Electric Circuits I “Dr. Ahmed El-Shenawy”
<Dr Hadi El-Helw><Dr Ahmed El-Shenawy>
Electrical Circuits I Lecture 12
Electric Circuits I “Dr. Ahmed El-Shenawy”
The response of the basic elements R,L, and C to a sinusoidal voltage or current
ResistanceThe resistance is unaffected by the frequency of the applied sinusoidal voltage or currentFor purely resistive circuit V and I were in phase
RVI m
m or RIV mm
Electric Circuits I “Dr. Ahmed El-Shenawy”
Inductance
For the inductor the voltage across an inductor is directly related to the rate of change of current through the coil. Consequently, the higher the frequency, the greater will be the rate of change of current through the coil, and the greater the magnitude of the voltage
Electric Circuits I “Dr. Ahmed El-Shenawy”
tLItILdtdiLv
tItIdtd
dtdi
dtdiLv
mmL
L
mmL
LL
cos)cos(
cos)sin(
or
)90sin( tVv mL Where mm LIV
Therefore, for the inductor, vL leads iL by 90o, or iL lags vL by 90o
The quantity ωL, called the reactance of an inductor, is symbolically represented by XL and is measured in ohms
Electric Circuits I “Dr. Ahmed El-Shenawy”
fLLX L 2 (ohms)
In an ohm’s law format, its magnitude can be determined from:
m
mL I
VX
Inductive reactance is opposition to the flow of current, which results in the continual interchange of energy between the source and the magnetic field of the inductor
Electric Circuits I “Dr. Ahmed El-Shenawy”
Capacitance
The voltage across the capacitor is limited by the rate at whichcharge can be deposited on, or released by, the plates of the capacitor during the charging and discharging phases, respectively. In other words, an instantaneous change in voltageacross a capacitor is opposed by the fact that there is an element of time required to deposit charge on the plates of a capacitor
Electric Circuits I “Dr. Ahmed El-Shenawy”
The fundamental equation relating the voltage across a capacitor to the current of a capacitor is:
dtdvci c
c
applying differentiation
tVtvdtd
dtdv
mmc cos)sin(
Therefore
tcVtVcdtdvci mm
cc cos)cos(
or
)90sin( tIi mc where mm CVI
Electric Circuits I “Dr. Ahmed El-Shenawy”
For a capacitor ic leads vc by 90o or vc lags ic by 90o
The quantity 1/ωc called the reactance of a capacitor, is symbolically by Xc and is measured in ohm
Capacitor reactance is the opposition to the flow of charge, which results in the continual interchange of energy between the source and the electric field of a capacitor.
cX c
1
Electric Circuits I “Dr. Ahmed El-Shenawy”
Series RL circuit
vmVV imII
Where
22 )( LRVI m
m
RL
vi 1tan
Electric Circuits I “Dr. Ahmed El-Shenawy”
Series RC circuit
vmVV imII
Where22 )1(
CR
VI mm
CRvi 1tan 1
Electric Circuits I “Dr. Ahmed El-Shenawy”
Series RLC circuit
vmVV imII
Where22 )1(
CLR
VI mm
RcL
vi
1
tan 1
Electric Circuits I “Dr. Ahmed El-Shenawy”
Example 1
A coil is connected to supply voltage as shown in the figure below calculate the following
1.the inductive reactance(XL)2.the total impedance (Z)3.the current(I)4.the phase angle Ө between the current and the supply voltage
Electric Circuits I “Dr. Ahmed El-Shenawy”
Example 2
Two coils are connected in series across a voltage supply (V) as shown in the figure below. Calculate the value of the voltage V1,V2 across each coil. each coil has its own resistance
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