Electromagnetics (ENGR 367) The Complete Sourced & Loaded T-line

T-line Power: Big or Small While T-lines carry from the high voltage of Power Generators to the miniscule voltage detected by Communications Receivers, the load can only absorb as much signal power as is transmitted, and that depends initially on the source and input impedances. While T-lines carry from the high voltage of Power Generators to the miniscule voltage detected by Communications Receivers, the load can only absorb as much signal power as is transmitted, and that depends initially on the source and input impedances. To address this issue adequately, we must consider the fully sourced and loaded T-line. To address this issue adequately, we must consider the fully sourced and loaded T-line.

Outline of Lecture Model the Fully Sourced & Loaded T-line Model the Fully Sourced & Loaded T-line Define Wave & Input Impedance of T-line Define Wave & Input Impedance of T-line Form an Equivalent Circuit at the Input Form an Equivalent Circuit at the Input Observe Special Input Impedance Cases Observe Special Input Impedance Cases –For a half-wavelength line –For a quarter-wavelength line Work Out Examples of T-line Problems Work Out Examples of T-line Problems

Introduction: Practical T-lines Connect to Both Source and Load Practical T-line may be mismatched at both ends Practical T-line may be mismatched at both ends If fully unmatched, reflections occur at both ends If fully unmatched, reflections occur at both ends

Sourced & Loaded T-line (lossless) (in case of lossy T-line, replace j  with  ) Express the Phasor line Voltage & Current as Express the Phasor line Voltage & Current as

Introduce Wave Impedance Define Wave Impedance as the Ratio of Total Voltage to Current Anywhere on line Define Wave Impedance as the Ratio of Total Voltage to Current Anywhere on line In terms of the voltage reflection coeff. (  ) In terms of the voltage reflection coeff. (  )

Wave Impedance & Input Impedance In terms of line and load impedances In terms of line and load impedances Define Input Impedance as the Wave Impedance at the front (source) end of line Define Input Impedance as the Wave Impedance at the front (source) end of line

Equivalent Circuit The Practical Sourced & Loaded T-line with Z in The Practical Sourced & Loaded T-line with Z in Acts like the equivalent circuit: Acts like the equivalent circuit:

Special Case I Half-Wavelength ( /2) Line Half-Wavelength ( /2) Line –Also known as the Half-Wave Transformer (HWT) –Electrical Length –Input Impedance Notes on the HWT Notes on the HWT –Ideally, this T-line section is transparent to the source and acts as if it were absent from the circuit –Actually, any line loss results in less than ideal performance, so the shortest possible section is usually preferred in application –Results depend on frequency:

HWT Behavior Due to the length of the HWT: Due to the length of the HWT: Provided: Provided:

Special Case II Quarter-Wavelength ( /4) Line Quarter-Wavelength ( /4) Line –Also known as the Quarter-Wave Transformer (QWT) –Electrical Length –Input Impedance Notes on the QWT Notes on the QWT –Ideally, can be used to match impedances at a junction –Affected by line losses so shortest section preferred –Results depend on frequency:

QWT Matching Desire: Desire: Provided: Provided:

Examples of T-line Problems Consider a lossless 2-wire 300-  line such as the lead-in wire from the antenna to a TV or FM radio communications receiver Consider a lossless 2-wire 300-  line such as the lead-in wire from the antenna to a TV or FM radio communications receiver

T-line Problem: Example A Given: Z G = Z 0 = Z L = (R in of Rx)= 300  v p = 2.5 x 10 8 m/s and l = 2.0 m Given: Z G = Z 0 = Z L = (R in of Rx)= 300  v p = 2.5 x 10 8 m/s and l = 2.0 m Find: , SWR,, , , Z in, V in, I in, V L, I L, P L Find: , SWR,, , , Z in, V in, I in, V L, I L, P L Solution: line is matched at both ends! Solution: line is matched at both ends!

T-line Problem: Example A Solution: (continued) Solution: (continued)

T-line Problem: Example B

Given: a 2 nd identical Rx in parallel with 1 st so that Z G = Z 0 = 300  ; Z L = R in1  R in2 Given: a 2 nd identical Rx in parallel with 1 st so that Z G = Z 0 = 300  ; Z L = R in1  R in2 Find: , SWR,, , , Z in, V in, I in, V L, I L, P L with locations & values of V min, V max Find: , SWR,, , , Z in, V in, I in, V L, I L, P L with locations & values of V min, V max Solution: Z L = (300  )  (300  ) =150  Solution: Z L = (300  )  (300  ) =150 

T-line Problem: Example B Solution: (continued) Solution: (continued) Good News: Z in may be calculated or checked by MATLAB Zx on a lossless line to save complex arithmetic labor and show a graphical result! Good News: Z in may be calculated or checked by MATLAB Zx on a lossless line to save complex arithmetic labor and show a graphical result!

T-line Problem: Example B Solution: (continued) Solution: (continued)

T-line Problem: Example B Solution: (continued) Solution: (continued) Check with the graphical output of MATLAB Zx Check with the graphical output of MATLAB Zx

T-line Problem: Example B Solution: (note that for all real, resistive Z L & Z 0 ) Solution: (note that for all real, resistive Z L & Z 0 ) –The voltage min. load if Z L <Z 0 –The voltage max. load if Z L >Z 0 For the phase of the voltage at the input versus the load For the phase of the voltage at the input versus the load

T-line Problem: Example C

T-line Problem: Example C (based on Ex. 11.8, H&B, 7/e, p. 363) Given: same T-line but add –j300  at load Given: same T-line but add –j300  at load Find: , VSWR, Z in, I in, P L Find: , VSWR, Z in, I in, P L Solution: now Z L = (150  )  (–j300  ) Solution: now Z L = (150  )  (–j300  )

T-line Problem: Example C Solution: (continued) Solution: (continued)

T-line Problem: Example D

T-line Problem: Example D (based on Ex. 11.9, H&B, 7/e, pp. 363, 364) Given: same T-line as before but now use the purely capacitive load Z L = -j300  Given: same T-line as before but now use the purely capacitive load Z L = -j300  Find: , VSWR, Z in Solution: Find: , VSWR, Z in Solution:

T-line Problem: Example E (based on D11.4&5, H&B, 7/e, p. 364) Given: a 50-  lossless line, l = 0.4 long, operating at 300 MHz with Z L = 40+j30  connected at z = 0, and a Thevenin Equiv. source of 12e j0 V at z = - l in series with Z Th = 50+j0  Given: a 50-  lossless line, l = 0.4 long, operating at 300 MHz with Z L = 40+j30  connected at z = 0, and a Thevenin Equiv. source of 12e j0 V at z = - l in series with Z Th = 50+j0  Find: , VSWR, Z in, V in, V L, P L Find: , VSWR, Z in, V in, V L, P L Solution: Solution:

T-line Problem: Example E Solution: (continued) Solution: (continued)

T-line Problem: Example E Solution: (continued) Solution: (continued)

Methods of Solving T-line Problems Analytical Expressions (as illustrated here) Analytical Expressions (as illustrated here) + No special tools needed other than calculator –Complex number calc’s may become tedious Software Tools Software Tools 1) MATLAB Zx for lossless lines (have access) 2) Other commercial programs for calc. & sim. Graphical Tools: Smith Chart Graphical Tools: Smith Chart

Conclusions Practical T-line problems must address both source and load conditions; if fully unmatched, reflections occur at both ends. Practical T-line problems must address both source and load conditions; if fully unmatched, reflections occur at both ends. Input impedance (Z in ) is the complex wave impedance at the source end of the T-line evaluated by analytical formula, MATLAB, other software tools or the Smith Chart. Input impedance (Z in ) is the complex wave impedance at the source end of the T-line evaluated by analytical formula, MATLAB, other software tools or the Smith Chart.

Conclusions The input impedance (Z in ) reduces any sourced and loaded T-line to an equivalent circuit for the purpose of voltage, current and power analysis. The input impedance (Z in ) reduces any sourced and loaded T-line to an equivalent circuit for the purpose of voltage, current and power analysis. Certain T-line lengths ( l ) transform the load impedance (Z L ) at the input in a special way Certain T-line lengths ( l ) transform the load impedance (Z L ) at the input in a special way –QWT: l = /4  Z in = Z 0 2 /Z L (good for matching) –HWT: l = /2  Z in = Z L (good for transparency)

Conclusions To find average power delivered to the load (P L ) of a sourced and loaded T-line To find average power delivered to the load (P L ) of a sourced and loaded T-line –If one knows the Thevenin equivalent source (V th, Z th ) Thevenin equivalent source (V th, Z th ) Characteristic Line & Load impedances (Z 0, Z L ) Characteristic Line & Load impedances (Z 0, Z L ) T-line length, frequency & prop. velocity ( l, f, v p ) T-line length, frequency & prop. velocity ( l, f, v p ) –then one can calculate T-line electrical length (  =  l ) T-line electrical length (  =  l ) Input impedance (Z in ) Input impedance (Z in ) Power input (P in ) via equivalent circuit at the source and real power absorbed in the load Power input (P in ) via equivalent circuit at the source and real power absorbed in the load

Conclusions If the load is not matched to a T-line, some power reflects back to the source and may damage a generator if excessive If the load is not matched to a T-line, some power reflects back to the source and may damage a generator if excessive If the source is not matched to a T-line (not addressed in examples here!), power may reflect at both ends and calculations become more involved If the source is not matched to a T-line (not addressed in examples here!), power may reflect at both ends and calculations become more involved

References Hayt & Buck, Engineering Electromagnetics, 7/e, McGraw Hill: New York, Hayt & Buck, Engineering Electromagnetics, 7/e, McGraw Hill: New York, Kraus & Fleisch, Electromagnetics with Applications, 5/e, McGraw Hill: New York, Kraus & Fleisch, Electromagnetics with Applications, 5/e, McGraw Hill: New York, 1999.