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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
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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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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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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.