chapter 33 rlc

7/31/2019 Chapter 33 Rlc

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What is AC circuit?

It involves power supplies that can generate thesine wave voltages

V(t) = Vo Sin (2 ft) -------------(1)

How to generate Sine wave voltage?

When a coil is rotated in a magnetic field, at a

constant angular frequency,

, a sinusoidal

voltage (EMF) is induced in the coil.

In physics, angular frequency (also referred toangular speed, radial frequency, circular

frequency, orbital frequency, and radian

frequency) is a scalar measure of rotation rate

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Frequency is the number of

occurrences of a repeating event

per unit time. It is also referred toas temporal frequency. The

period is the duration of one cycle

in a repeating event, so the period

is the reciprocal of the frequency.

Sine waves of various frequencies; the

bottom waves have higher frequencies

than those above. The horizontal axis

represents time
http://en.wikipedia.org/wiki/Timehttp://en.wikipedia.org/wiki/Turn_(geometry)http://en.wikipedia.org/wiki/Reciprocal_(mathematics)http://en.wikipedia.org/wiki/Reciprocal_(mathematics)http://en.wikipedia.org/wiki/Turn_(geometry)http://en.wikipedia.org/wiki/Timehttp://en.wikipedia.org/wiki/File:Sine_waves_different_frequencies.svghttp://en.wikipedia.org/wiki/File:AngularFrequency.gif


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I (t) = IoSin (2 ft) -------------------(2)

Where Io= Vo/R (maximum current), f= frequencyThe motion of the actual charges in AC circuit

Charges wiggle with large amplitudes at low

frequencies and with small amplitudes at high

frequencies.

the movement of electric charge periodically reverses

direction. In direct current (DC), the flow of electric

charge is only in one direction. High voltage transmission lines deliver power from

electric generation plants over long distances using

alternating current.

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http://en.wikipedia.org/wiki/Direct_currenthttp://en.wikipedia.org/wiki/Electric_generationhttp://en.wikipedia.org/wiki/Electric_generationhttp://en.wikipedia.org/wiki/Electric_generationhttp://en.wikipedia.org/wiki/Electric_generationhttp://en.wikipedia.org/wiki/Direct_currenthttp://en.wikipedia.org/wiki/Direct_currenthttp://en.wikipedia.org/wiki/Direct_current


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The properties of

Resistor, R in AC circuit

Inductor, L in AC circuit

Capacitor, C in AC circuit

RLC circuit

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Current, I Potential, V

Energy, UPower deliver,

P


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It consists of a combination of circuit element and ac

generator or source.

The output of AC generator is sinusoidal & varies

with time.

V (t) = Vo Sin ( t) ------------------(3)

Ohms law, V= IR, I= V/R

I(t) = Io

Sin ( t) -------------------(4)

Where = 2 f

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The graphs ofV(t) vs t andI(t) vs t


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The current and the voltage reach their maximum

values at the same time. The current and the voltage are said to be in phase

Phasor diagram

The length of the arrows correspond to Vmax and Imax The direction of the current has no effect on the

behavior of the resistor. The average value of the current over one cycle is

zero.

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No matter current is in positive direction or negative

direction, it also possesses the same amount of time

and same magnitude. The rate at which the electrical energy is dissipated in

the circuit is given by

P = work / time

= q V/ t

=I V

Since V= IR =I2R

And I= V/R = V2/R ------------------------(5)I= Instantaneous current

Electrical energy convert to heat in resistor

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The heating process produced by an AC current with

a maximum value of Imax is not the same as of that

in DC. The maximum current (Imax) occurs for a small

amount of time.

Average value of current= rms current (ammeter)

Irms = Imax/ 2 = 0.707 Imax ----------------(6)

Alternating voltages can also be discussed in terms of

rms values

Vrms = Vmax/ 2 = 0.707 Vmax --------------(7) The average power dissipated in a resistor that carries

AC current is

Pav = I2rmsR ------------------------(8)

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The output of a generator is given by V=200Sin t.

Find the rms current in the circuit when this generatoris connected to a 100 resistor. Find the maximum

current in the circuit.

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Vrms = Vmax/2

Irms = Vrms/R

Imax = Irms 2


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rms values will be used when discussing AC currents

and voltages

AC ammeters and voltmeter are designed to read rms

values

Many of the equations will be in the same form as inDC circuits, like

Vrms = IrmsR -------------------(9)

Vmax= ImaxR ------------------(10)

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From trigonometry equations

Sin ( t + /2) = sin ( t) cos ( /2) + cos ( t) sin ( /2)

= cos ( t)

Sin ( t - /2) = sin ( t) cos ( /2)cos ( t) sin ( /2)

= - cos ( t)

cos ( t + /2) = cos ( t) cos ( /2)sin ( t) sin ( /2)

= -sin ( t)

cos ( t - /2) = cos ( t) cos ( /2) + sin ( t) sin ( /2)

= sin ( t)

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Based on Kirchhoff's loop rule

VVc = 0

V= Vc

Vc = Vmax sin( t) ----------------(11)

Since Q= VC, so Q= Cvmax sin( t)

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AndI= dQ/dt, soI= CVmax cos( t) -----------------(12)

Ic= CVmax sin( t + /2) ----------------(13)

The graphs ofV(t) vs t andI(t) vs t

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For a sinusoidally applied emf, the current (Ic)

always leads the voltage (Vc) across the capacitor by

90 o. OR

The voltage across the capacitor lags behind thecurrent by 90 o.

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Phasordiagram


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The impeding effect of a capacitor on the current in

an AC circuit we called it Capacitive Reactance, Xc.

It is inversely proportional to the capacitance, C.

Xc 1/CXc = 1/ C= 1/ 2 fC ---------------(14)

Where f is in hertz (Hz) and C is in farad (F), then Xc

in ohms ( )

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What is inductors?

It has a self inductance that prevents the current from

increasing or decreasing instantaneously.

A type of energy storage elements.

When switch is close at t = 0, thecurrent begins to increase, and

because of the increasing current,

the inductor produces a back emf.

The current in the circuit above is impeded by the

back emf from the inductor.

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VL dI/dt

VL = L (dI/dt)

From Kirchhoff's Loop rule

VVL = 0

V= VL

V=L (dI/dt) = Vmax sin ( t) ---------(15)From (15), to get current I, we have to integrate it.

L (dI/dt) = Vmax sin ( t)

dI/dt = (Vmax/ L) sin ( t) dtI = (Vmax/ L) (- cos( t)/ )

= Vmax/ L (- cos( t))

I = (Vmax/ L) [sin ( t - /2)] -----(16)

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The effective resistance of a coil in an AC circuit is

called inductive reactance and is given by

XL L

XL = L = 2 fL ----------------------(17)

when frequency, fis in hertz (Hz) and inductance, L

is in Henry (H), XL will be in Ohms ( )

Vrms = Irms XL ----------------------(18)

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An RLC circuit (or LCR circuit) is an electrical

circuit consisting of a resistor, an inductor, and a

capacitor, connected in series or in parallel.

There are many applications for this circuit, like

oscillator circuit.

An electronic oscillator is an electronic circuit thatproduces a repetitive electronic signal, often a sine

wave or a square wave.

They are widely used in electronic devices.

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Common examples of signals generated by oscillators

include signals broadcast by radio and television

transmitters, clock signals that regulate computersand quartz clocks, and the sounds produced by

electronic beepers and video games..

Another important application is for tuning, such asin radio receivers or television sets, where they are

used to select a narrow range of frequencies from the

ambient radio waves.

In this role the circuit is often referred to as a tunedcircuit.

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http://en.wikipedia.org/wiki/Radio_transmitterhttp://en.wikipedia.org/wiki/Television_transmitterhttp://en.wikipedia.org/wiki/Television_transmitterhttp://en.wikipedia.org/wiki/Quartz_clockhttp://en.wikipedia.org/wiki/Video_gamehttp://en.wikipedia.org/wiki/Tuner_(electronics)http://en.wikipedia.org/wiki/Receiver_(radio)http://en.wikipedia.org/wiki/Television_sethttp://en.wikipedia.org/wiki/Television_sethttp://en.wikipedia.org/wiki/Television_sethttp://en.wikipedia.org/wiki/Television_sethttp://en.wikipedia.org/wiki/Receiver_(radio)http://en.wikipedia.org/wiki/Receiver_(radio)http://en.wikipedia.org/wiki/Receiver_(radio)http://en.wikipedia.org/wiki/Tuner_(electronics)http://en.wikipedia.org/wiki/Video_gamehttp://en.wikipedia.org/wiki/Video_gamehttp://en.wikipedia.org/wiki/Video_gamehttp://en.wikipedia.org/wiki/Quartz_clockhttp://en.wikipedia.org/wiki/Quartz_clockhttp://en.wikipedia.org/wiki/Quartz_clockhttp://en.wikipedia.org/wiki/Television_transmitterhttp://en.wikipedia.org/wiki/Television_transmitterhttp://en.wikipedia.org/wiki/Television_transmitterhttp://en.wikipedia.org/wiki/Radio_transmitter


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The input voltage Vdrives a current I through the

circuit. By Kirchhoffs Law,

V= VL + VC + VR

IZ= I( L + 1/ C+ R)

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To find the total resistance (Impedance, Z) in RLC

circuit,

Vmax2= VR

2 + (VL - VC)2

Z = [ R2 + (XL - XC)2]1/2 ------------ (19)

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Imax

VRImax

VL

Imax

VC

Imax

VL

VC

VR

VR Imax

VL - VC


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Where VR= ImaxR; VL= ImaxXL; VC= ImaxXC At resonance frequency, current is maximum where

the total resistance (impedance, Z) is minimum.XL = XC

Equation (19) becomes

Z= [R2]1/2

Z= R --------------------------- (20)

When XL = XCL = 1/ C

2 = 1 / LC4 2f2 = 1 / LC

f = 1 /[2 LC] ----------------------(21)

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Where is the phase angle between the current and

the maximum voltage.

The impedance triangle for a series RLC circuit, so

we can find by using equation

tan = ( XL XC ) /R -----------------(22)

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VRImax

VL - VC

Vmax


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In a series RLC circuit, the applied voltage has a

maximum value of 120 V and oscillates at 60 Hz. The

circuit contains a resistor (R= 800 ), a capacitor (C=

4F) and an inductor, L, which inductance can be

varied. If = 60o, Find the L?

The quality factor Q is defined as the following

equation,

Q = oL /R= 1 / oCR ----------------- (23)

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No power losses are associated with capacitors and

pure inductors in an AC circuits. In a capacitor, during first half of the cycle, energy is

stored. Whereas, during second half of the cycle,energy is returned.

In an inductor, the source does work against theback emf of the inductor and energy is stored in theinductor. But when the current begins to decrease inthe circuit, the energy is returned to the circuit.

In resistor, the average power delivered by thegenerator is converted to internal energy in theresistor.

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V= Vmax sin ( t) -------------------(24)

I=Imax sin ( t - ) ------------------(25)

Pav=I V

=Imax sin ( t - ) Vmax sin ( t)

=Imax Vmax [sin ( t) sin ( t - ) ] (26)

sin ( t - ) = sin ( t) cos ( ) - cos ( t) sin ( ) ( )

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Imax

VL - VC

Vmax


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Subs ( ) into (26)

Pav= Imax Vmax [sin ( t)[sin ( t) cos ( ) - cos ( t) sin ( )]]

=Imax Vmax [sin2( t)cos ( )sin ( t)cos ( t) sin ( )]

----------------------------------- (27)

Where,

sin ( t) cos ( t) = sin (2 t) andthe average value of [sin2 ( t)]av =

Subs these 2 equation into (27)

Pav= Imax Vmax [sin2( t)cos ( )0]

=Imax Vmax [1/2 cos ]

= Imax Vmax cos ------------------(28)

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Where,

Irms = Imax / 2 and Vrms = Vmax / 2

So, Pav= Imax Vmax cos

= Irms Vrms cos -----------------------(29)

The average power delivered by the source isconverted to internal energy in the resistor, it can befound by,

Pav= Irms2 R -----------------------(30)

When = 0 (purely resistive),

Pav= Irms Vrms ----------------------(31)

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A series RLCcircuit has a resistance of 45.0 and an

impedance of 75.0 . What average power is

delivered to this circuit when Vrms = 210 V?

A power supply with is connected between points aand d in Figure P33.26. At what frequency will it

deliver a power of 250 W?

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In physical terms, EMF can be regarded as circuit

elements which put energy into a circuit and a resistoras an element which removes energy from a circuit.

The energy is dissipated in the resistor as heat. In ACcircuit, we have additional circuit elements,

capacitance C and inductance L, which store energyin electric and magnetic fields respectively.

C and L are referred to as reactive elements while Ris resistive element.

There is no power loss in an ideal inductor andcapacitor.

Real circuits always have some resistance.

chapter 33 rlc

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