Transmission Lines - Models and Characteristic Impedance

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Transmission Lines
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In this section, we'll form a model for transmission lines and then analyze the equations that govern their behaviour. We'll then introduce the key property of characteristic impedance.

To understand transmission lines, we'll set up an equivalent circuit to model and analyze them. To start, we'll take the basic symbol for a transmission line of length L and divide it into small segments:

transmission line model divided into small segments

Then we'll model each small segment with a small series resistance, series inductance, shunt conductance, and shunt capcitance:

small section of transmission line modelled with inductance, resistance, conductance, capacitance

The parameters in the above figure are defined as follows:

R' - resistance per unit length for the transmission line (Ohms/meter)

L' - inductance per unit length for the tx line (Henries/meter)

G' - conductance per unit length for the tx line (Siemans/meter)

C' - capacitance per unit length for the tx line (Farads/meter)

We will use this model to understand the transmission line. All transmission lines will be represented via the above circuit diagram. For instance, the model for coaxial cables will differ from microstrip transmission lines only by their parameters R', L', G' and C'.

To get an idea of the parameters, R' would represent the d.c. resistance of one meter of the transmission line. The parameter G' represents the isolation between the two conductors of the transmission line. C' represents the capacitance between the two conductors that make up the tx line; L' represents the inductance for one meter of the tx line. These parameters can be derived for each transmission line. An example of deriving the paramters for a coaxial cable is given here.

Assuming the +z-axis is towards the right of the screen, we can establish a relationship between the voltage and current at the left and right sides of the terminals for our small section of transmission line:

voltage and current on small segment of transmission line for deriving telegraphers equations

Using oridinary circuit theory, the relationship between the voltage and current on the left and right side of the transmission line segment can be derived:

voltage and current on tx line

Taking the limit as dz goes to zero, we end up with a set of differential equations that relates the voltage and current on an infinitesimal section of transmission line:

telegraphers equations

These equations are known as the telegraphers equations. Manipulation of these equations in phasor form allow for second order wave equations to be made for both V and I:

wave equations for voltage and current

The solution of the above wave-equations will reveal the complex nature of transmission lines. Using ordinary differential equations theory, the solutions for the above differential equations are given by:

solution of wave equations

The solution is the sum of a forward traveling wave (in the +z direction) and a backward traveling wave (in the -z direction). In the above, amplitude of forward voltage wave is the amplitude of the forward traveling voltage wave, amplitude of 
backward voltage wave is the amplitude of the backward traveling voltage wave, amplitude of the forward current wave is the amplitude of the forward traveling current wave, and amplitude of the backward current wave is the amplitude of the backward traveling current wave.

Characteristic Impedance - Z0

We're now ready to introduce a fundamental parameter of every transmission line: its characteristic impedance. This is defined as the ratio of the magnitude of the forward traveling voltage wave to the magnitude of the forward traveling current wave:

characteristic impedance

In terms of the transmission line per-length parameters, the characteristic impedance is given by:

characteristic impedance Z0 for transmission line

Z0 will be extensively used in determining other transmission line parameters.

This page will end with special cases of the characteristic Impedance. If R'=G'=0, then the conductors of the transmission line are perfectly conducting (so R'=0) and the dielectric medium that separates the conductors has zero conductivity (so that G'=0). In this case, the line is referred to as a Lossless Line. The characteristic impedance becomes:

characteristic impedance for lossless lines

Another type of line of interest is the distortionless line This type of line may contain loss (so that the voltage dies off somewhat as it propagates down the line), but the magnitude of the attenuation is frequency-independent, and the phase constant varies linearly with frequency. This is desirable; similar to filter theory, this would be considered "linear phase" - that is, signals that come out of the transmission line might be attenuated, but have the same shape. The criteria for this is:

characteristic impedance for distortionless line

The Propagation Constant: propagation constant for transmission lines, important to antenna tutorials

The propagation constant shows up in the solution for the spatial variation of the voltage and current waves along the line (see above). The real part is given by ; this represents the rate of decay of the wave as it travels down the transmission line. The larger is, the more "lossy" the line is, and the faster the wave decays. If =0, then the line is lossless, and the voltage and current waves do not die (shrink) as they travel down the line.

The imaginary part of the propagation constant is given by . This represents the rate at which the waves oscillate as a function of position on the line. In contrast, frequency represents the rate of change of oscillation as function of time. For a lossless line, can be determined from the speed of propagation along the line (u). In general, can be determined from:

determining propagation constant for a transmission line

In the above, epsilon is the permittivity of the line, and mu is the permeability of the transmission line. Note that lambda in the above equation is the wavelength within the transmission line: it is not necessarily the wavelength of a wave of frequency f in free space. For an "air line", the speed u is equal to the speed of light c.

In the next section, we'll look at further solving for the wave equation, and move towards determining the input impedance for a transmission line.

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