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Electromagnetics (ENGR 367)PowerPoint Presentation

Electromagnetics (ENGR 367)

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### Electromagnetics(ENGR 367)

T-line Power, Reflection & SWR

T-line Theory: Something New or Not?!

- Power, Reflection and Standing Waves in T-lines act just like Uniform Plane Waves (UPW) in unbounded and layered media!
- Once you understand UPWs, you can also see by analogy how waves behave in T-lines with a few simplifications!

Traveling Waves on T-lines

- Space-time phenomena may be described by phasor functions representing either
- Voltage and current disturbances
- Electromagnetic wave disturbances

T-line Traveling Waves

- Analagous to waves on a string or
sound waves in a tube since all these waves

- carry real power
- reflect at boundaries and discontinuities
- exhibit impedance at each point in the medium

- Unique from waveguides since on T-lines they propagate in the (quasi-) Transverse Electromagnetic (TEM) mode: ~plane waves

Power in T-lines via Circuit Model during time harmonic oscillation

- Instantaneous Power over a fixed line length z
- Express the real parts of V, I in the (+) direction only
- Apply Euler’s Identity
- Thus

Power in T-lines via Circuit Model during time harmonic oscillation

- Time Averaged Power

Power Loss due to Attenuation oscillation

- Explicitly
- In decibel (dB) units

Power Loss due to Attenuation oscillation

- In terms of Voltage

Example of Calculating oscillationT-line Power Loss

- Exercise 1 (based on D11.2, H&B, 7/e, p. 350)
Given:two T-lines joined end-to-end by an adaptor. Line 1 is 30 m long and is rated at 0.1 dB/m, whereas line 2 is 45 m long and is rated at 0.15 dB/m. Due to a poor adaptor, the joint imparts another 3 dB loss.

Find: the percentage (%) of the input power that reaches the output of this combination

Solution:

Example of Calculating oscillationT-line Power Loss

- Exercise 1 (continued)
Solution:

Wave Reflection at oscillationT-line Discontinuity

- T-line discontinuity may consist of
- an actual load termination: device with complex input impedance (e.g., antenna or display)
- a junction between lines: connector and/or line mismatch

- Schematic model

Wave Reflection at oscillationT-line Discontinuity

- Energized T-line with discontinuity
- Incident Voltage phasor
- Reflected Voltage phasor
(where the time dependence ejt has been supressed)

Wave Reflection at oscillationT-line Discontinuity

- Consider the situation at the load junction (z=0):
- Voltages of opposite going waves add
- Currents of opposite going waves add
where the – sign arises due to neg. z-going current wave

Wave Reflection at oscillationT-line Discontinuity

- Define Voltage Reflection Coefficient ()
- Solving for in terms of impedances only

Wave Transmission at oscillationT-line Discontinuity

- Define Voltage Transmission Coefficient ()
- Solving in terms of impedances only

Matching Condition oscillationat a T-line Junction

- An impedance match becomes a desired design condition for most practical T-line systems because it
- Maximizes power transferred to the load
- Minimizes power reflected back to generator

- In terms of ZL and Z0

Power Reflected and Transmitted at a T-line Junction oscillation

- Ratio of Reflected to Incident Power
- Ratio of Transmitted to Incident Power

Calculating Power oscillationIn Case of a Line-Load Mismatch

- Exercise 2 (Ex. 11.5, H&B, 7/e, p. 352)
Given:a 50 lossless T-line terminated by a load impedance, ZL=50-j75 . Power incident from the T-line to the load is 100 mW.

Find: the power dissipated by the load

Solution: first calculate the reflection coefficient

Calculating Power oscillationIn Case of a Line-Load Mismatch

- Exercise 2 (continued)
Solution: next calculate the transmitted power in terms of incident power and

Calculating Power In Case of Both Line Loss and Line-Load Mismatch

- Exercise 3 (Ex. 11.6, H&B, 7/e, pp. 352, 353)
Given: two lossy lines joined end-to-end.

Line 1 is 10 m long and has a 0.20 dB/m loss. Line 2 is 15 m long and has a 0.10 dB/m loss. At the junction of these two lines = 0.30. Power input to line 1 is Pi1 = 100 mW

Find: a) the total loss of the line combination in dB.

b) the power transmitted to the output of line 2.

Calculating Power In Case of Both Line Loss and Line-Load Mismatch

- Exercise 3 (Ex. 11.6, H&B, 7/e, pp. 352, 353)
Solution: a) first calculate the dB loss of the joint from

then calculate the total loss of the link

b) now calculate the output power as

Voltage Standing Wave Ratio (VSWR) for Terminated T-lines Mismatch

- The status of waves on a T-line depends on the termination: 3 possibilities exist
1) Matched termination (ZL = Z0 = 0)

- All waves travel from source to load
- No waves reflect back to the source
- No standing waves exist, only pure traveling waves
2) Perfectly reflective termination ( = 1)

- All waves travel from source to load and back again
- All waves completely reflect
- A pure standing wave pattern exists with fixed null and maximum voltage locations along the line

Voltage Standing Wave Ratio (VSWR) for Terminated T-lines Mismatch

- The status of waves on a T-line depends on the termination: 3 possibilities exist
3) A partially reflective termination (0<<1)

- Some waves travel from source to load and back
- Some waves reflect; others pass to the load
- A partial standing wave pattern exists with fixed minimum and maximum locations along the line mixed with traveling waves!
(animated partial standing wave pattern)

Terminated Lossless T-line Mismatch

- Total voltage wave phasor (w/load @ z=0)
- Complete space-time voltage wave function

Terminated Lossless T-line Mismatch

- After applying Euler’s Identity and taking the real part the total voltage wave function becomes

Terminated Lossless T-line Mismatch

- Where are maximum and minimum voltages located?
- In terms of wavelengths () between successive
- Vmax locations
- Vmin locations
- Vmax to Vmin locations

Graphical Standing Wave Patterns Mismatch

- Voltage Standing Wave Patterns for
Real Reflection Coefficient Complex

VSWR: Terminated Lossless T-line Mismatch

- Now define as
- Note special cases
- Matched termination:
- Perfectly reflective termination:

- Range:
- Significance: indicates the degree of standing waves vs. traveling waves present on the T-line

VSWR Calculations Mismatchfor a Lossless Terminated T-line

- Exercise 4
Given: = 3/5

Find: VSWR = ?

Solution:

- Exercise 5
- Given: for a good match, we desire VSWR < 2.5
- Find: the condition on
- Solution:

Conclusions Mismatch

- Traveling waves on T-lines carry power subject to the losses of attenuation over distance and any mismatch of impedance at junctions
- The power output expected from a T-line may be computed from the input power by taking into account any dB loss factors

Conclusions Mismatch

- The reflection (or transmission) coefficient ( or ) at any T-line discontinuity
- Indicates how much voltage and power will be reflected (or transmitted) at the junction
- May be computed from the line impedance (Z0) on the source side and the effective input impedance (ZL = Zin) on the load side

Conclusions Mismatch

- The Voltage Standing Wave Ratio (VSWR) for a terminated T-line
- Indicates the degreeof standing waves versus traveling waves present on the line
- Serves as a figure of merit for the quality of impedance match at a junction
- Represents the max. to min. voltage ratio along the line, but may be calculated directly from the reflection coefficient at a junction

References Mismatch

- Hayt & Buck, Engineering Electromagnetics, 7/e, McGraw Hill: New York, 2006.
- Kraus & Fleisch, Electromagnetics with Applications, 5/e, McGraw Hill: New York, 1999.

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