Basic Theory - Transmission lines

Transmission lines are continuous borderline case of a network consisting of discrete LC elements. In pulsed power system, as energy is extracted rapidly, its regarded as a distributed parameter element. Various transmission line geometries and the corresponding distributed parameters are listed in the table below :

Here, L' = Inductance per unit length C' = Capacitance per unit length R' = Resistance per unit length G' = Shunt conductance per unit length

Now, we will discuss the case of only Coaxial transmission line. In further text, transmission line will always refer to Coaxial transmission line, unless mentioned other wise. Using the equivalent capacitance and inductance formula discussed above, following equivalent circuit can be drawn for each small section of transmission line of length 'dx' :

Here G' is due to leakage from one conductor to the other through the insulation. This is the case of a non-ideal transmission line. But for designing PFN, we will be considering ideal transmission line i.e. Loss-less or Low-loss transmission line. So R'=0, G'=0. This assumption gets us most of the important results.

Now, the equivalent circuit reduces to :

And for an infinite transmission line, equivalent circuit will be :

Transmission line Equations :

For section of transmission line of length 'dx', equivalent circuit is :

f() and g() are generally any differentiable functions, where f() is forward propagating function and g() is backward propagating function.

Equivalent distributed parameter network of transmission line :

As is clear from the equivalent circuit of ideal transmission line discussed before that transmission line is basically an infinite LC ladder network. So to calculate different equivalent networks with different arrangements of L and C, we firstly need to find one port LC network driving point impedance function.

Here, Z(s) is an impedance function of a passive one-port network, thus for the network to be physically realizable, causal and stable, Z(s) is a positive real function. Therefore,

1. Z(s) is real when s is real.

2. Re Z(s) ≥ 0, when Re s ≥ 0.

These are the basic properties of impedance function. Moving on to deriving a general expression for Z(s). For this, we will first discuss general properties of LC impedance function based on it being positive real function and being physically realizable :

1. Consider numerator and denominator of Z(s) as sum of odd and even power functions of s as described below :

(M is even N is odd)

As we know, when the input current is I, the average power dissipated by purely reactive one-port network is zero:

Thus, Average Power = = 0

(for pure reactive network)

2. The poles and zeros are simple and lie on the axis.

Since both M and N are Hurwitz, they have only imaginary roots, and it follows that the poles and zeros of Z(s) or Y(s) are on the imaginary axis.

In order for the impedance to be positive real, the coefficients must be real and positive. Impedance function cannot have multiple poles or zeros on the axis. And the highest powers of the numerator and the denominator polynomials can differ by, at most, unity.

3. The poles and zeros interlace on the axis, i.e. Highest power: 2n, next highest power must be 2n-2. They can't be missing term.

4. The highest powers of the numerator and denominator must differ by unity; the

OR

Z(s) or Y(s) is the ratio of even to odd or odd to even!!

lowest powers also differ by unity.

5. There must be either a zero or a pole at the origin and infinity.

Based on above properties, we can write general impedance function for infinite LC network as :

Since these poles are all on the axis, the residues must be real and positive in order for Z(s) to be positive real .

Since all the residues Ki are positive, it is easy to see that for an L-C function;

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