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Capacitor
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qC
:capacitora of dimension physical theon dependsonly eCapacitanc
A circuit element that stores electric energy and electric charges
A capacitor always consists of two separated metals, one stores +q, and the other stores –q. A common
capacitor is made of two parallel metal plates.
Capacitance is defined as: C=q/V (F); Farad=Colomb/volt
Once the geometry of a capacitor is determined, the capacitance (C) is fixed (constant) and is independent of voltage V. If the voltage is increased, the charge will increase to keep q/V constant
Application: sensor (touch screen, key board), flasher, defibrillator, rectifier, random access memory RAM, etc.
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Capacitor: cont.
• Because of insulating dielectric materials between the plates, i=0 in DC circuit, i.e. the braches with Cs can be replaced with open circuit.
• However, there are charges on the plates, and thus voltage across the capacitor according to q=Cv.
• i-v relationship:
i = dq/dt = C dv/dt
• Solving differential equation needs an initial condition
• Energy stored in a capacitor: WC =1/2 CvC(t)2
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Capacitors in
V=V1=V2=V3
q=q1+q2+q3
321321 CCC
V
qqq
V
qCeq
parallel series
V=V1+V2+V3
q=q1=q2=q3
321
321
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1
CCC
q
VVV
q
V
Ceq
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Inductor
i-v relationship: vL(t)= LdiL/dt
L: inductance, henry (H)Energy stored in inductors
WL = ½ LiL2(t)
In DC circuit, can be replaced with short circuit
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Sinusoidal waves
• Why sinusoids: fundamental waves, ex. A square can be constructed using sinusoids with different frequencies (Fourier transform).
• x(t)=Acos(t+)• f=1/T cycles/s, 1/s, or Hz =2f rad/s 2t / rad
=360 t / deg.
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Average and RMS quantities in AC Circuit
01
0
T
dttxT
tx
It is convenient to use root-mean-square or rms quantities to indicate relative strength of ac signals rather than the magnitude of the ac signal.
rmsrmsavermsrms VIPV
VI
I ,2
,2
T
rms dttxT
x0
21
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Complex number review
A
Ae
jA
ba
bj
ba
abajba
j
sincos
2222
22
Euler’s indentity
ab
11
2
1
2
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A
Ae
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A
c
c
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AeAcAeAc
j
j
jj
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Phasor
How can an ac quantity be represented by a complex number?Acos(t+)=Re(Aej(t+))=Re(Aejtej )
Since Re and ejt always exist, for simplicity
Acos(t+) AejPhasor representation
Any sinusoidal signal may be mathematically represented in one of two ways: a time-domain form
v(t) = Acos(t+)
and a frequency-domain (or phasor) formV(j) = Aej
In text book, bold uppercase quantity indicate phasor voltage or currents
Note the specific frequency of the sinusoidal signal, since this is not explicit apparent in the phasor expression
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AC i-V relationship for R, L, and C
Resistive Load Source vS(t) Asint
tR
A
R
vRi
tAtvv
R
SR
sin
sin
vR and iR are in phase
Phasor representation: vS(t) =Asint = Acos(t-90°)= A -90°=VS(j)
IS(jw) =(A / R)-90°
Impendence: complex number of resistance Z=VS(j)/ IS(j)=R
Generalized Ohm’s law VS(j) = Z IS(j)
Everything we learnt before applies for phasors with generalized ohm’s law
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Capacitor Load
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jj
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CC
CC
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tCAdt
dqi
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tAv
CC
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cos
sin
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tC
AiC
ICE
VC(j)= A -90°
Notice the impedance of a capacitance decreases with increasing frequency
o
cC X
AjI 0
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Inductive Load
tL
Adt
L
Ai
dt
diLv
tAv
L
LL
L
cossin
sin
90sin tL
AiL
Phasor: VL(j-90°IL(j)=(A/L) -180°
ZL=jL
ELI
Opposite to ZC, ZL increases with frequency
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AC circuit analysis
• Effective impedance: example
• Procedure to solve a problem– Identify the sinusoidal and note the excitation frequency.
– Covert the source(s) to phasor form
– Represent each circuit element by its impedance
– Solve the resulting phasor circuit using previous learnt analysis tools
– Convert the (phasor form) answer to its time domain equivalent.
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Ex. 4.21 P188
R1=100 R2=75 C= 1F, L=0.5 H, vS(t)=15cos(1500t) V.Determine i1(t) and i2(t).
Step 1: vS(t)=15cos(1500t), =1500 rad/s.Step 2: VS(j)=15 0Step 3: ZR1=R1, ZR2=R2, ZC=1/jC, ZL=jLStep 4: mesh equation
0)()( -
)()( )(
221
211
jIZZZjIZ
jVjIZjIZZ
RLCC
SCCR
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SCCR
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R1=100 , R2=75 , C= 1F, L=0.5 H, vS(t)=15cos(1500t) V