Transcript V–2 AC Circuits 6. 8. 2003 1

V–2 AC Circuits
6. 8. 2003
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Main Topics
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Power in AC Circuits.
R, L and C in AC Circuits. Impedance.
Description using Phasors.
Generalized Ohm’s Law.
Serial RC, RL and RLC AC Circuits.
Parallel RC, RL and RLC AC Circuits.
The Concept of the Resonance.
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The Power
• The power at any instant is a product of the
voltage and current:
P(t) = V(t) I(t) = V0sin(t)I0sin(t + )
• The mean value of power depends on the
phase shift between the voltage and the
current:
<P> = VrmsIrmscos
• The quality cos is called the power factor.
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AC Circuit with R Only
• If a current I(t) = I0sint flows through a
resistor R Ohm’s law is valid at any instant.
The voltage on the resistor will be in-phase:
V(t) = RI0sint = V0sint
V0 = RI0
<P> = VrmsIrms = RIrms2 = Vrms2/R
• We define the impedance of the resistor :
XR = R
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AC Circuit with L Only I
• If a current I(t) = I0sint supplied by some
AC power-source flows through an
inductance L Kirchhoff’s law is valid in any
instant:
V(t) – LdI(t)/dt =0
• This gives us the voltage on the inductor:
V(t) = LI0cost = V0sin(t+/2)
V0 =  LI0
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AC Circuit with L Only II
• There is a phase-shift between the voltage
and the current on the inductor. The current
is delayed by  = /2 behind the voltage.
• The mean power now will be zero:
<P> = VrmsIrms cos = 0
• We define the impedance of the inductance:
XL = L  V0 = I0 XL
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AC Circuit with L Only III
• Since the impedance, in this case the
inductive reactance, is a ratio of the peak
(and also rms) values of the voltage over
current we can regard it as a generalization
or the resistance.
• Note the dependence on ! The higher is
the frequency the higher is the impedance.
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AC Circuit with C Only I
• If a current I(t) = I0sint supplied by some
AC power-source flows through an
capacitor C Kirchhoff’s law is valid in any
instant:
V(t) – Q(t)/C =0
• This is an integral equation for voltage:
V(t) = –I0/C cost = V0sin(t – /2)
V0 = I0/C
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AC Circuit with C Only II
• There is a phase-shift between the voltage
and the current on the inductor. The voltage
is delayed by  = /2 behind the current.
• The mean power now will be again zero:
<P> = VrmsIrms cos = 0
• We define the impedance of the capacitor:
XC = 1/C  V0 = I0 XC
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AC Circuit with C Only III
• Since the impedance, in this case the
capacitive reactance, is a ratio of the peak
(and rms) values of the voltage over current
we can regard it again as a generalization or
the resistance.
• Note the dependence on ! Here, the higher
is the frequency the lower is the impedance.
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A Loudspeaker Cross-over
• The different frequency behavior of the
impedances of an inductor and a capacitor
can be used in filters and for instance to
simply separate sounds in a loud-speaker.
• high-frequency speaker ‘a tweeter’ is
connected is series with an capacitor.
• low-frequency speaker ‘a woofer’ is
connected is series with an inductance.
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General AC Circuits I
• If there are more R, C, L elements in an AC
circuit we can always, in principle, build
appropriate differential or integral equations
and solve them. The only problem is that
these equations would be very complicated
even in very simple situations.
• There are, fortunately, several ways how to
get around this more elegantly.
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General AC Circuits II
• AC circuits are a two-dimensional problem.
• If we supply any AC circuit by a voltage
V0sint, the time dependence of all the
voltages and currents in the circuit will also
oscillate with the same t but possibly
different phase.
• So it is necessary and sufficient to describe
any quantity by two parameters its phase
and magnitude.
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General AC Circuits III
• There are two mathematical tools
commonly used:
• Two-dimensional vectors, so called, phasors in
a coordinate system which rotates with t so all
the phasors, which also rotate, are still
• Complex numbers in Gauss plane. This is
preferred since more operations (e.g. division,
roots) are defined for complex numbers.
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General AC Circuits IV
• The description by both ways is similar:
The magnitude of particular quality (voltage
or current) is described by a magnitude of a
phasor (vector) or an absolute value of a
complex number and the phase is described
by the angle with the positive x-axis or a
real axis.
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General AC Circuits V
• The complex number approach:
• Describe voltages V, currents I, impedances Z
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and admittances Y = 1/Z by complex numbers.
Then a general complex Ohm’s law is valid:
V = ZI or I=YV
Serial combination: Zs = Z1 + Z2 + …
Parallel combination Yp = Y1 + Y2 + …
Kirchhoff’s laws are valid for complex I and V
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General AC Circuits VI
• The table of complex impedances and
admitancess of ideal elements R, L, C,
• j is the imaginary unit j2 = -1:
• R:
ZR = R
YR = 1/R
• L:
ZL = jL
YL = -j/L
• C:
ZC = -j/C
YC = jC
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RC in Series
• Let’s illustrate the complex number
approach on a serial RC combination:
• Let I, common for both R and C, be real.
Z = ZR + ZC = R – j/C
|Z| = (ZZ*)1/2 = (R2 + 1/2 C2)1/2
tg = –1/RC < 0 … capacity like
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RL in Series
• Let’s have a R and L in series:
• Let I, common for both R and L, be real.
Z = ZR + ZC = R + jL
|Z| = (ZZ*)1/2 = (R2 + 2L2)1/2
tg = L/R > 0 … inductance like
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RC in Parallel
• Let’s have a R and L in parallel:
• Let V, common for both R and C, be real.
Y = YR + YC = 1/R + jC
|Y| = (YY*)1/2 = (1/R2 + 2C2)1/2
tg = –[C/R] < 0 … again capacity like
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RLC in Series I
• Let’s have a R, L and C in series:
• Let again I, common for all R , L, C be real.
Z = ZR + ZC + ZL = R + j(L - 1/C)
|Z| = (R2 + (L - 1/C)2)1/2
• The circuit can be either inductance-like if:
L > 1/C …  > 0
• or capacitance-like:
L < 1/C …  < 0
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RLC in Series II
• New effect of resonance takes place when:
L = 1/C  2 = 1/LC
• Then the imaginary parts cancel and the
whole circuit behaves as a pure resistance:
• Z, V have minimum, I maximum
• It can be reached by tuning L, C or f !
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RLC in Parallel I
• Let’s have a R, L and C in parallel:
• Let now V, common for all R , L, C be real.
Y = YR + YC + YL = 1/R + j(C - 1/L)
|Y| = (1/R2 + (C - 1/L)2)1/2
• The circuit can be either inductance-like if:
L > 1/C …  > 0
• or capacitance-like:
L < 1/C …  < 0
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RLC in Parallel II
• Again the effect of resonance takes place
when the same condition is fulfilled:
L = 1/C  2 = 1/LC
• Then the imaginary parts cancel and the
whole circuit behaves as a pure resistance:
• Y, I have minimum, Z,V have maximum
• It can be reached by tuning L, C or f !
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Resonance
• General description of the resonance:
• If we need to feed some system capable of
oscillating on its frequency 0 then we do it
most effectively if the frequency our source
 matches 0 and moreover is in phase.
• Good mechanical example is a swing.
• The principle is used in e.g. in tuning
circuits of receivers.
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Impedance Matching
• From DC circuits we already know that if
we need to transfer maximum power
between two circuits it is necessary that the
output resistance of the first one matches
the input resistance of the next one.
• In AC circuits we have to match (complex)
impedances the same way.
• Unmatched phase may lead to reflection!
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Homework
• Chapter 31 – 1, 2, 3, 4, 7, 12, 13, 24, 25, 40.
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Things to read and learn
• This lecture covers:
The rest of Chapter 31
• Try to understand the physical background
and ideas. Physics is not just inserting
numbers into formulas!
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The Mean Power I
• We choose the representative time interval  = T:
1
 P 
T
T
 V (t ) I (t )dt 
0
V0 I 0
T
T
V0 I 0
T
2
[sin
t cos   sin t cos t sin  ]dt

 sin( t   ) sin tdt 
0
The Mean Power II
T
V0 I 0
dt
 P 
{cos  [ 

T
2
0
cos 2t
sin 2t
0 2 dt ]  sin  0 2 dt} 
T
T
V0 I 0
cos   Vrms I rms cos 
2
• Since only the first integral in non-zero.
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AC Circuit with C I
• From definition of the current I = dQ/dt and
relation for a capacitor Vc = Q(t)/C:
t
t
I0
1
V (t ) 
dQ 
sin tdt 


C 0
C 0
 I0
I0
cos t 
sin( t  2 )
C
C
• The capacitor is an integrating device.
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LC Circuit I
• We use definition of the current I = -dQ/dt
and relation of the charge and voltage on a
capacitor Vc = Q(t)/C:
d 2Q Q(t )

0
2
dt
LC
• We take into account that the capacitor is
discharged by the current. This is
homogeneous differential equation of the
second order. We guess the solution.
LC Circuit II
Q(t )  Q0 cos(t   )
• Now we get parameters by substituting into
the equation:
1
  Q(t ) 
Q(t )  0   
LC
• These are un-dumped oscillations.
2
1
LC
LC Circuit III
• The current can be obtained from the
definition I = - dQ/dt:
I (t )  Q0 sin( t   ) 
I 0 sin( t   )
• Its behavior in time is harmonic.
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LC Circuit IV
• The voltage on the capacitor V(t) = Q(t)/C:
Q0
V (t ) 
cos(t   )
C
• is also harmonic but note, there is a phase
shift between the voltage and the current.
^