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Lect23TLineTransients

Lect23TLineTransients - ECE 3030 Electromagnetic Fields and...

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1 ECE 3030 Electromagnetic Fields and Waves Fall 2009 Lecture 23 2009/10/21 Transmission Lines: Time Domain Transients 1 Swartz 09/10/20 Electromagnetic Fields and Waves – Fall 2009 Lecture 23: Instructor: Dr. Wesley E. Swartz 0 = z = z () t z V , + t z V , t z V , 2 2 0 = z = z t z V , + t z V , t z V , 2 2 v t 2 = v t 2 3 = 1 3 Time Domain Analysis – Basics (1) Question : How are transmission line signals treated that are NOT time harmonic and are NOT in a sinusoidal steady state? The concept of complex impedance has meaning only for the sinusoidal 0 = z t V s s R L R o Z = z 2 Swartz 09/10/20 Electromagnetic Fields and Waves – Fall 2009 Lecture 23: steady state . Arbitrary source voltages V s ( t ) , require solutions in the time domain which start from the basic time-domain equations: t t z I L z t z V = , , t t z V C z t z I = , , 2 2 2 2 2 , 1 , t t z V v z t z V = 2 2 2 2 2 , 1 , t t z I v z t z I = LC v 1 = Time Domain Analysis – Basics (2) The equation: has forward propagating solutions of the form: and backward propagating solutions of the form: o Z 2 2 2 2 2 , 1 , t t z V v z t z V = ( ) vt z V t z V = + , ( ) vt z V t z V + = , LC v 1 = 3 Swartz 09/10/20 Electromagnetic Fields and Waves – Fall 2009 Lecture 23: Examples: vt z V + z v vt z V + z v Suppose is as shown in the plot: Then what is the forward moving voltage on the line at t = / 2 v ? Space and time are related: Visualizing Propagation vt z V + t 1 T t V , + t V , + Leading point. 4 Swartz 09/10/20 Electromagnetic Fields and Waves – Fall 2009 Lecture 23: 0 = z = z v t z V 2 , = + 2 = z vT Voltages and Currents The current is related to the voltage via: and having solutions: o Z t t z I L z t z V = , , t t z V C z t z I = , , ( ) ( ) vt z V vt z I vt z V vt z I + = + = + + and + ++ + - -- - v 5 Swartz 09/10/20 Electromagnetic Fields and Waves – Fall 2009 Lecture 23: vt z V + z v Voltage: vt z I + z v Corresponding Current: o o Z Z Current is proportional to voltage since a higher voltage means more surface charges and more surface charges mean more current flow. Load End Boundary Condition The boundary condition at the load end, for all time is: o Z 0 = z t V s = z s R L R ( ) ( ) L R t z I t z V , 0 , 0 = = = + - t z V , 0 = t z I , 0 = 6 Swartz 09/10/20 Electromagnetic Fields and Waves – Fall 2009 Lecture 23: t z V t z V t z V , 0 , 0 , 0 = + = = = + o o Z t z V Z t z V t z I t z I t z I , 0 , 0 , 0 , 0 , 0 = = = = + = = = + + L t z V t z V Γ = = = + , 0 , 0 o L o L L Z R Z R + = Γ Substituting for the totals gives: The total voltage and current are: where

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2 Source End Boundary Condition The boundary condition at the source end, for all time is: o Z 0 = z () t V s = z s R L R + - ( ) t z V R t z I t V s s , , = + = = t z V , = t z I , = 7 Swartz 09/10/20 Electromagnetic Fields and Waves – Fall 2009 Lecture 23: t z V t z V t z V , , , = + = = = + o o Z t z V Z t z V t z I t z I t z I , , , , , = = = = + = = = + + ()( ) o s o s s Z R Z t V t z V t z V + + Γ = = = + , , o s o
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Lect23TLineTransients - ECE 3030 Electromagnetic Fields and...

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