Voltage and Current on a transmission line: Look for V=ZI Alan Murray

Also … can a “real” Line with R>0, G>0 (slightly lossy) be truly distortionless? It turns out that it can, provided that It turns out that it can, provided that This is known as the Heaviside Condition and allows a line to be designed to have zero distortion, even with R>0, G>0 This is known as the Heaviside Condition and allows a line to be designed to have zero distortion, even with R>0, G>0 Oliver Heaviside ( ) - English physicist and electrical engineer Also known for predicting the "Heaviside Layer" in the atmosphere

Current on a transmission line So far we have calculated the voltage So far we have calculated the voltage Instantaneous voltage at any point = Instantaneous voltage at any point = Forward wave + backward wave …Forward wave + backward wave … The current has (almost) the same form The current has (almost) the same form I=V÷R, but I+ and I- in opposite directionsI=V÷R, but I+ and I- in opposite directions I+→I+→ I-←I-←

Current on a transmission line … more non-examinable maths! Telegrapher’s Equations relate I to V From last lecture Look for V=ZI, I=V/Z

Current on a transmission line … more non-examinable maths! The characteristic impedance of the line

In the lossless case, Z o is purely REAL, i.e. resistive Sanity check – Characteristic Impedance of Lossless Line? Sanity check – Characteristic Impedance of Lossless Line?

Characteristic Impedance of a Slightly Lossy Line Characteristic Impedance of a Slightly Lossy Line Series expansions “small” x “small” ÷”large” = “very small”!

Characteristic Impedance of a Slightly Lossy Line Characteristic Impedance of a Slightly Lossy Line This is complex, but if () = 0 … i.e. So although the line has losses, its characteristic impedance is the same as that of a lossless line so no dispersion → no distortion occurs. This is known as the Heaviside condition for a distortionless line.

Characteristic Impedance – Some General Points The characteristic impedance relates: The characteristic impedance relates: the forward voltage wave V + to the forward current wave I +the forward voltage wave V + to the forward current wave I + or or the backward voltage wave V - to the backward current wave I -the backward voltage wave V - to the backward current wave I - It does not link the total current and voltage (unless V - =I - =0) It does not link the total current and voltage (unless V - =I - =0)  X

Example Characteristic impedance of a parallel-wire line. Calculate the characteristic impedance of a lossless, air-spaced, two-wire transmission line for which the wire radius is 0.5 mm and the spacing is 5mm. Calculate the characteristic impedance of a lossless, air-spaced, two-wire transmission line for which the wire radius is 0.5 mm and the spacing is 5mm. 5mm 0.5mm “it can be shown that …”

Example Characteristic impedance of a parallel-wire line. 5mm 0.5mm

Example 4.2 Characteristic impedance of a coaxial line. A coaxial, lossless transmission line with an inner conductor of diameter 2mm and internal diameter for the exterior conductor of 7.5 mm is filled with polythene dielectric (ε r = 2.56, µ r = 1). Calculate the characteristic impedance of the line. A coaxial, lossless transmission line with an inner conductor of diameter 2mm and internal diameter for the exterior conductor of 7.5 mm is filled with polythene dielectric (ε r = 2.56, µ r = 1). Calculate the characteristic impedance of the line. a=1mm b=3.75mm

Example 4.2 Characteristic impedance of a coaxial line. “it can be shown that …” a=1mm b=3.75mm

Reflections on Transmission Lines Waves can be reflected. Waves can be reflected. They are reflected by discontinuities They are reflected by discontinuities a load?a load? change from one type of line to another?change from one type of line to another? A line with discontinuities will also have backward (or reflected) waves A line with discontinuities will also have backward (or reflected) waves

Reflections on Transmission Lines The total voltage and current = The total voltage and current = V T = V + +V - and I T = I + –I -V T = V + +V - and I T = I + –I - The impedance “seen” by the source will depend on their magnitude and phaseThe impedance “seen” by the source will depend on their magnitude and phase V T /I T = Z IN V T /I T = Z IN Power delivered to the end of the line is reduced by reflections. Power delivered to the end of the line is reduced by reflections. When there are no reflections the power delivered to the end of the line, and thus the load, is maximised When there are no reflections the power delivered to the end of the line, and thus the load, is maximised

Infinitely Long Transmission Lines No discontinuities and so no reflections No discontinuities and so no reflections the total voltage and total current at any point on the line (including the input) →the total voltage and total current at any point on the line (including the input) → The line just “looks like” a load Z 0 The line just “looks like” a load Z 0IV ZoZoZoZoI Infinite line with characteristic impedance Z o V =

NB Z 0 ≠ impedance of finite line from an impedance meter This would give the open-circuit impedance, Z OC. This would give the open-circuit impedance, Z OC. We will see later that Z 0 = (Z OC Z SC ) ½ We will see later that Z 0 = (Z OC Z SC ) ½ where Z SC is the short-circuit impedance.where Z SC is the short-circuit impedance.