7C&L&Phasor

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Transcript 7C&L&Phasor

Capacitor
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
Capacitanc e only depends on the physical dimension of a capacitor : C 
q A

V
d
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
parallel
series
V=V1+V2+V3
q=q1=q2=q3
1
V

Ceq q
V=V1=V2=V3
q=q1+q2+q3
Ceq 
q q1  q2  q3

 C1  C2  C3
V
V
V1  V2  V3

q
1
1
1



C1 C 2 C3
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(wt+f)
• f=1/T cycles/s, 1/s, or Hz
 w=2pf rad/s
 f 2p (Dt / T) rad
=360 (Dt / T)
deg.
Average and RMS quantities in AC Circuit
T
1
x(t )   x(t )dt  0
T 0
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.
T
1 2
xrms 
x (t )dt

T 0
I
V
I rms 
,Vrms 
, Pave  I rmsVrms
2
2
Complex number review
Euler’s indentity

a
b

a  jb  a  b 
j
2
2
a 2  b2
 a b
 A(cos   j sin  )
2
2



 Ae j
 A
b
a
c1  A1e j1  A11 , c2  A2e j2  A2 2
c1  c2  A1 A2e j (1 2 )  A1 A2(1  1 )
c1 A1 j (1 2 ) A1

e

(1  1 )
c2 A2
A2
Phasor
How can an ac quantity be represented by a complex number?
Acos(wt+)=Re(Aej(wt+))=Re(Aejwtej )
Since Re and ejwt always exist, for simplicity
Acos(wt+)  AejA Phasor representation
Any sinusoidal signal may be mathematically represented in one of two
ways: a time-domain form
v(t) = Acos(wt+)
and a frequency-domain (or phasor) form
V(jw) = AejA
In text book, bold uppercase quantity indicate phasor voltage or currents
Note the specific frequency w of the sinusoidal signal, since this is not
explicit apparent in the phasor expression