Maxwell's Equations for different Medium

In the later half of the nineteenth century, Maxwell demonstrated that all previously established experimental facts regarding electric and magnetic fields could be summed up in just four equations. Nowadays these equations are generally known as Maxwell’s equation. Maxwell did not proved the equations. He only gave above opinion and some mathematical verification and modification of others. These equations can be represented as integral and differential form.

Let’s go

Here, 

H= magnetic field strength

E= electric field intensity

J= conduction current density

D= electric flux density

B= magnetic flux density

ρ_v = volume charge density

dL is the differential length and dS is the differential area whose direction is always outward normal to the surface.

When the field are static all field terms which have time derivatives  become zero, that is:

   dB/dt = 0

  dD/ dt = 0

So the Maxwell’s equation’s for static fields become:

 

Now we know the characteristics of free space that:

Relative permittivity ϵ_r = 1

Relative permeability µ_r = 1

Conductivity σ = 0

Conduction current density J = 0

Volume charge density ρ_v = 0

Characteristics impedance = 377Ω or 120π

So the Maxwell’s equation’s for the free space but no static field become

    Differential form                   Integral from

                              

Also the Maxwell’s equation’s for the free space with static field become:

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What is bilateral and unilateral circuit element?

Bilateral element: The element in which the voltage current relationship is same for current flowing in either direction( means changing input and output terminal no impact on the operation), is known as bilateral element. Example- transformer, resistor, inductance, capacitance etc.

Unilateral element: The element in which the voltage current relationship is not same for current flowing in either direction (means changing input and output terminal impact on the operation) is known as unilateral element. Example- vaccum tubes, diodes, transistors etc.

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Difference between active and passive components of an electric circuit

Circuit Components: 

An circuit component is any basic discrete device or physical entity in a system used to affect electrons or their associated fields. These may be two types:

Active Components: 

Those devices or components which produce energy in the form of Voltage or Current are called as Active Components. For more precise, an electrical element is called active, if it generates power and an electronic component is called active, if it can amplify or control power Example- generators, batteries, and operational amplifiers, transistors etc.

Passive Components:  

Those devices or components which store or maintain or attenuate but can not generate or amplify energy in the form of Voltage or Current are known as Passive Components. Example- capacitor, inductor, resistor etc.

Difference between active and passive components:

1. Active devices inject power to the circuit, whereas passive devices are incapable of supplying  any energy.

2. Active devices are capable of providing power gain (like amplifier), and passive devices are incapable of  providing power gain.

3. Active device requires external source for its operations and passive device doesn't require any external source for its operation. 

4. Sometimes some elements can be assumed as active and passive both. Then actually  it depends on the context in which the component is used . This doesn't make things easier. Especially for diodes there are so many conflicting or different arguments: "In most cases (rectifier, Zener diode etc.) a diode is no doubt a passive device. Only in some special cases like with a tunnel diode, when its negative resistance region is used, it can be considered as an ACTIVE device." Because- it is an active device since it requires an external power source, to operate it in forward or reverse bias. It is an active device, since it can be used as an waveform generator (half wave rectifier). If the i-v characterisitic of the diode are in region I and III, then it is a passive device (always dissipating power). I think most diodes fall into this category. That's why the pn-junction device may be considered as an active device.

Another type of component is also used. This may be named as electromechanical components. Electromechanical components can carry out electrical operations by using moving parts or by using electrical connections. Example- Piezoelectric devices, crystals, resonators etc.

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General Relation between voltage and current of transmission line: Derivation

We  know from the equivalent circuit of a transmission line (mentioned in a post before ) per unit length:

Series impedance, Z = R + jωL

Shunt admittance, Y = G + jωC

 

Let us consider a short section of transmission line PQ of length dx, at a distance x from the sending end A.

At   P,          voltage = V

                       Current = I

Then at Q,     voltage = V + dV

                            Current = I + dI 

Now we get for short section dx,

                          Series impedance = ( R + jωL )dx

                          Shunt admittance = ( G + jωC )dx

Therefore, Voltage drop from P to Q may be considered to be due to the current I flowing through the impedance  ( R + jωL )dx

                         V – ( V + dV ) = I ( R + jωL )dx

                   Or ,    - dV | dx   =    I ( R + jωL )   --------(i)

And the decrease in current from P to Q may be considered due to the voltage V applied to 
the shunt admittance   ( G + jωC )dx

                       I – ( I + dI ) = V ( G + jωC )dx

                   Or,  - dI | dx  = V  ( G + jωC ) ---------(ii)

Now differentiating (i) with respect to x, we get –

- d ²V | dx ²   =    ( R + jωL ) dI | dx

Or,    d ²V | dx ²   =    ( R + jωL )  ( - dI | dx )

                                  =  ( R + jωL )    ( G + jωC )V   [ from (ii) ]

So,  d ²V | dx ²   = P ²V   ------------(iii)

Where, P ² = ( R + jωL )    ( G + jωC )

P is propagation constant and a complex quantity may represent as

 P  = œ  + jß

Similarly from (ii) we get,  d ²I | dx ²   = P ²I   ----------(iv)

The equations (iii) and (iv) are standard linear differential equations and their solution in exponential form can be written as, 

V  =  A e ^-Px   + B e ^Px      ------------(v)

I   =   C e ^-Px  +  D e ^Px      ---------(vi)

Where A, B, C, D are arbitrary constants

              A, B having dimensions of voltage

             C, D having dimensions of current

So,     V = A e ^–œx  e ^–jßx   + B e ^œx  e ^jßx      

                          [ as  P  = œ  + jß ]

             I =  C e ^–œx  e ^–jßx    + D e ^œx  e ^jßx     

                        ---I term---         ---2^nd term ----

The first term in the equation of voltage and current are called incident waves which travels from source to load and suffer phase shift ß . The second term represents the reflected wave, which travels from load to source. The two are the general equation of voltage and current from sending end.

Now the complete general solution for transmission line, we consider the following  figure  in which load impedance is Z_R ( not equal to characteristic impedance).

  Now differentiating  (v) we get –

 

    dV | dx  =  - PA e ^-Px   + PB e ^Px  

     

   or,     I ( R + jωL )   = P (A e ^-Px  -  B e ^Px )             [from (i)]

or,   I  =   P (A e ^-Px  -  B e ^Px )  |  ( R + jωL )   

         

Since        cosh Px  =  ( e^Px  +  e ^–Px )| 2

                    sinh Px  =  ( e^Px  -  e ^–Px )| 2

therefore,  cosh Px   -  sinh Px    =   e  ^-Px 

                               cosh Px   +  sinh Px    =   e  ^Px

Now putting these values into (v) and (vii) –

V = A (cosh Px   -  sinh Px  ) + B (cosh Px   +  sinh Px ) 

V  =  coshPx ( A + B ) – sinhPx ( A – B )    --------(viii)

And   I =  {coshPx ( A - B ) – sinhPx ( A + B ) }  | Z_o   ------(ix)

Now condition at the input end is 

                  x = 0

                 V = V_s

                 I  =  I_s 

                coshPx = 1

               sinhPx = 0

Hence we get,    V_s  = A + B

                                 I_s = ( A  -  B ) | Z_o

Putting the values in (viii)  and (ix)  we get  – 

V  =  V_s  coshPx   –   I_s  Z_o  sinhPx 

I    =  I_s coshPx     -    sinhPx   V_s | Z_o

These  are the completely general equations of transmission line for voltage and current.

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