**Transverse Electromagnetic Waves:** Propagation of energy in a transmission line takes place such that electric and magnetic fields transverse to one another and also to direction of propagation. The resultant wave is termed as TEM wave.

Consider a small section n of a parallel wire.

Length of this line is dx.

Voltage at input is V, at the other end is V+dV; Similarly current is I, I+dI.

Assume **Primary Line Constants** of the line are R,L,G and C

Note: w= 2.pi.f

For small dx, dI is zero.

Potential drop across the line is V – (V+dV) = R.dx.I + jwL.dx.I

à -V’ = I(R+jwL) ….. eq.1

Note: V’ = (d/dx)V

In a similar method, assuming dV is zero,

à -I’ = V(G+iwC) ….. eq.2

Differentiate eq.1,2

V”= V(R+jwL)(G+jwC) ….. eq.3

I” = I(R+jwL)(G+jwC) …..eq.4

Let (gamma)^2 = (R+jwL)(G+jwC)

Gamma = (alpha) + j(beta)

alpha is attenuation constant

beta is phase constant

gamma is **propagation constant**

Now eq.3,4 becomes

V” = V. (gamma)^2

I” = I. (gamma)^2

Solving the above equations,

V=A.exp(-gamma.x) + B.exp(gamma.x) …..eq.5

I= C.exp(-gamma.x) + D.exp(gamma.x) …..eq.6

Note: exp(x) = e^x

The first terms in eq.5,6 are called **incident component**(magnitude of V or I decreases from source towards load, whereas the second terms are called **reflected component**(magnitude of V or I decreases form load towards source)

** **

**Hypothetical infinite line:** Voltage at distant end approaches zero(i.e no reflected component)

At x=0, V=Vs

Substitute in eq.5, Vs=A+B

At x=infinity, V=0

à B=0, V= Vs.exp(-gamma.x)

V’=-gamma.Vs.exp(-gamma.x)=-(R+jwL)I

Simplifying,

I=(Vs.exp(-gamma.x))/Z0

Where Z0=[(R+jwL)/(G+jwC)]^(1/2)

Z0 the input impedance of such infinite line is commonly referred to as **Characteristic Impedance** of the line. Z0 and propagation constant are termed as **secondary constants** (or coefficients) of the line

**Line Terminated in a Load Impedance Zr:**

At a distance x from the source, voltage and current are Vx, Ix.

-V’=(R+jwL).Ix

Substitute the above after differentiating eq.5,6 and then simplifying,

Voltage, current at source are Vs=A+B, Is=(A-B)/Z0

Vx=Vs.cosh(gamma.x)-Is.Z0.sinh(gamma.x) …..eq.7

Ix=Is.cosh(gamma.x)-(Vs/Z0)sinh(gamma.x) …..eq.8

For load impedance Zr, length of transmission line is l.

Ix=Ir, Vx=Vr such that Vr=Ir.Zr

Substitute the above in eq.7,8, and solve for Vs/Is

Input Impedance Zin = Vs/Is = Z0.N/D

Where N = Zr.cosh(gamma.l) + Z0.sinh(gamma.l)

D= Z0.cosh(gamma.l) + Zr.sinh(gamma.l)

**Line Terminated in Load Impedance Z0:**

Zr=Z0 in N,D

à Zin=Z0

A line terminated in its characteristic impedance has input impedance equal to Z0. In such a line, there is no reflected component and at any point x distant from the signal source, the voltage and current are same as that for infinite length transmission line.

** **

**Low frequency transmission line: **

R is very bigger than wL

G is very lesser than wC

Z0 = (R/jwC)^(1/2)

**High frequency line:**

R is very lesser than wL

G is very lesser than wC

Z0 = (L/C)^(1/2)

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