Star to Delta and Delta to Star Transformations:

Star to Delta and Delta to Star Transformations:

 In a pattern series and parallel connections, electrical components may be connected in Star or Delta configurations as shown in the figure below (with some Resistances). Many a times circuits have to be transformed from Star to equivalent Delta and Delta to equivalent Star configurations such that the network terminal Resistances (  or Impedance's ) across the terminals are the same. We will show this transformation methodologies and the resulting configurations for both Delta to Star and Star to Delta one by one.

 

 

 

Delta to Star Transformation: 

The circuit configurations are identically provided then the network net resistances across the terminal pairs XY , YZ and ZX in both connections are the same. 

In Star Connection they are:

 RX-Y = RX+RY ----------------------(a) 

 RY-Z = RY+RZ -----------------------(b) 

 RZ-X = RZ+RX -----------------------(c)

 Similar way in Delta connection they are:

 RX-Y = R1//(R2+R3) = R1(R2+R3) / [R1+R2+R3] ------------------(d)  

 RY-Z= R2//(R1+R3) = R2(R1+R3) / [R1+R2+R3]  -------------------(e)

 RZ-X= R3//(R1+R2) = R3(R1+R2) / [R1+R2+R3] --------------------(f) 

 By equating these respective equation's, we get

RX+ RY = R1(R2+R3) / [R1+R2+R3] ------------------(g) 

RY+ RZ = R2(R1+R3) / [R1+R2+R3]  -------------------(h)

 RZ+ Rx = R3(R1+R2) / [R1+R2+R3] --------------------(i)

 By subtracting equation h from equation g given above,we get

 

𝑅𝑋−𝑅𝑍= [[𝑅1𝑅2+𝑅1𝑅3]/[𝑅1+𝑅2+𝑅3]]− [[𝑅2𝑅1+𝑅2𝑅3]/[𝑅1+𝑅2+𝑅3]]---------(j)

 Then adding this equation to equation i above i.e. (RZ+RX) we get

 2𝑅𝑋= [𝑅1𝑅2+𝑅1𝑅3−𝑅2𝑅1−𝑅2𝑅3+𝑅3𝑅1+𝑅3𝑅2] / [𝑅1+𝑅2+𝑅3] 

2𝑅𝑋=2𝑅1𝑅3 / [𝑅1+𝑅2+𝑅3]

𝑅𝑋=𝑅1𝑅3 / [𝑅1+𝑅2+𝑅3]

And in a similar way we can get:

 𝑅𝑌=𝑅1𝑅2 /[𝑅1+𝑅2+𝑅3]

𝑅𝑍=𝑅2𝑅3 /[𝑅1+𝑅2+𝑅3]

Where RX , RY and RZ are the equivalent resistances in the Star Circuit connection corresponding to the Delta Circuit connection with resistances R1, R2 and R3.

 Star to Delta Transformation: 

Now we have to get the equivalent values of R1, R2 and R3 in Delta connection in terms of the three resistances RX, RY and RZ in Star connection. Let us, use the equations we got earlier i.e. RX, RY and RZ in terms of R1, R2 and R3 and get the sum of the three product pairs i.e. RXRY+ RYRZ+ RZRX as

 𝑅𝑋𝑅𝑌+𝑅𝑌𝑅𝑍+𝑅𝑍𝑅𝑋 = 𝑅1²𝑅2𝑅3+ 𝑅2²𝑅1𝑅3+ 𝑅3²𝑅1𝑅2 / (𝑅1+𝑅2+𝑅3)²

 Here let us divide this equation by RX to get

 𝑅𝑌+𝑅𝑍+ [𝑅𝑌𝑅𝑍] / 𝑅𝑋 = 𝑅1𝑅2𝑅3(𝑅1+𝑅2+𝑅3) / 𝑅𝑋(𝑅1+𝑅2+𝑅3)²

                            =𝑅1𝑅2𝑅3 / 𝑅𝑋(𝑅1+𝑅2+𝑅3)

Now substituting the value of RX= (R1+R2+R3) / R1.R3 from the earlier equations into the above equation we will get

 𝑅𝑌+𝑅𝑍+𝑅𝑌𝑅𝑍𝑅𝑋=[𝑅1𝑅2𝑅3 / (𝑅1+𝑅2+𝑅3)] × [(𝑅1+𝑅2+𝑅3) / 𝑅1𝑅3] =𝑅2

 Then after similarly dividing the same equation by RY and RZ we get the other two relations as:

 𝑅𝑋+𝑅𝑍+𝑅𝑋𝑅𝑍 / 𝑅𝑌 = R3

 𝑅𝑌+𝑅𝑋+𝑅𝑋𝑅𝑌 / 𝑅𝑍 = 𝑅1

Thus we get the three equivalent resistances R1, R2 and R3 in Delta connection in terms of the three resistances RX, RY and RZ in Star connection as 

 𝑅𝑌+𝑅𝑋+𝑅𝑋𝑅𝑌 / 𝑅𝑍 = 𝑅1 

𝑅𝑌+𝑅𝑍+𝑅𝑌𝑅𝑍 / 𝑅𝑋 = 𝑅2 

𝑅𝑋+𝑅𝑍+𝑅𝑋𝑅𝑍 / 𝑅𝑌 = R3

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