ch 21 pt 2 and ch 22
if the resistance R is equal to the inductive reactance X_L in an RL circuit, what is the power factor?
0.707
what is the power factor at the resonant frequency of an RLC circuit?
1
omega =
2pif
the current lags the voltage by a phase angle of
90 degrees
purely resistive AC: I= V/R becomes
I= V/R = Vo/r x sin(2pif) = Io x sin(2pif)
Vo=
NBA x omega
when phi=0 (such as at resonant f), R/Z=
R/Z equals 1 cos(0)=1
the power factor is
cos(phi)
in the phasor model, the voltage and current are represented by arrows called phasors that rotate
counterclockwise in an xy coordinate system
when a capacitor is connected to a DC source,
current doesn't flow once its charged
purely resistive AC circuit: when voltage increases,
current increases (directly related)
the magnetic flux through the inductor changes as the
current is changing
B changes as a function of time because
current is changing as function of time
the *vertical component* of the phasor is the instantaneous value of the
current or voltage V(t)=Vosin(2pift) I(t)=Iosin(2pift)
why?
current switches directions (I don't quite understand)
review: electric charges generate
electric fields (stationary charges generate only E fields)
two straight wires connected to the terminals of an AC generator can create an
electromagnetic (EM) wave
the net result of these interacting changing fields is a wave of E and B fields that can propagate through space. This wave is called an
electromagnetic wave
the change in magnetic flux induces an
emf
review: moving magnets generate
emf (since emf is a voltage, it basically gnerates E-field)
Irms depends on the frequency. it will be a max when Xc=Xl; the frequency this occurs at is ____ and is called the ____
f_0= (1/2pi)*sqrt[(1/LC)]; resonant frequency (Z will have its minimum value)
phasors rotate at a constant
frequency (or angular speed of 2pif)
XL is proportional to the
frequency of the AC voltage
a bulb is connected to a 60 Hz voltage source and glows with a certain brightness (1). A capacitor is then placed in series with the bulb (2). will the brightness of the bulb:
how is the current changed? 1. Irms= Vrms/R 2. Irms=Vrms/(Z) R1<R2 I1>I2 bulb dims
the current leads the voltage across the capacitor by 90 while being
in parallel with V_R
for a purely resistive circuit, the current and voltage are
in phase
Xc unit
is ohm (resistance)
the charge Q on the capacitor Q=CV, so
it is in phase with the voltage
how will the total impedance of an RC circuit change if the frequency is decreased?
it will increase
if the frequency of the AC voltage is increased, how will the current and voltage phasors change for a capactior?
length increases
maxwell also discovered: changing electric field (not just current) gives rise to
magnetic field
review: magnets generate
magnetic fields (stationary magnets- only B field)
An AC generator converts ____ into _____
mechanical energy into electric energy
phasor model can be used to determine voltage current relationship in
more complex circuits
the field is created first near the wires and then, like the effect of a pebble dropped into a pond,
moves outward
In an RL circuit, the voltage across the resistor and inductor are
not in phase
in an RC circuit, the voltage across the resistor and the voltage across the capacitor are
not in phase (note: Vr and I are in phase while Vc and I are out of phase)
Z (impedence) has a unit of
ohm
unit of XL is
ohm -inductive resistance
for a capacitor, the phasors remain
perpendicular while rotating because the phase angle between the current and the voltage is 90. Also, the current is ahead of the voltage.
for an inductor, phasors remain
perpendicular. voltage leads.
the current leads the voltage across the capacitor by a
phase angle of 90 degrees (1/4 of cycle or T)
most importantly, maxwell's four equations can be put together to
predict waves
James Clerk Maxwell (1831-1879) was the first to put together the four equations that
quantitatively relate electric charge, electric current, electric fields, and magnetic fields
the resistance in a purely resistive circuit has the ____ value at all frequencies
same
the induced emf in a rotating coil. varies
sinusoidally emf= NBAomega x sin(omega x t) *radians/second
Z also =
sqrt[R^2 + (1/2pifC)^2]
RLC: total impedance Z=
sqrt[R^2 + (Xl-Xc)^2]
Z=
sqrt[R^2 + Xc^2]
the voltages across the inductor and across the resistor are at 90 in the diagram; if they are added as vectors we find the total voltage: Vo=
sqrt[Vro^2 +Vlo^2] = Io*sqrt[R^2 + Xl^2]
RLC Circuit Vo=
sqrt[Vro^2+(Vlo-Vco)^2] = Io*Z
the average power used by a capacitor in an AC circuit is zero,
the capacitor alternately absorbs and release energy --> 1/2 of time P is +, 1/2 of time P is - --> 1/2 time capacitor absorbs E, 1/2 time capactior releases E --> avg P dissipated by capactior= 0
the magnetic fields created at earlier times propagate outward as a wave, just as
the electric fields do
what is not true of a capacitor in an AC circuit: its instantaneous voltage equals
the instantaneous current time the capacity reactance
the length of the voltage (current) phasor corresponds to
the max voltage (current)
An electric field is generated when
there are opposite charges on the two wires *good explanation in 3/5 video 2* *diagram of switching current and charges*
electromagnetic waves can be produces in situations that do not involve a
wire antenna
the power is as often positive as it is negative: the average power used by an inductor in an ac circuit is
zero (half stored E, half released E)
what will not generate EM waves?
a steady direct current (DC) -bc charge moves in constant speed AND in one direction
when a capacitor is connected to an AC source,
an alternating current will flow continuously
there is a phase angle between the overall voltage and the current cos(phi)=
V_Ro/V_o cos(phi)= IoR/IoZ cos(phi)= R/Z
the voltage across the capacitor and across the resistor are at 90 in the diagram; if they are added as vectors, we find the total voltage:
Vo= sqrt[VR0^2+Vc0^2] Vo=sqrt[(IoR)^2+ (IoXc)^2] Vo=Io[R^2 + Xc^2]
RLC: phase angle between current and total voltage tan(phi)=
(Vl-Vc)/Vr tan(phi)= (Xl-Xc)/R
emf= -N(delta phi/ delta t)... for an inductor in a circuit is
-L(delta phi/ delta t)
When Xl=Xc, tan(phi)=
0 and phi=0 total voltage and current phasor are in phase (parallel) -Vl and Vc cancel -total V = V across resistor
Xc=
1/(2pifC)
The current reaches its peak _________ the voltage does
1/4 cycle AFTER
1 period is
360 degrees
A 10 ohm resistor is connected to a source of AC voltage. if the peak current through the resistor is 1 A, what is the average power dissipated by the resistor
5 W
by going point by point in this manner, the time dependent voltage and current curves can be constructed
:)
review: moving charge (current) generates a
B field
if a changing B-field produces an E-field, that E-field is itself changing. This changing E-field will, in turn, produce
B-field, which will be changing, and so it too will produce a changing E-field and so on
in extreme case when f=0, V is not changing as function of time, T= infinity-->
DC circuit w/ capacitor! (Xc=> infinity) (current becomes zero)
an inductor is usually a coil of wire, and the basis of its operation is
Faraday's law of electromagnetic induction
RL: Vo=
Io*Z (Vrms= Irms*Z)
therefore, Vo=
Io*Z -same form as V=IR for DC circuits (Vrms=Irms*Z)
if the value of the capacitance is doubled but the peak voltage remains unchanged, the peak current is
Io= Vo/Xc Xc= 1/(2pifc) Xc is halved Current is doubled!
RLC: avgP=
Irms^2*R = Irms(Vrms/Z)R = Irms(Vrms)cos(phi)
what will happen to the light bulb if the frequency is doubled? Circuit 1: F, L, AND R Circuit 2: 2F, L, AND R
P=Irms^2*R Irms= Vrms/Z Irms= Vrms/sqrt[R^2 + 2pifL)^2] Irms decreases, P dec, I dec -> light dims
according to the above equation, the current I is increasing most rapidly when
V has its peak value V0 (point d). The current I will be decreasing most rapidly when V= -Vo (point b)
kirchoff's loop rule for a circuit with just L and an AC generator
V+emf=0 V=-emf= L(delta I/delta t) delta I/delta t= V/L Vapplied= Vosin2pift
XL=
V0/I0 = sqrt[2]*Vrms/sqrt[2]*Irms =2pifL -must be peak or rms not valid at any point
resonant frequency Irms=
Vrms/sqrt[R^2] Z=R -minimum Z and max Irms
the relationship between RMS V and RMS I for Capacitor AC circuits
Vrms= Irms*Xc (also Vo= Io * Xc) --> this is not valid at any instant --> depends on the Xc value and frequency of oscillating voltage
if no inductor, L=0 means Z= ? and cos(phi)=? phi=?
Z=R, cos(phi)=1, phi=0
the current used to generate the electric wave creates
a magnetic field
the electric current I through the inductor produces
a magnetic field through itself
in general, any electric charge that is _________ emits an electromagnetic wave
accelerating
the average power of a purely resistive AC circuit
avg P= IrmsVrms avg P= (IoVo)/2 (power is always positive bc V and I are in phase)
the magnetic field oscillates perpendicular to
both electric field and direction of wave travel
Xc is
capacitive reactance
the graph shows the voltage across the current through a single circuit element connected to an ac generator. what is the circuit element? -curve shows a higher peak voltage than current -voltage curve moves "slower" than the current -differ by 1/4 period
capacitor
as the oscillating current changes, the magnetic field
changes accordingly
inductive reactance relates the peak (and rms) voltage to the
peak (and rms) current
V/L=
delta I/ delta t
for a resistor, the current and voltage phasors remain ____ while rotating bc they're in phase
parallel