capacitor a circuit element that stores electric energy and electric charges a capacitor always...
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Capacitor
d
A
V
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.
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
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
111
1
CCC
q
VVV
q
V
Ceq
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
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.
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
Complex number review
A
Ae
jA
ba
bj
ba
abajba
j
sincos
2222
22
Euler’s indentity
ab
11
2
1
2
1
2
1
11212121
22221111
21
21
21 ,
A
Ae
A
A
c
c
AAeAAcc
AeAcAeAc
j
j
jj
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
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
Capacitor Load
CjCj
jj
C
j
XjI
jVZ o
CC
CC
1
90
tCAdt
dqi
Cvq
tAv
CC
CC
C
cos
sin
90sin1
tC
AiC
ICE
VC(j)= A -90°
Notice the impedance of a capacitance decreases with increasing frequency
o
cC X
AjI 0
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
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.
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
2
21
2
2
1
21
)(0
)(
)(CRLCCR
SRLC
RLCC
CCR
RLC
CS
ZZZZZZ
jVZZZ
ZZZZ
ZZZ
ZZZ
ZjV
jI
0)()( -
)()( )(
221
211
jIZZZjIZ
jVjIZjIZZ
RLCC
SCCR
63.012.09.3512.08.4785500
0157.83687
8.4785500
63300575504450683005000456007500
4450683757.66100
01568375
)(7505.01500
7.667.66
101500
11
1
1
6
jI
jjj
jj
jjI
jjLjZ
jjjCj
Z
L
C
)(63.01500cos12.01 Atti
R1=100 , R2=75 , C= 1F, L=0.5 H, vS(t)=15cos(1500t) V