With the transmission line clearly defined as a circuit element, it can now be analyzed when a load is attached. Using the characteristic impedance, we can define the current in terms of the voltage. 11 leads to the conclusion thatĪnd is defined as the characteristic impedance of the transmission line. 8, we can get the relationshipĬomparing the terms in eq. Where $$e^$$ for the reflection in the negative direction. Solving equations 7 & 8 for I(z) and V(z) give Is for a single line and is a function of frequency. Where the $$\gamma$$ is the complex propagation constant:
We can then solve these equations simultaneously to find I(z) and V(z).Įquations 7 and 8 are commonly known as the telegraph equations. Simplify equations 3 and 4 using Cosine phasors. For initial simplicity, the model is two parallel lines with one conductor and one ground.Įxamining a section of the lumped element model as a single mesh with Kirchoff’s voltage law, and using circuit elements whose values are per-unit length:Īnother analysis of the lumped element model with Kirchoff’s current law with one node at the top, gives the equation:ĭivide both sides by $$\Delta z$$ and take the limit as $$\Delta z\rightarrow 0$$ (note that the last terms become derivatives). These elements can be thought of as being distributed along the length of the transmission line. To quantify this analytically, consider the familiar passive parasitics affecting a circuit’s performance: inductance, capacitance, resistance, and conductance (L, C, R, G). So if the propagation delay of a wire or trace is 5ns, then any signal with a rise time of less than 10ns will be affected due to transmission line effects. If the rise time is less than twice the propagation delay, transmission line effects must be considered. The ratio of wavelength to wire length can be considered as low as 0.01.Īs a mental shortcut, so as not having to analyze the harmonic components of a signal, compare the rise time of the signal to the propagation delay. When the frequency and lengths become comparably large, the impedance becomes non-negligible. This is due to the effects being frequency dependent and typically having very small values that are linearly dependent on wire/trace length. When the physical dimension of a circuit approaches the magnitude of a wavelength of the signal, wires and circuit traces begin to affect circuit performance. What's a transmission line and why does it exist? Find out!
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