INTRODUCTION TO ELECTRICAL CIRCUITS -2 ,Source Transformation ,R-L-C Parameters,Voltage -Current relationships for Passive Elements .

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

 

INTRODUCTION TO ELECTRICAL CIRCUITS -1 , Concept of Network and circuit , Types of Elements,Types of Sources

INTRODUCTION TO ELECTRICAL CIRCUITS -1

•Concept of Network and circuit

•Types of Elements

•Types of Sources

•Source Transformation

•R-L-C Parameters

•Voltage -Current relationships for Passive Elements 

 

INTRODUCTION: 

An Electric circuit is an inter-connection of various element's in which there is at least one closed path in which current can flow. An Electric circuit is used as a component for any engineering system.The performance of any electrical device or machine is always studied by drawing its electrical equivalent circuit. By simulating an electric circuit, any type of system can be studied for ex., mechanical, hydraulic thermal, traffic flow, weather prediction etc.All control systems are studied by representing of  them in the form of electric circuits. The analysis, of any system can be learnt by mastering the techniques of circuit theory. 

Elements of an Electric circuit:

 An Electric circuit consists of following types of elements. 

Active elements: 

Active elements are the elements of a circuit which possess energy of their own and can impart it to other element of the circuit.Active elements are of two types 

a) Voltage source    b) Current source 

     A Voltage source has a specified voltage across its terminals, independent of current flowing through it.

  A current source has a specified current through it independent of the voltage appearing across it.

Passive Elements: 

 The passive elements of an electric circuit do not possess energy of their own. They receive energy from the sources. The passive elements are the resistance, the inductance and the capacitance. 

 When electrical energy is supplied to a circuit element, it will respond in one and more of the following ways. 

If the energy is consumed or dissipated, then the circuit element is a pure resistor.

 If the energy is stored in the form of a magnetic field, the element is a pure inductor.

 And if the energy is stored in the form of an electric field, the element is a pure capacitor.

Linear and Non-Linear Elements:

 Linear elements show the linear characteristics of the voltage & current. That is its voltage-current characteristics are at all-times a straight-line through the origin.

 For example, the current passing through a resistor is proportional to the voltage applied through its and the relation is expressed as VI or V = IR. A linear element or network is one which satisfies the principle of superposition, i.e., the principle of homogeneity and additive.

 Resistors, inductors and capacitors are the examples of the linear elements and their properties do not change with a change in the applied voltage and the circuit current. 

   Non linear element’s V-I characteristics do not follow the linear pattern i.e. the current passing through it does not change linearly with the linear change in the voltage across it. Examples are the semiconductor devices such as diode, transistor.

Bilateral and Unilateral Elements: 

An element is said to be bilateral, when the same relation exists between voltage and current for the current flowing in both directions.Ex: Voltage source, Current source, resistance, inductance & capacitance.The circuits containing them are called bilateral circuits.

 An element is said to be unilateral, when the same relation does not exist between voltage and current when current flowing in both directions. The circuits containing them are called unilateral circuits.Ex: Vacuum diodes, Silicon Diodes, Selenium Rectifiers etc .

Lumped and Distributed Elements Lumped elements:

 Lumped elements are those elements which are very small in size & in which their simultaneous actions takes place. Typical lumped elements are capacitors, resistors, inductors. 

Distributed elements are those which are not electrically separable for analytical purposes.For ex: a transmission line has distributed parameters along its length and may extend for hundreds of Kms.

Types of Sources: 

Independent & Dependent sources: 

If the voltage of the voltage source is completely independent source of current and the current of the current source is completely independent of the voltage, then the sources are called as independent sources. 

The kind of sources in which the source voltage or current depends on some other quantity in the circuit which may be either a voltage or a current anywhere in the circuit are called Dependent sources or Controlled sources.

 There are four possibility dependent sources: 

a.Voltage dependent Voltage source 

b.Current dependent Current source 

c.Voltage dependent Current source

 d.Current dependent Current source

 

 

 The constants of proportionalities are written as B, g, a, r in which B & a has no units,

r has units of ohm & g units of mho . 

 Independent sources actually exist as physical entities such as battery, a dc generator & an alternator. But dependent sources are used to represent electrical properties of electronic devices such as diode,Transistors etc.,

Ideal & Practical sources: 

1.An ideal voltage source is one which delivers energy to the load at a constant terminal voltage, irrespective of the current drawn by the load at any time.

 2.An ideal current source is one, which delivers energy with a constant current to the load, irrespective of the terminal voltage across the load at any time. 

3.A Practical voltage source always possesses a very small value of internal resistance r. The internal resistance of a voltage source is always connected in series with it & for a current source; it is always connected in parallel with it.As the value of the internal resistance of a practical voltage source is very small, its terminal voltage is assumed to be almost constant within a certain limit of current flowing through the load.

 4.A practical current source is also assumed to deliver a constant current, irrespective of the terminal voltage across the load connected to it.

 

     

  

   

The equivalent single ideal voltage sum is given by V= V1 + V2 

   Any n number of ideal voltage sources connected in series can be represented by a single ideal voltage sum taking in to account the polarities connected together in to consideration.

 

When two ideal voltage sources are of e.m.f's V1 & V2 are connected in parallel,what voltage appears across its terminals is ambiguous.Hence such connections should not be made.However if V1 = V2= V, then the equivalent voltage sum is represented by V.In this case also, such a connection is unnecessary as only one voltage source serves the purpose.

 

When ideal current sources are connected in series, what current flows through the line is ambiguous. Hence such a connection is not permissible.However, it I1 = I2 = I, then the current in the line is I.But, such a connection is not necessary as only one current source serves the purpose.

 

 

Two ideal current sources in parallel can be replaced by a single equivalent ideal current source.