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Electric Notes (Natural Response of RC circuits (V(0^-)=V(0^+)=V0, z=R.C,…
Electric Notes
Natural Response of RC circuits
V(0^-)=V(0^+)=V0
z=R.C
V(t)=V0e^-t/z
i(t)=(Vo/R)
e^-t/e
P=(V0^2/R)e^-2t/c
Wdiss=1/2CV0^2(1-e^-2t/c)
Resistive Circuits
Series Connection
Current is same
Req=ERi
Parallel Connection
Voltages are same
1/Req=E1/Req
Voltage Divider
Vj=Rj/Req.Vs
With only 2 resistors= Vi=Ri/(R1+R2).Vs
Current Divider
İj=(Req/Rj).İs
With only 2 resistors=İ1=R2/(R1+R2).İs
Delta-Y Equivalent Circuit
Delta-Y
Rx=(Neighbour sides)/(Sum of the Delta sides)
Y-Delta
Rx=(Sum of the multiplication of the Y sides)/(Pointing resistor)
Natural Response of RL circuits
İ(0^-)=İ(0^+)=I0
İ(t)=İ0e^-t/z
z=L/R
P=I^2Re^-2t/z
Wdissipated=1/2LIo^2(1-2e^-2t/z)
Wdiss=
INTEGRAL(0->t)
Pdt
Fundamental Definition of Circuit Parameters
Voltage(V)
[dW/dq] Energy per unit charge created by seperation [W(J), q(c)]
Current (A)
[dq/dt(s)] The rate of charge flow [Deltaq=Integrali.dt]
Power (W)
[dW(J)/dt(s)] Energy per unit time (P=v.i) [DeltaW=IntegralP.dt]
Ideal Basic Circuit Element
If P>0 circuit absorbing power
If P<0 circuit delivering power
P=v.i
Ohm's Law
i=V/R
P=v.i=i^2.R=v^2/R
Power in resistive circuit
Wdissipated=IntegralP.dt
Wtotal dissipated=Integral(0 to infinity)P.dt
Method of using Kirchoff's Laws
2)Write KCL equations at nodes and KVL equations in loops as many as the number of unknowns
3)Solve n equations with n unknowns
Be careful about the direction of currents and polarity of voltage
When there are dependent sources, express their values in terms of the unknowns in your equations
1)Determine unknown currents
Determination of Thevenin Equaivalent
General Method:
Calculate open circuit voltage with request to intercoated terminals
Using source tranformation:
(Finds Vth and Rth) but works with only independent sources
Deactivation Mode:
(Find Rth)
Replace voltage sources with short cicuit
Replace current sources with open circuit
(Work only with independent sources)
Kirchoff's Laws
Sum of the currents at a node is 0
KCL=Eİnode=0
Eİin=Eİout
Leaving current=+
Entering currents=-
Sum of the voltages around any closed path is 0
DeltaVloop=0
Inductive and Capacitive Circuits
Inductive
V=L(di/dt)
i=1/L
INTEGRAL
Vdt+i(0)
P=v.i=L.i.di/dt
Wstored=1/2Li^2
Winitial=1/2Li(0)^2
Wremaining=1/2[Li(t->0)]^2
Inductors behaves as short circuit when full
Capacitors
i=c(dV/dt)
V=1/c
INTEGRAL
İdt+V(0)
P=v.i=c.V.(dV/dt)
Wstored=1/2cV^2
Winitial=1/2c(V(0))^2
Wremaining=1/2c(V(t->0))^2
Capacitors behaves as open circuit when full
Techniques of circuit analysis
1)Node-Voltage Method
Application of KCL
b)Pick a reference node (The node connects the most elements)
a)Determine essential node (Connects at least of 3 circuit elements)
c)Define node voltages (Voltage across essential nodes and reference node)
f)Solve linear equation set to obtain node-voltage (Classically or matrix evaluation)
If you have dependent source/s, define their value in terms of node-voltages so obtain "constraint equations"
If you have a voltage source (only voltage source either ind or dep) in terms of essential nodes. Use the technique of super node=Write a combined equation for both essential nodes)
d)write node-voltage equations (Sum of
leaving
cureents at essential nodes from KCL Eleaving=0)
e)Reorganize your equations in the form of AVnv=B)
2)Mesh-Current method
Application of KVL
1)Define mesh currents
2)Write mesh-current equations (sum of voltages around each meshes according to KVL DeltaVmesh=0)
4)Solve equation to obtain İmesh
If you have dependent sources, define their values in terms of mesh-current, so obtain "constraint equations"
If you have a
only
current source(Dep/Ind) between two meshes use supermesh, combine mesh equations
!!
Watch out, do not define new currents, rather use predefined mesh-currents
3)Reorganize your equations as Aİmesh=B
Thevenin and Norton Equivalents
Any resistive network containing ind. and dep. sources can be replaced by its Thevenin equivalent
Maximum Power Transfer
To obtain maximum power from a circuit on a load
RL=Rth
Pmax=Vth^2/4Rth
Superposition Technique
The response of a circuit can be obtained as the
sum
of responses from individual sources.
1)Pick a source and deactivate the others. Find solution 1.
2)Repeat for each sources
General Solution
=Esolutioni
Source Transformation