Transmission lines are used at frequencies from dc to about 50 or 60 GHz, but anything above 5 GHz only short runs are practical, because attenuation increases dramatically as frequency increases. There are three types of losses in conventional transmission lines: ohmic, dielectric, and radiation. An idealized transmission line consists of two parallel conductors of uniform cross-sectional area. The conductors possess a capacitance per unit length, C, and an inductance per unit length, L.
- In traditional low-frequency circuit analysis, the transmission line would not matter. As a result, the current that flows in the circuit would simply be:
- However, in the high frequency case, the length L of the transmission line can significantly affect the results. To determine the current that flows in the circuit, we would need to know what the input impedance is, Zin, viewed from the terminals of the transmission line:
- Examples of common transmission lines include the coaxial cable, the micro-strip line which commonly feeds patch/micro strip antennas, and the two wire line.
2. Voltage and Current Relationship
Using ordinary circuit theory, the relationship between the voltage and current on the left and right side of the transmission line segment can be derived:
- 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:
- Using ordinary differential equations theory, the solutions for the above differential equations are given by:
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, V+ is the amplitude of the forward traveling voltage wave,V- is the amplitude of the backward traveling voltage wave, I+ is the amplitude of the forward traveling current wave, and I- is the amplitude of the backward traveling current wave.
3. Characteristic impedance (Z0)
- It is defined as the ratio of the magnitude of the forward traveling voltage wave to the magnitude of the forward traveling current wave:
- In terms of the transmission line per-length parameters, the characteristic impedance is given by:
4. Lossless Transmission line and Distortionless Transmission line
- 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:
- 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:
- The characteristic impedance becomes:
5. Propagation constant
- 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 α; this represents the rate of decay of the wave as it travels down the transmission line. The larger is α, the more "lossy" the line is, and the faster the wave decays. If α=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 β. 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, β can be determined from the speed of propagation along the line (u). In general, β can be determined from:
- The Propagation Constant:
- Therefore final relation would be:
6. Voltage Reflection Coefficient
We are now aware of the characteristic impedance of a transmission line, and that the tx line gives rise to forward and backward travelling voltage and current waves. We will use this information to determine the voltage reflection coefficient, which relates the amplitude of the forward travelling wave to the amplitude of the backward travelling wave.
- To begin, consider the transmission line with characteristic impedance Z0 attached to a load with impedance ZL:
- At the terminals where the transmission line is connected to the load, the overall voltage must be given by:
- Putting this relation in voltage and current relationship expressions and we get:
- The ratio of the reflected voltage amplitude to that of the forward voltage amplitude is the voltage reflection coefficient. This can be solved for via the above equation:
6. Standing Waves
- We'll now look at standing waves on the transmission line. Assuming the propagation constant is purely imaginary (lossless line), We can re-write the voltage and current waves as:
- After plotting graphs on above equations, we came on to a conclusion that the ratio of Vmax to Vmin becomes larger as the reflection coefficient increases. That is, if the ratio of Vmax to Vmin is one, then there are no standing waves, and the impedance of the line is perfectly matched to the load. If the ratio of Vmax to Vmin is infinite, then the magnitude of the reflection coefficient is 1, so that all power is reflected. Hence, this ratio, known as the Voltage Standing Wave Ratio (VSWR) or standing wave ratio is a measure of how well matched a transmission line is to a load. It is defined as:
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