Question

11.14 For a lossless two-wire transmission line, show that (a) The phase velocity u = c = VLC (b) The characteristic impedance Z = - Vecosh-1 d. Is part (a) true of other lossless lines?

Answer 1

Q. ans.

Equations for a “lossless” Transmission Line A transmission line has a distributed inductance on each line and a distributed capacitance between the two conductors. We will consider the line to have zero series resistance and the insulator to have infinite resistance (a zero conductance or perfect insulator). We will consider a “Lossy” line. Define L to be the inductance/unit length and C to be the capacitance/unit length. Consider a transmission line to be a pair of conductors divided into a number of cells with each cell having a small inductance in one line and having small capacitance to the other line. In the limit of these cells being very small, they can represent a distributed inductance with distributed capacitance to the other conductor. Consider one such cell corresponding to the components between position x and position x + ∆x along the transmission line.

The small series inductance is L.∆x and the small parallel capacitance is C.∆x. Define the voltage and current to the right on the left side to be V and I. Define the voltage and current to the right on the right side to be V + ∆V and I + ∆I. We now can get two equations. 1. The current increment ∆I between the left and right ends of the cell is discharging the capacitance in the cell. The charge on the cell’s capacitance = capacitance x voltage = C.∆x.V and so the current leaving the capacitance to provide ∆I must be; ∆I = − ∂ ∂t(Charge) = − ∂ ∂t(C.∆x.V ) The minus sign is due to the current leaving the capacitor. ∆I = −C.∆x. ∂V/ ∂t ∆I ∆x = −C. ∂V/ ∂t Note the minus sign. 2. The voltage increment ∆V between the left and right ends of the cell is due to the changing current through the cell’s inductance. (Lenz’s Law) ∆V = −Inductance. ∂I ∂t = −∆x.L. ∂I ∂t ∆V/ ∆x = −L. ∂I/ ∂t . Now take the limit of the cell being made very small so that the inductance and capacitance are uniformly distributed. The two equations then become ∂I ∂x = −C. ∂V ∂t Equation 1. ∂V ∂x = −L. ∂I ∂t Equation 2. Remember that L and C are the inductance/unit length measured, in Henries/meter and are the capacitance/unit length measured in Farads/meter. Differentiate equation 2 with respect to the distance x.

∂ ∂x ( ∂V /∂x ) = −L. ∂ ∂x ( ∂I/ ∂t ) ∂ 2V /∂x2 = −L. ∂ ∂x ( ∂I /∂t ) x and t are independent variables and so the order of the partials can be changed. ∂ 2V /∂x2 = −L. ∂ ∂t( ∂I /∂x ) Now substitute for ∂I ∂x from equation 1 above ∂ 2V /∂x2 = −L. ∂ ∂t(−C. ∂V /∂t ) ∂ 2V /∂x2 = LC. ∂ 2V /∂t2 Equation 3 This is usually called the Transmission Line Differential Equation. Notes • L and C are NOT just the inductance and the capacitance. They are both measured per unit length. • The Transmission Line Differential Equation 3 above does NOT have a minus sign. The Transmission Line Differential Equation 3 above is a normal 1 dimensional wave equation and is very similar to other wave equations in physics. From experience with such wave equations, we can try the normal solution of the form V = V (s) where s is a new variable s = x + ut. Substituting this into the two sides of the Transmission Line Differential Equation 3 above we get the two sides being

∂ 2V/ ∂x2 and 1 /u2 . ∂ 2V/ ∂t2

Thus the form V (x + ut) can satisfy the Transmission Line Differential Equation 3 if and only if 1 u2 = LC Equation 4. Both roots of this satisfy the Equation 3.

u = ± √ 1/ LC

In a long transmission line the line constants are uniformly distributed over the entire length of line. This is because the effective circuit length is much higher than what it was for the former models (long and medium line) and hence we can no longer make the following approximations:

1. Ignoring the shunt admittance of the network, like in a small transmission line model.
2. Considering the circuit impedance and admittance to be lumped and concentrated at a point as was the case for the medium line model.

Rather, for all practical reasons, we should consider the circuit impedance and admittance being distributed over the entire circuit length as shown in the figure below. The calculations of circuit parameters, for this reason, are going to be slightly more rigorous as we will see here. For accurate modelling to determine circuit parameters let us consider the circuit of the long transmission line as shown in the diagram below.

AV Amzam V + AV YAX Je - - AX - - - - - - - - -- Long Transmission Line

Here a line of length l > 250km is supplied with a sending end voltage and current of V_S and I_S respectively, whereas the V_R and I_R are the values of voltage and current obtained from the receiving end. Lets us now consider an element of infinitely small length Δx at a distance x from the receiving end as shown in the figure where.
V = value of voltage just before entering the element Δx.
I = value of current just before entering the element Δx.
V+ΔV = voltage leaving the element Δx.
I+ΔI = current leaving the element Δx.
ΔV = voltage drop across element Δx.
zΔx = series impedance of element Δx
yΔx = shunt admittance of element Δx
Where, Z = z l and Y = y l are the values of total impedance and admittance of the long transmission line.

Therefore, the voltage drop across the infinitely small element Δx is given by

ΔΙ = IzΔε Or I2 = ΔΙ Δ.r Or Iz = ........ (1)

Now to determine the current ΔI, we apply KCL to node A.

ΔΙ = (V + ΔV)yΔz = VyΔr + ΔVyΔα

Since the term ΔV yΔx is the product of 2 infinitely small values, we can ignore it for the sake of easier calculation.
Therefore, we can write
In = Vy ........ (2)

Now derivating both sides of eq (1) w.r.t x,

“P = гр АР

Now substituting
Vy
from equation (2)

dv , 2 = cyᏙ oᎢ d2e , - zyᏙ = 0 ........ (3)

The solution of the above second order differential equation is given by.

Derivating equation (4) w.r.to x.

Now comparing equation (1) with equation (5)

Now to go further let us define the characteristic impedance Z_c and propagation constant δ of a long transmission line as

Ze=J and 8 = vyz

Then the voltage and current equation can be expressed in terms of characteristic impedance and propagation constant as

Now at x=0, V= V_R and I= I_r. Substituting these conditions to equation (7) and (8) respectively.

Solving equation (9) and (10),
We get values of A₁ and A₂ as,

Now applying another extreme condition at x = l, we have V = V_S and I = I_S.
Now to determine V_S and I_S we substitute x by l and put the values of A₁ and
A₂ in equation (7) and (8) we get

By trigonometric and exponential operators we know

Therefore, equation (11) and (12) can be re-written as

Thus compared with the general circuit parameters equation, we get the ABCD parameters of a long transmission line as,

from above analysis we can obtained the value of characteristics impedance as shown in the given question which is

Z_o=Z_c=120/($\varepsilon$_r)^0.5*cosh^-1(d/2a)

yes part (a ) is also true for other lossless lines.