P. Piot, PHYS 375 – Spring 2008 Alternating & Direct Currents – AC versus DC signals – AC characterization – Mathematical tools: • Complex number • Complex representation of an AC signal – Resistor in an AC circuit – Capacitors – Reactance and Impedance – RC circuits – High and low-pass filters
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P. Piot, PHYS 375 – Spring 2008
Alternating & Direct Currents
– AC versus DC signals– AC characterization– Mathematical tools:
• Complex number• Complex representation of an AC signal
– Resistor in an AC circuit– Capacitors– Reactance and Impedance– RC circuits– High and low-pass filters
P. Piot, PHYS 375 – Spring 2008
Alternating Current (AC) versus Direct Current (DC)
• With AC it is possible to build electric generators, motors and power distribution systems that are far more effcient than DC.
• AC is used predominately across the world of high power
P. Piot, PHYS 375 – Spring 2008
Alternating Current (AC): waveforms
)()( tSTtS =+• AC signal are periodic:
periodπω2
1==
Tf
frequency pulsation
Heinrich Rudolf Hertz (1857-1894)
UNITS:f: in Hertz (Hz)ω: in rad.s-1
P. Piot, PHYS 375 – Spring 2008
• Can an AC waveform be characterized by a few parameters?
• Peak-to-peak (PP)
• Peak
• Average
• Practical Average
• Root-mean-square
Alternating Current (AC): characterization
)max(SP =
∫+
=Tt
t
dttST
S )(1
∫+
=Tt
t
dttST
AVG )(1
2/1
222 )(1
=−≡ ∫
+Tt
t
dttST
SSRMS ∫+
=Tt
t
nn dttST
S )(1where
)min()max( SSPP −=
P. Piot, PHYS 375 – Spring 2008
Alternating Current (AC): characterization
22)(
21)(sin11)(sin1)(
2
0
222
=⇒
=== ∫∫+
RMS
dT
dttT
RMSTt
t
π
ϑϑω
ω
• For some analytical waveform, there exits relation between the different parameters
• Take a sinusoidal waveform with amplitude 1 then
πϑϑ
π
ϑϑω
ω
π
π
2)sin(212
|)sin(|11|)sin(|1)(
0
2
0
==
==
∫
∫∫+
d
dT
dttT
AVGTt
t
P. Piot, PHYS 375 – Spring 2008
Alternating Current (AC): characterization
• It matters what waveform is considered
• For instance for the same peak value, a square waveform will result in higher power than a triangular waveform.
P. Piot, PHYS 375 – Spring 2008
• In the following we will consider sinusoidal-type waveform (in principle any waveform can be synthesized as a series of sine wave (Fourier)
• We will write (in real notation)
• It often better to use complex notation:
• And will often do calculation in complex notation and at the end recall that our physical signal is the real part of the complex results
• We can associate a vector in the complex plane to this complex number
Alternating Current (AC): mathematical tools
][)( )(0
φω +ℜ= tieStS
V
Real
Imag
inar
y
φ)cos()( 0 φω += tStS
P. Piot, PHYS 375 – Spring 2008
Resistor in an AC circuit
VTVR
VRIR
Real
Imag
inar
y
RR RIV =
• R is a real number. So in the complex plane, all quantities are along real axis
• Current and Voltage are said to be in phase
• When instantaneous value of current is zero corresponding instantaneous value of voltage is zero
• Note power > 0 at all time resistor always dissipates energy
PR
IRVR
P. Piot, PHYS 375 – Spring 2008
Capacitors: voltage versus current relation
tP
tE
tDJ
∂∂
+∂∂
=∂∂
= ε
CQ
AQLV
AQEQdSE
≡=⇒
=⇒=∫
0
00
.
ε
εε
• Current induced by electric displacement:.
• Assume a simple model of two plate separated by a small distance. Gauss’s law gives:
capacitance
tVCII
CAIdJd
tV
tEJ
∂∂
=⇔===∂∂
⇒∂∂
=1
000 εεε
P. Piot, PHYS 375 – Spring 2008
Capacitors: technical aspects
M. Faraday (1791-1867)
• Unit for Capacitance is Farad (in honor to Faraday)
• Capacitor symbol:
•
• Real world capacitors also introduce a resistance (we will ignore this effect)
P. Piot, PHYS 375 – Spring 2008
Capacitor
• A capacitor either acts as a load or as a source
• A capacitor can therefore store energy.
P. Piot, PHYS 375 – Spring 2008
Capacitor in an AC circuits
VT
VC
ICReal
Imag
inar
y
IC
VC• Capacitors do not behave the same as resistors
• Resistors allow a flow of e- proportional to the voltage drop
• Capacitors oppose change by drawing or supplying current as they charge or discharge.
CC
C CVidt
dVCI ω==
PC
P. Piot, PHYS 375 – Spring 2008
• The general linear relation between V and I is of the form
Z is called impedance.
• For a resistor Z=R is a real number.
• For a capacitor is an imaginary number
• Generally Z will be a complex number (if V and I are written in their complex forms)
• For instance if a circuit has both capacitor(s) and resistor(s) we expect Z to generally be a complex number
• For a capacitor the quantity is called reactance and is in Ohm (Ω)
Reactance and Impedance
IVZ /≡
CiZ
ω−
=
CX C ω
1−=
P. Piot, PHYS 375 – Spring 2008
Z
VT
Example: Impedance of a series RC Circuits
C
RCT
iXRCiRZ
ICiRRII
CiVVV
+=−=⇒
−=+−
=+=
ω
ωω)(
VT
• Let’s compute the total impedance of the RC circuit:
• The impedance can be written as:
• NA: Z=5-26.52i or |Z|=29.99 and Ξ=-79.325 degreeRC
witheC
RZ i
ωω1tan,1
222 −=Ξ+= Ξ
VC
IVR
XC
R
P. Piot, PHYS 375 – Spring 2008
XC
IR
Example: Impedance of a parallel RC Circuits
VT
I
RXR
CiZ
VR
CiIII
C
RC
1111
1
1
+=
+=⇒
+=+=
−
ω
ω
• Let’s compute the total impedance of the RC circuit:
• NA: Z=4.83-0.91i or |Z|=4.91 and Ξ=-10.68 degree
IC I
I
R
P. Piot, PHYS 375 – Spring 2008
General Analysis of an RC series circuits
( )tieVtVV ωω 00 cos ℜ==
V
dtdV
RI
RCdtdI
dtCIRIVVV CR
11=+⇔
+=+= ∫• Let’s write the ODE for the current
• How do we solve?
P. Piot, PHYS 375 – Spring 2008
Solving the differential equation for the RC series circuit
• Previous equation is of the form:
• First find the solution for the homogeneous equation
• Then find a particular solution of the inhomogeneous equation
• The general solution is of the form
• So finally we have
P. Piot, PHYS 375 – Spring 2008
General Analysis of an RC series circuits
t
CRCR
Vt
CRRVe
CRRVtI RC
t
ωω
ωωω
ω
ω
ω sin1cos111)(22
220
222
20
222
20
+−
++
+−=
−
• Applying previous results to RC series circuits gives:
• Or in real notations:
−
+
++=
+−111)(
222
2
0 RCtti
RCt
e
CR
RCi
eRVtI
ω
ω
ωω
P. Piot, PHYS 375 – Spring 2008
General Analysis of an RC series circuits
I/(V/
R)
Time (sec)
f=200 Hz
1,100
10,100 1000,100
(R[Ω],C[µF])
P. Piot, PHYS 375 – Spring 2008
RC series circuits as frequency filters: low pass
VCR
iRCV
ZV
CiI
CiV
c
c
22211
ωωωω
+−
=⇒
−=−=
)arctan(;1
1||
11
222
222
ωω
ωω
RCCR
A
CRiRC
VVA C
−=Θ+
=⇒
+−
==
C
RCT
XRCiRZ
ICiRVVV
+=−=⇒
−=+=
ω
ω)( • The voltage across capacitor is
• The gain A is defined as:
0lim;90lim
1||lim;0||lim
/1/1
/1/1
=Θ−=Θ
==
<<>>
<<>>
RCRC
RCRCAA
ωω
ωω
• Note the limits
Θ= ieAA ||
P. Piot, PHYS 375 – Spring 2008
RC series circuits as frequency filters: low passG
ain
Phas
e (r
d)ωRC
ωRC
−π/4
0.707
• Signal with frequencies below1/RC are unaltered,
• Signal with frequency above1/RC are attenuated
P. Piot, PHYS 375 – Spring 2008
RC series circuits as frequency filters: high pass
=Θ
+=⇒
++
==
ωωω
ωωω
RCCRRCA
CRiRCCR
VVA C
1arctan;1
||
1
222
222
222
• The gain A is defined as:
90lim;0lim
||lim;1||lim
/1/1
/1/1
=Θ=Θ
==
<<>>
<<>>
RCRC
RCRCRCAA
ωω
ωωω
• Note the limits
Θ= ieAA ||
V
CiR
RV
ZVRRIV
c
R
ω−
=⇒
==
1
• The voltage across capacitor is
P. Piot, PHYS 375 – Spring 2008
RC series circuits as frequency filters: high passG
ain
Phas
e (r
d)ωRC
ωRC
π/4
0.707
• Signal with frequencies above1/RC are unaltered,