INTRODUCTION TO ELECTRICAL CIRCUITS -2
•Source Transformation
•R-L-C Parameters
•Voltage -Current relationships for Passive Elements
Source transformation:
A current source or a voltage source drives current through its load resistance and the magnitude of the current depends on the value of the load resistance.Consider a practical voltage source and a practical current source connected to the same load resistance RL as shown in the figure
R1’s in figure represents the internal resistance of the voltage source VS and current source IS.Two sources are said to be identical, when they produce identical terminal voltage VL and load current IL.The circuit in figure represents a practical voltage source & a practical current source respectively, with load connected to both the sources.The terminal voltage VL and load current IL across their terminals are same. Hence the practical voltage source & practical current source shown in the dotted box of figure are equal.The two equivalent sources should also provide the same open circuit voltage & short circuit current.
IL=VS/[Ri+ RL] and IL=I.r/[R+RL]
VS=IR or I = VS/R
Hence a voltage source Vs in series with its internal resistance R can be converted into a current source I = VS/R with its internal resistance R connected in parallel with it. Similarly a current source I in parallel with its internal resistance R can be converted into a voltage source V = IR in series with its internal resistance R.
R-L-C Parameters:
Resistance:
Resistance is that property of a circuit element which opposes the flow of electric current and in such a way converts electrical energy into heat energy.It is the proportionality factor in ohm’s law relating to voltage and current.
Ohm’s law states that the voltage drop across a conductor of given length and area of cross section is directly proportional to the current flowing through it .
R∝ i , V=R.i , i=V/R= G.V
Where the reciprocal of resistance is called conductance G. The unit of resistance is ohm and the unit of conductance is mho or Siemens.
When current flows through any resistive material, heat is generated by the collision of electrons with other atomic particles. The power absorbed by the resistor is converted to heat and is given by the expression
P= vi = i²R
where i is the resistor in amps, and v is the voltage across the resistor in volts. Energy lost in a resistance in time t is given by
Inductance:
Inductance is the property of a material by virtue of which it opposes any change of
the magnitude and direction of the electric current passing through the conductor. A wire of certain length, when twisted into a coil becomes a basic conductor. A change
in the magnitude of the current changes the electromagnetic field.
Increase in current expands the field & decrease in current reduces it. A change in current produces change in the electromagnetic field. This induces a voltage across the coil according to Faraday's laws of Electromagnetic Induction.
Induced Voltage V = L[ di/dt]
V = Voltage across inductor in volts , I = Current through inductor in amperes .
di = 1/L .[v] dt
Integrating both sides,
Power absorbed by the inductor P = VI = Li [di/dt]
Energy stored by the inductor
Therefore: a) V=L [di/dt] The induced voltage across an inductor is zero if the current through it is a constant. i.e,. an inductor acts as short circuit to dc.b) For time change the current within zero time(dt = 0) gives an infinite voltage across the inductor which is physically not at all feasible. In an inductor, the current cannot change suddenly. An inductor behaves as open circuit just after switching across dc voltage.c) The inductor can store finite amount of energy, even if the voltage across the inductor is zero. d) A pure inductor never dissipates energy and only stores it. Hence it is also called as a non–dissipation passive element. However, physical inductor dissipates power due to internal resistance.
Capacitance:
A capacitor consists of two conducting metallic surfaces separated by a dielectric medium.
It is a circuit element which is capable of storing electrical energy in its electric field.
Capacitance is its units of capacity to store electrical energy.
Capacitance is the proportionality constant relating the charge on the conducting plates to the potential.
Charge on the capacitor q∝ V
q = CV
Where C is the capacitance in farads, if q is charge in coulombs and V is the potential difference across the capacitor in volts.
i= dq/dt = C. [dv/dt ]
i=C .[dv/dt]
The capacitance of a capacitor is depends on the dielectric medium and their physical dimensions. For a parallel plate capacitor, the capacitance.
C= Є [A/D] = Є0.Єr .[A/D]
where A is the surface area of plates , D is the separation between two metallic plates.
Є - is the absolute permeability of medium ,
Є0-is the absolute permeability of free space ,
Єr- is the relative permeability of medium .
i = dq/dt = C [dv/dt]
dv/dt = i/C
integrating on both side ,
V = 1/C ∫ i . dt
The power absorbed by the capacitor P = Vi = v.c. [dv/dt] .
Energy stored in the capacitor W = ∫ P dt = ∫ VC [dv/dt]
= C∫ V dv = 1/2 . CV² Joules.
Thus the energy is stored in the electric field set up by the voltage across capacitor.
Therefore:
a.The current in a capacitor is zero, if the voltage across it is constant, i.e,. the capacitor acts as an open circuit in dc .
b.A small change in voltage across a capacitance within zero time gives an infinite current through the capacitor, which is physically impossible.In a fixed capacitor, the voltage cannot change suddenly . A capacitor behaves as short circuit just after switching across dc voltage.
c.The capacitor can be store a finite amount of energy,even if the current through it is zero.
d.A pure capacitor never dissipates energy but only stores energy hence it is called non-dissipative element of the circuit.
Voltage-Current Relationship For Passive Elements
There are Three Passive Elements they are Resistance,Inductance and capacitance.Thebehavior of these three elements along with the respective voltage-current relationship is given in the table
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