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Chapter 28: AC Circuits
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Terms in this set (112)
AC current
involve time varying electrical quantities
rotational motion in electric generators
naturally lead to voltage and current that vary sine w/ time
studying circuits w/ sine varying electrical quantities
insight to AC circuits
sine. AC voltage and current
characterized by:
1. amp (Vp, Ip)
2. freq (w)
3. phase constant (Ѳ)
same quantities describing SHM
V = Vp sin (wt + Ѳv) AND I = Ip sin (wt + Ѳi)
**i and v subscript = phase constant need not have same phase
amp
peak value (Vp, Ip) or room mean square value
root mean square value (rms)
how amp. is specificed
rms
it's an avg.
V(rms) = Vp/√2
I(rms) = Ip/√2
example of rms value
12 v housing wire
usually describe frequency f
in cycles per second or Hz
angular freq (w)
radians per second or inverse seconds
relation between frequency and angular freq
w = 2nf
same for rotational and SHM: full cycle contains 2n radian
phase constant of AC signal
tells when sine curve crosses zero w/ pos slope
separately study AC of
resistors
capacitors
inductors
ideal resistor
current and voltage are prop: I = V/R
resistor ACROSS an AC generator
making voltage across the resistors = generator voltage
generator voltage when Ѳ = 0, the current is
I = V/R = Vp sin wt/R = Vp/R (sin wt)
voltage and current are in phase => peak at the same time
current has same frequency as voltage, since phase constant as zero
peak current and voltage divided by resistance
Ip = Vp/R
both voltage and current are sine
so their rms values are same ratio as peak values => Irms = Vrms/R
capacitor ACROSS AC generator
voltage and charge directly prop. in capacitor => q = CV (diff. q and V)
dq/dq of q = CV
current flowing to capacitor plates (even though charge doesn't ACTUALLY flow through space between plates)
I = Cd/dt(Vpsin(wt)
=wCVpcos(wt) = wCVpsin (wt + n/2)
cosine curve
is like sine curve, but shifted to left by pi/2
in capacitors
CURRENT leads VOLTAGE by 90 degrees
wCVp
in wCVpcos(wt) => is peak current Ip = wCVp =>
Ip = Vp/1/wC = Vp/Xc
where Xc = 1/wC
resistance in capacitor
Xc = 1/wC
relation between peak voltage and current
capacitor
phase diff. between voltage and current => also difference between resistor and capacitors
resistor dissipates electric energy as heat
capacitor stores and releases electric energy
agent turning generator does no net work
agent turning generator with resistive load does work => gets dissipated as heat in resistor (Xc)
capacitive reactance Xc
Like resistance, this is measured in ohms
Xc
depends on frequency
frequency
goes from zero Xc to infinity
at zero = nothing is changing, no charge moving on or off the plates, =. capacitor might as well be an open circuit
as this inc => large currents flow to move charge on and off plates in ever shorter times => capacitor looks like a short circuit
capacitor at low freq
open circuit
capacitor at high freq
like short circuit
why does current lead voltage in capacitor
since capacitor voltage is proportional to its charge => takes current to MOVE charge onto capacitor plates => therefore THIS flows before THAT changes siginificantly.
inductor
across an AC generator
voltage across an inductor
leads the inductor current by 90
peak current (Ip)
Ip = V(p)/wL = V(p)/X(L)
inductive reactance X(L)
wL
inductors in ac circuits
no power is dissipated
energy is alternately stored and released as the inductor's magnetic field builds and decays
inductive reactance
increases w/ w and L
induced back emf => an inductor opposes changes in current => greater inductance => greater opposition => rapidly current changing => more inductor opposes change => inductive reactions inc. at high freq => inductor like open circuit
low freq => like short circuit => until w/ direct current (zero freq) => inductors as zero reactance since current is changing
why does inductor voltage lead current
CHANGING CURRENT in inductor => induces an emf
BEFORE CURRENT CAN BUILD UP, there must FIRST be voltage across the inductor
phasor diagrams
summarize phase and amp relations in AC circuit
phasor
arrow whose length represents the amp of an AC VOLTAGE or CURRENT, rotating counterclockwise with ANGULAR FREQUENCY w of the AC quantity
we use VERTICAL AXIS (represents sie. varying AC quantity)
PHASORS for CURRENT and VOLTAGE in a resistor
in phase => two phasors point in the same direction
PHASORS in CAPICITORS and INDUCTORS
current and voltage are at right angles (90 phase diff)
phasor magnitude related V(p) = I(p)X (X = appropriate reactance)
capacitors and inductors
are complementary
complement between e and m field
capacitors
oppose instant changes in voltage
inductors
opposes instant changes in current
in RC circuit
voltage builds up across the capacitor
in RL circuit
currents builds up in the inductor
similar curves
capacitor voltage and inductor current over time
capacitor stores electric energy
(1/2)CV^(2)
inductor stores MAG energ y
(1/2)LI^(2)
capacitor
open circuit at low freq
short circuit at high freq
inductor
short circuit at low freq
open circuit at high freq
LC Circuit
charge capacitor to some voltage V(p) and charge q(p) => connect across an inductor => capacitor has stored electric energy, but no current in INDUCTOR (so no mag. energy) => capacitor begins to discharge through inductor, but slowly at first because the inductor opposes the CHANGES in current => gradually current rises, and with it the magnetic energy => capacitor voltage, charge, and stored energy dec. => some time, initial energy divided equally between capacitor and inductor => capacitor keeps discharging => reach zero change => NOW all energy in E field of capacitor is in mag field of inductor
CURRENT IN INDUCTOR, and inductor current can't change instanteneously => current keeps flowing, putting pos charge on bottom plate of capacitor => stored electric energy inc AND current and magnetic energy both decrease => eventually capacitor fully charged => ALL energy is in capacitor => capacitor begins to disharge => REPEAT with COUNTERclockwise current => all energy is transferred to inductor
PROVIDED NO ENERGY LOSS => oscillations repeats indefinitely
LC oscillation like mass-spring system
KE of mass and PE of spring
oscillates w/ freq by mass and spring constant k
LC oscillation (like mass spring)
energy between magnetic energy of inductor and electric energy of capacitor
frequency determined by inductance L and capacitance C
C = k, L = m, current = velocity
total energy of LC circuit
sum of magnetic and electric energy
(1/2)LC^(2) + (1/2)CV^(2)
remains constant
in ideal LC circuit
quanitity of energy doesn't change => derivative is zero
dU/dt = 0
sine. oscillation
q = q(p) cos wt (capacitor charge as function of time)
w/. angular FREQ:
w = 1/sq rt (LC)
real inductors, capacitors and wires have
resistance
if low enough that fraction of energy lost in each cycle => like THIS
circuit oscillates
at a frequency near w = 1/sq rt (LC) BUT THIS amp. slowly declines as energy is dissipated in the resistance
RLC circuit
dU/dt => not zero, but rate of energy dissipation (-I^2R), where minus indicates energy loss from circuit
dU/dt = -I^2R
charge as a function of time ?
q(t) = q(p)e^-(Rt/2L)cos wt
RLC circuit damping
voltage and current behave similarly, w/ oscillation amp decaying exponentially w/ time constant 2L/R
RLC circuit
like damped harmonic motion
RLC circuit
as resistance increases => oscillations decay more rapidly and the frequency of oscillation DEC??
critical damping
when the time constant 2L/R = inverse of frequency => all circuit quantities decay to zero, w/o oscillation
RLC circuit connected across an AC generator
adding a generator => like adding external driving force on mech. oscillator
call generator freq. w(d), the driving frequency
driven RLC circuit (w/ AC generator added
exhibit resonant behavior => operation of radio, TV
in RLC circuit, we vary generator frequency wd, while keeping the generator's peak voltage constant
at low freq => capacitor open circuit (high reactance) => little current
at high freq => inductor open circuit (high reactance) => little current
some int. freq.=> current max => RESONANT FREQUENCY
resonant frequency
some int. freq in which the current must be max
is undamped natural freq. w(o) = 1/sq rt (LC)
RLC series circuit resonant freq.
same current through all components
capacitor => voltage lags
inductor => voltage leads
since same current both => voltage out of phase => cancel (only when two voltage have the same peak value)
resonant freq (w(0) = 1/sq rt(LC)
Ip = Vp/Xc and Ip = Vp/XL=> peak voltages are the same when capacitance Xc = XL = > GIVING THIS
AKA UNdAMPED NATURAL FREQUENCY
resonant frequ
aka undamped natural freq
at resonace in RLC circuit
the capacitor and voltages completely cancel
the voltages across the pair together is zero => AND AT RESONANT FREQ => pair like a wire
resistance alone determines circuit current
at any other frequency, the capacitor and inductor voltages don't cancel and the current is lower
phasor diagram to find CURRENT in RLC circuit as a function of DRIVING FREQ.
same current through all components (series) => single phasor of length I(p) represents the current
resistor voltage in phase w/ current => phasor, V(Rp) is in same direction as I(p)
Inductors voltage and capacitors leads and lags => phasors of VLp and VCp are perpendicular to the current phasor => each instant three voltages sum to give generator voltage (see formula P. 500) => see 2nd formula in terms of I, R, and reactance
****relates the peak current and voltage in RLC circuit
impadence (Z)
generalization of resistance including frequency-dependent effects of capacitance and inductance
Z = see formula (reciprocal of Ip
impedence
lowest when XL = Xc or w = 1/(LC)^(1/2) => equal to the resistance alone
becomes large at high freq, when XL = wl and low freq Xc = 1/wC (BIG RESISTANCE)
peak current vs. freq w/ 3 resistances
at low resistance => curve peaks sharp => high Q (high quality) => good job distinguishing resonance freq from nearby freq
at higher resistance => resonance curve broadens => circuit responds to range of frequencies => low Q
I(p) equation
phase difference vs. freq w
current and voltage out of phase by angle o (phase difference between voltage and current) => tan
angle o (phase difference between voltage and current)
pos => voltage leads
neg => current leads
at resonance
XL = XC and angle = 0 => capacitor and inductor voltages cancel => circuit behaves like pure resistance
Phase versus freq
at low freq => capacitance reactance dominates => angle is negative and current leads
at high freq => inductance reactance dominates dominates => pos and voltage leads
can't analyze AC circuits
by treating resistors, capacitors and inductors all as "resistors" with resistances R, Xc, XL, because each component has different phase relation between current and voltage
peak current vs frequency
the lower the resistance => sharper the peak value
phase constant vs frequency
lower resistance => higher amp. of phase constant
pos - inducance (voltage leads)
neg - capacitance (current leads)
when dealing with RLC circuits,
look in terms of the effects of capactiance and inductance to CURRENT via FREQUENCY and PHASE CONSTANT and AMP (MAYBE)
capacitors and inductors
don't dissipate energy, they alternately store and release it => average power consumption over one cycle is ZERO in a purely reactive circuit (one containing only capacitance and/or inductance)
capacitor
absorbs energy during part of cycle (power is positive) => it returns the same amount later (neg. power) => giving zero net power over the cycle (because current and voltage are out of phase => so product can be negative or positive at diff times)
resistor
V and I are in phase=> so power is ALWAYS positive and THIS takes energy from the circuit
I and V slightly out of phase
net power consumption but less than with a pure resistance
time average product of V and I w/ arbitrary phase diff angle
see formula p. 502
<> = time avg over cycle
average of (sin wt)(cos wt)
zero for two signals 90 out of phase
(sin w^2t) swings symm. from 0 to 1 => average is 1/2
voltage and current are in phase, avg. power
I(rms)V(rms)
voltage and current are out of phase, avg power
is lower, at 90 phase difference is zero
cos o is power factor
purely resistive circuit has a power factor 1
circuit w/ circuit w/ only inductance and capacitance has THIS as 0.
generally, THIS depends on frequency (1 at resonance but lower at other frequencies)
transformer
is a pair of wire coils, often wound on an iron core to concentrate magnetic flux
primary
a changing current in the THIS coil results in a changning magnetic flux through the SECONDARY => induces an emf in the 2ndary => drives current in any circuit connected across the secondary => device transferes electric power between two circuits w/o direct electrical contact
secondary
a changing current in the PRIMARY coil results in a changning magnetic flux through this => induces an emf in THIS
step up transformer
transfomer has more turns in its secondaryh
since each turn encircles the same changing magnetic flux => each gets same induced emf => emf across 2ndary is greather than across the primary
step down transfomer
interchanging primary and secondary would give this
V(2) = (N2)(V1)/(N1)
ratio of peak (or rms) secondary voltage V2 to the peak (or rms) primary voltage V1 is that same as the ratio of turns in two coils
step up transformer
increases voltage, but not power
ideal transfomer
passes all power applied to its primary on to the secondary => so I1V1 = I2V2
real transformers
have losses
transformers
work only w/ AC since use electromagnetic induction, so therefore require changing current
readily handle the voltage conversions in AC power systems
AC power
ease of changing voltage levels
low voltage leves are safer to user
BUT since P = IV => using higher VOLTAGE in long DISTANCE means lower CURRENT
power dissipated in conductors
I^2R => less power lost in transmission
changning voltage from a DC source
requires first interuppting DC => produce a changing current
diode
PN junction that serves as a one way valve for electric current
ideal diode
act like a short circuit in the preferred direction and like an open circuit in the opposite direction
DC power supply using transformer, diode and capacitor and delivering power to a load symbolized by the resistor R
transfomer steps voltage to desired level, while diode passes current only in preferred direction, chopping off negative half of AC cycle => capacitor smoothes (or filters) the remaining half to produce nearly steady DC
AC voltage rises => capacitor charges rapid through low resistance of the diode in its "on" state => but diode turns off when AC voltage drops => only resistor as a discharge path for the capacitor
** if RC time constant is long enough => capacitor voltage hardly drops before the next cycle again sends in a surge of charge.
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