diagram

• To simplify the analysis of AC circuits, a graphical constructor called a phasor diagram can be used

• A phasor is a vector whose length is proportional to the maximum value of the variable it represents

Phasors

• The vector rotates counterclockwise at an angular speed equal to the angular frequency associated with the variable

• The projection of the phasor onto the vertical axis represents the instantaneous value of the quantity it represents

rms Current and Voltage

• The average current in one cycle is zero

• The rms current is the average of importance in an AC circuit

• rms stands for root mean square

• Alternating voltages can also be discussed in terms of rms values

0 7072

maxrms max

II . I

07072

maxmax.rms

VV V

DD D

Inductors in an AC Circuit

• Kirchhoff’s loop rule can be applied and gives:

0 or

0

max

,

sin

Lv v

div L

dt

div L V ωt

dt

D D

D

D D

Current in an Inductor

• The equation obtained from Kirchhoff's loop rule can be solved for the current

• This shows that the instantaneous current iL in the inductor and the instantaneous voltage ΔvL across the inductor are out of phase by (p/2) rad = 90o

max sin

2

max

max maxmax

cos

sin I

L

L

V Vi ωt dt ωt

L ωL

V π Vi ωt

ωL ωL

D D

D D

Phase Relationship of Inductors in an AC Circuit

• The current is a maximum when the voltage across the inductor is zero

– The current is momentarily not changing

• For a sinusoidal applied voltage, the current in an inductor always lags behind the voltage across the inductor by 90° (π/2)

Phasor Diagram for an Inductor

• The phasors are at 90o

with respect to each other

• This represents the phase difference between the current and voltage

• Specifically, the current lags behind the voltage by 90o

Inductive Reactance

• The factor ωL has the same units as resistance and is related to current and voltage in the same way as resistance

• Because ωL depends on the frequency, it reacts differently, in terms of offering resistance to current, for different frequencies

• The factor is the inductive reactance and is given by:– XL = ωL

Inductive Reactance

• Current can be expressed in terms of the inductive reactance

• As the frequency increases, the inductive reactance increases– This is consistent with Faraday’s Law:

• The larger the rate of change of the current in the inductor, the larger the back emf, giving an increase in the reactance and a decrease in the current

max rmsmax rms

L L

V VI or I

X X

D D

Voltage Across the Inductor

• The instantaneous voltage across the inductor is

max

max

sin

sin

L

L

div L

dt

V ωt

I X ωt

D

D

Capacitors in an AC Circuit

• The circuit contains a capacitor and an AC source

• Δv = ΔvC = ΔVmax sin ωt– Δvc is the instantaneous

voltage across the capacitor

Capacitors in an AC Circuit

• The charge is q = CΔVmax sin ωt

• The instantaneous current is given by

• The current is p/2 rad = 90o out of phase with the voltage

max

max

cos

or sin2

C

C

dqi ωC V ωt

dt

πi ωC V ωt

D

D

More About Capacitors in an AC Circuit

• The current reaches its maximum value one quarter of a cycle sooner than the voltage reaches its maximum value

• The current leads the voltage by 90o

Phasor Diagram for Capacitor

• The phasor diagram shows that for a sinusoidally applied voltage, the current always leads the voltage across a capacitor by 90o

Capacitive Reactance

• The maximum current in the circuit occurs at cos ωt = 1 which gives

• The impeding effect of a capacitor on the current in an AC circuit is called the capacitive reactance and is given by

maxmax

1which givesC

C

VX I

ωC X

D

maxmax max

(1 )

VI ωC V

/ ωC

D D

Voltage Across a Capacitor

• The instantaneous voltage across the capacitor can be written as ΔvC = ΔVmax sin ωt = Imax XC sin ωt

• As the frequency of the voltage source increases, the capacitive reactance decreases and the maximum current increases

• As the frequency approaches zero, XC approaches infinity and the current approaches zero

– This would act like a DC voltage and the capacitor would act as an open circuit

The RLC Series Circuit

• The resistor, inductor, and capacitor can be combined in a circuit

• The current and the voltage in the circuit vary sinusoidally with time

The RLC Series Circuit

• The instantaneous voltage would be given by Δv = ΔVmax sin ωt

• The instantaneous current would be given by i = Imax sin (ωt - φ)– φ is the phase angle between the current and the

applied voltage

• Since the elements are in series, the current at all points in the circuit has the same amplitude and phase

i and v Phase Relationships –Graphical View

• The instantaneous voltage across the resistor is in phase with the current

• The instantaneous voltage across the inductor leads the current by 90°

• The instantaneous voltage across the capacitor lags the current by 90°

i and v Phase Relationships –Equations

• The instantaneous voltage across each of the three circuit elements can be expressed as

max

max

max

sin sin

sin cos 2

sin cos 2

R R

L L L

C C C

v I R ωt V ωt

πv I X ωt V ωt

πv I X ωt V ωt

D D

D D

D D

Voltage in RLC Circuits

• ΔVR is the maximum voltage across the resistor and ΔVR

= ImaxR

• ΔVL is the maximum voltage across the inductor and ΔVL

= ImaxXL

• ΔVC is the maximum voltage across the capacitor and ΔVC = ImaxXC

• The sum of these voltages must equal the voltage from the AC source

• Because of the different phase relationships with the current, they cannot be added directly

Phasor Diagrams

• To account for the different phases of the voltage drops, vector techniques are used

• The phasors are rotating vectors

• The phasors for the individual elements are shown

Resulting Phasor Diagram

• The individual phasor diagrams can be combined

• Here a single phasor Imax

is used to represent the current in each element– In series, the current is the

same in each element

Vector Addition of the Phasor Diagram

• Vector addition is used to combine the voltage phasors

• ΔVL and ΔVC are in opposite directions, so they can be combined

• Their resultant is perpendicular to ΔVR

Total Voltage in RLC Circuits

• From the vector diagram, ΔVmax can be calculated

22

max

22

max max max

22

max max

( )

R L C

L C

L C

V V V V

I R I X I X

V I R X X

D D D D

D

Impedance

• The 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 ohms

max maxmax

22

L C

V VI

ZR X X

D D

22

L CZ R X X

Phase Angle

• The right triangle in the phasor diagram can be used to find the phase angle, φ

• The phase angle can be positive or negative and determines the nature of the circuit

1tan L CX Xφ

R

Determining the Nature of the Circuit

• If f is positive– XL> XC (which occurs at high frequencies)

– The current lags the applied voltage

– The circuit is more inductive than capacitive

• If f is negative– XL< XC (which occurs at low frequencies)

– The current leads the applied voltage

– The circuit is more capacitive than inductive

• If f is zero– XL= XC

– The circuit is purely resistive

Power in an AC Circuit

• The average power delivered by the AC source is converted to internal energy in the resistor

– av = ½ Imax ΔVmax cos f = IrmsΔVrms cos f

– cos f is called the power factor of the circuit

• We can also find the average power in terms of R

– av = I2rmsR

Power in an AC Circuit

• The average power delivered by the source is converted to internal energy in the resistor

• No 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 capacitor

– In 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 Phase

• The power delivered by an AC circuit depends on the phase

• Some applications include using capacitors to shift the phase to heavy motors or other inductive loads so that excessively high voltages are not needed

Resonance in an AC Circuit

• Resonance occurs at the frequency ωo where the current has its maximum value

– To achieve maximum current, the impedance must have a minimum value

– This occurs when XL = XC

– Solving for the frequency gives

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

1oω

LC

Resonance

• Resonance occurs at the same frequency regardless of the value of R

• As R decreases, the curve becomes narrower and taller

• Theoretically, 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

rms

av

o

V Rω

R ω L ω ω

D

Quality Factor

• The sharpness of the resonance curve is usually described by a dimensionless parameter known as the quality factor, Q

• Q = ωo / Δω = (ωoL) / R

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

• These points are called the half-power points

Quality Factor

• 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 frequencies

Transformers

• An AC transformerconsists of two coils of wire wound around a core of iron

• The side connected to the input AC voltage source is called the primary and has N1 turns

Transformers

• The other side, called the secondary, is connected to a resistor and has N2 turns

• The 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 core

Transformers

• Assume an ideal transformer

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

• In the primary,

• The rate of change of the flux is the same for both coils

1 1Bd

v Ndt

D

Transformers

• The voltage across the secondary is

• The voltages are related by

• When N2 > N1, the transformer is referred to as a step-up transformer

• When N2 < N1, the transformer is referred to as a step-down transformer

22 1

1

Nv v

ND D

2 2Bd

v Ndt

D

Transformers

• The 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

2

L

NR R

N

Transformers

• A transformer may be used to match resistances between the primary circuit and the load

• This 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

Rectifier

• The process of converting alternating current to direct current is called rectification

• A rectifier is the converting device

• The 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 diode– The diode has low resistance to current flow in this direction

• Because 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 resistor

• It 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 circuit

• The circuit is now a simple DC power supply

• The time variation in the circuit is close to zero

– It is determined by the RC time constant of the circuit

– This is represented by the dotted lines in the previous graph

Filter Circuit, Example

• A filter circuit is one used to smooth out or eliminate a time-varying signal

• After rectification, a signal may still contain a small AC component

– This component is often called a ripple

• By filtering, the ripple can be reduced

• Filters can also be built to respond differently to different frequencies

High-Pass Filter

• The circuit shown is one example of a high-pass filter

• A high-pass filter is designed to preferentially pass signals of higher frequency and block lower frequency signals

High-Pass Filter

• At low frequencies, ΔVout is 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 high

• As the frequency increases, the reactance and voltage decrease

• This is an example of a low-pass filter