AC Circuits.pptx

AC Circuits

Why AC power and not DC?

• _{In the earliest days Direct Current (DC), where the power flows in one}

direction like a water hose, was the standard for delivering electrical power. Now Alternating Current (AC), where the power flow is constantly

alternating direction, is the standard for delivering electrical power.

• _{The standard for delivering electrical power changed from Direct Current} (DC) to Alternating Current (AC) because Alternating Current (AC) delivers electrical power more efficiently over long distances.

• _{In the US, 60 Hertz (cycles per second) is the Alternating Current (AC)} frequency.

Alternating Voltages and Currents

Electrical fires can be started by improper or damaged wiring because of the heat caused by a too-large current or resistance.

A fuse is designed to be the hottest point in the circuit – if the current is too high, the fuse melts.

RMS voltage or current

• _{The amount of AC power that produces the same heating effect as an equivalent}

DC power

• _{The RMS value is the square root of the mean (average) value of the squared}

function of the instantaneous values. The symbols used for defining an RMS value are V_RMS or I_RMS.

• The term RMS, ONLY refers to time-varying sinusoidal voltages, currents or

complex waveforms where the magnitude of the waveform changes over time and is not used in DC circuit analysis or calculations where the magnitude is

always constant. When used to compare the equivalent RMS voltage value of an alternating sinusoidal waveform that supplies the same electrical power to a

RMS value continued

• _{In other words, the effective value is an equivalent DC value which}

tells you how many volts or amps of DC that a time-varying sinusoidal waveform is equal to in terms of its ability to produce the same

power. For example, the domestic mains supply in the United

RMS value continued

RMS Voltage Analytical Method

• _{A periodic sinusoidal voltage is constant and can be defined as V(t) =}

Dividing through further as ω = 2π/T, the complex

equation above eventually reduces down too:

• _{Integrating through with limits taken from 0 to 360}o _{or “T”, the period}

Average Voltage

• _{The process used to find the Average Voltage of an alternating}

waveform is very similar to that for finding its RMS value, the

difference this time is that the instantaneous values are not squared and we do not find the square root of the summed mean.

• _{The average voltage (or current) of a periodic waveform whether it is}

a sine wave, square wave or triangular waveform is defined as: “the quotient of the area under the waveform with respect to time”. In

• _{For a periodic waveform, the area above the horizontal axis is positive}

while the area below the horizontal axis is negative. The result is that the average or mean value of a symmetrical alternating quantity is zero because the area above the horizontal axis (the positive half

• _{Then the average or mean value of a symmetrical alternating quantity,}

such as a sine wave, is the average value measured over only half a cycle since over a complete cycle the average value is zero regardless of the peak amplitude.

• _{The electrical terms Average Voltage and Mean Voltage or even}

Average Voltage Analytical

Method

• _{As said previously, the average voltage of a periodic waveform whose}

two halves are exactly similar, either sinusoidal or non-sinusoidal, will be zero over one complete cycle. Then the average value is obtained by adding the instantaneous values of voltage over one half cycle

• _{The area under the curve can be found by various approximation}

methods such as the trapezoidal rule, the mid-ordinate rule or

Terms related to ac circuits

• _{Form factor of a wave =}

Alternating Voltages and Currents

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}

Alternating Voltages and Currents

Alternating Voltages and Currents

Since this circuit has only a resistor, the current is given by:

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.

•_{The vector rotates counterclockwise at an angular}

speed equal to the angular frequency associated with the variable.

A Phasor is Like a Graph

•_{An alternating voltage can be presented in different representations.} •_{One graphical representation is using rectangular coordinates.}

• _{The voltage is on the vertical axis.} • _{Time is on the horizontal axis.}

•_{The phase space in which the phasor is drawn is similar to polar coordinate graph}

paper.

• _{The radial coordinate represents the amplitude of the voltage.} • _{The angular coordinate is the phase angle.}

• _{The vertical axis coordinate of the tip of the phasor represents the instantaneous value of}

the voltage.

• _{The horizontal coordinate does not represent anything.}

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.

•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.

Inductors in an AC Circuit

•_{Kirchhoff’s loop rule can be applied}

and gives:

0 or 0 max , sin L v v di v L dt di

v L V ωt

dt

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 Δv_L across the inductor are out of phase by (p/2) rad = 90o.

max _sin 2 max max max max cos sin I L L V V

i ωt dt ωt

L ωL

V π V

i 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,}

Phasor Diagram for an Inductor

•_{The phasors are at 90}o _{with respect}

to each other.

•_{This represents the phase}

difference between the current and voltage.

•_{Specifically, the current lags behind}

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:}

Inductive Reactance, cont.

•_{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 rms

max rms

L L

V V

I or I

X X

D D

Voltage Across the Inductor

•_{The instantaneous voltage across the inductor is}

max max sin sin L L di v L dt V ωt

I X ωt

Inductors in AC Circuits

Inductors in AC Circuits

The power factor for an RL circuit is:

Capacitors in an AC Circuit

•_{The circuit contains a capacitor}

and an AC source.

•_{Kirchhoff’s loop rule gives:} Δv + Δv_c = 0 and so

Δv = Δv_C = ΔV_max sin ωt

• _{Δvc is the instantaneous voltage}

Capacitors in an AC Circuit, cont.

•_{The charge is q = CΔVmax sin ωt}

•_{The instantaneous current is given by}

•_{The current is} p_{/2 rad = 90}o _{out of phase with the voltage}

max max cos or sin 2 C C dq

i ωC V ωt

dt

π

i 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}

Phasor Diagram for Capacitor

•_{The phasor diagram shows that for}

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

max max 1 which gives C C V X I ωC X D   max max max (1 ) V I ωC V

/ ωC

D

Voltage Across a Capacitor

•_{The instantaneous voltage across the capacitor can be written as ΔvC =}

ΔV_max sin ωt = I_max X_C 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}

RC

Circuits

RC

Circuits

This phasor diagram illustrates the phase relationships. The voltages across the

RC

Circuits

RC

Circuits

RC

Circuits

The power in the circuit is given by:

The

RLC

Series Circuit

•_{The resistor, inductor, and}

capacitor can be combined in a circuit.

•_{The current and the voltage in the}

The

RLC

Series Circuit, cont.

•_{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}

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 π

v I X ωt V ωt

π

v I X ωt V ωt

D   D  

D  _  _  D  

 

More About 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}

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.}

•_{The resultant of all the individual voltages across the}

individual elements is Δv_max.

Total Voltage in RLC Circuits

•_{From the vector diagram, ΔVmax can be calculated}





max max max

2 2

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

  

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 max max ₂ 2 L C V V I Z

R X X

D D

 

 

 2

2

L C

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.

1

tan XL XC

φ

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}

Resonance in Electrical Circuits

Resonance in Electrical Circuits

Resonance in Electrical Circuits