RF Cafe Software

RF Cascade Workbook

About RF Cafe

Copyright

1996 -
2016

Webmaster:

Kirt Blattenberger,

BSEE
- KB3UON

RF Cafe began life in 1996 as "RF Tools" in an AOL screen name web space totaling 2 MB. Its primary purpose was to provide me with ready access to commonly needed formulas and reference material while performing my work as an RF system and circuit design engineer. The Internet was still largely an unknown entity at the time and not much was available in the form of WYSIWYG ...

All trademarks, copyrights, patents, and other rights of ownership to images and text used on the RF Cafe website are hereby acknowledged.

My Hobby Website:

AirplanesAndRockets.com

Try Using SEARCH

to
Find What You Need.

There are 1,000s of Pages Indexed on RF Cafe !

Module 10—Introduction to Wave Propagation, Transmission Lines,

and Antennas

Chapter 3: Pages 3-41 through 3-50

Pages i - ix, 1-1 to 1-10, 1-11 to 1-20, 1-21 to 1-30, 1-31 to 1-40, 1-41 to 1-47,

2-1 to 2-10, 2-11 to 2-20, 2-21 to 2-30, 2-31 to 2-40, 2-40 to 2-47,

3-1 to 3-10, 3-11 to 3-20, 3-21 to 3-30, 3-31 to 3-40, 3-41 to 3-50,

3-51 to 3-58, 4-1 to 4-10, 4-11 to 4-20, 4-21 to 4-30, 4-31 to 4-40,

4-41 to 4-50, 4-51 to 4-60, Index

Figure 3-32.—Voltage, current, and impedance on open line.

At all odd quarter-wave points (1/4λ, 3/4λ, etc.), the voltage is minimum, the current is maximum, and the impedance is minimum. Thus, at all odd quarter-wave points, the open-ended transmission line acts as a series-resonant circuit. The impedance is equivalent to a very low resistance, prevented from being zero only by small circuit losses.

At all even quarter-wave points (1/2λ, 1λ, 3/2λ, etc.), the voltages is maximum, the current is minimum, and the impedance is maximum. Comparison of the line with an LC resonant circuit shows that at an even number of quarter-wavelengths, an open line acts as a parallel-resonant circuit. The impedance is therefore an extremely high resistance.

In addition, resonant open lines may also act as nearly pure capacitances or inductances. The illustration shows that an open line less than a quarter-wavelength long acts as a capacitance. Also, it acts

3-41

as an inductance from 1/4 to 1/2 wavelength, as a capacitance from 1/2 to 3/4 wavelength, and as an inductance from 3/4 to 1 wavelength, etc. A number of open transmission lines, with their equivalent circuits, are shown in the illustration.

Follow figure 3-33 as we study the shorted line. At the odd quarter-wavelength points, the voltage is high, the current is low, and the impedance is high. Since these conditions are similar to those found in a parallel-resonant circuit, the shorted transmission line acts as a parallel-resonant circuit at these lengths.

Figure 3-33.—Voltage, current, and impedance on shorted line.

3-42

At the even quarter-wave points voltage is minimum, current is maximum, and impedance is minimum. Since these characteristics are similar to those of a series-resonant LC circuit, a shorted transmission line whose length is an even number of quarter-wavelengths acts as a series-resonant circuit.

Resonant shorted lines, like open-end lines, also may act as pure capacitances or inductances. The illustration shows that a shorted line less than 1/4 wavelength long acts as an inductance. A shorted line with a length of from 1/4 to 1/2 wavelength acts as a capacitance. From 1/2 to 3/4 wavelength, the line acts as an inductance; and from 3/4 to 1 wavelength, it acts as a capacitance, and so on. The equivalent circuits of shorted lines of various lengths are shown in the illustration. Thus, properly chosen line segments may be used as parallel-resonant, series-resonant, inductive, or capacitive circuits.

**STANDING WAVES ON A TRANSMISSION LINE**

There is a large variety of terminations for rf lines. Each type of termination has a characteristic effect on the standing waves on the line. From the nature of the standing waves, you can determine the type of termination that produces the waves.

Termination in Z

3-43

Figure 3-34.—Effects of various terminations on standing waves.

In an open-circuited rf line (figure 3-34, view C), the voltage is maximum at the end, but the current is minimum. The distance between two adjacent zero current points is 1/2λ, and the distance between alternate zero current points is 1. The voltage is zero at a distance of 1/4λ from the end of the line. This is true at any frequency. A voltage peak occurs at the end of the line, at 1/2λ from the end, and at each 1/2λ thereafter.

3-44

On the line terminated in a short circuit, shown in figure 3-34, view D, the voltage is zero at the end and maximum at 1/4λ from the end. The current is maximum at the end, zero at 1/4λ from the end, and alternately maximum and zero every 1/4λ thereafter.

When a line is terminated in capacitance, the capacitor does not absorb energy, but returns all of the energy to the circuit. This means there is 100 percent reflection. The current and voltage relationships are somewhat more involved than in previous types of termination. For this explanation, assume that the capacitive reactance is equal to the Z

When the line is terminated in an inductance, both the current and voltage shift in phase as they arrive at the end of the line. When X

Whenever the termination is not equal to Z

Where:

ER = the reflected voltage

Ei = the incident voltage

RR = the terminating resistance

Z_{0}= the characteristic impedance of the line

If you try different values of R

3-45

approaches an infinite value, E

When RL is greater than Z0, the end of the line is somewhat like an open circuit; that is, standing waves appear on the line. The voltage maximum appears at the end of the line and also at half-wave intervals back from the end. The current is minimum (not zero) at the end of the line and maximum at the odd quarter-wave points. Since part of the power in the incident wave is consumed by the load resistance, the minimum voltage and current are less than for the standing waves on an open-ended line. Figure 3-34, view G, illustrates the standing waves for this condition.

When R

A line does not have to be any particular length to produce standing waves; however, it cannot be an infinite line. Voltage and current must be reflected to produce standing waves. For reflection to occur, a line must not be terminated in its characteristic impedance. Reflection occurs on lines terminated in opens, shorts, capacitances, and inductances, because no energy is absorbed by the load. If the line is terminated in a resistance not equal to the characteristic impedance of the line, some energy will be absorbed and the rest will be reflected.

The voltage and current relationships for open-ended and shorted lines are opposite to each other, as shown in figure 3-34, views C and D. The points of maximum and minimum voltage and current are determined from the output end of the line, because reflection always begins at that end.

Q26. A nonresonant line is a line that has no standing waves of current and voltage on it and is considered to be flat. Why is this true?

Q27. On an open line, the voltage and impedance are maximum at what points on the line?

The measurement of standing waves on a transmission line yields information about equipment operating conditions. Maximum power is absorbed by the load when Z

You have probably noticed that the variation of standing waves shows how near the rf line is to being terminated in Z

3-46

The ratio of maximum voltage to minimum voltage on a line is called the VOLTAGE STANDING- WAVE RATIO (vswr). Therefore:

The vertical lines in the formula indicate that the enclosed quantities are absolute and that the two values are taken without regard to polarity. Depending on the nature of the standing waves, the numerical value of vswr ranges from a value of 1 (ZL = Z0, no standing waves) to an infinite value for theoretically complete reflection. Since there is always a small loss on a line, the minimum voltage is never zero and the vswr is always some finite value. However, if the vswr is to be a useful quantity, the power losses along the line must be small in comparison to the transmitted power.

The square of the voltage standing-wave ratio is called the POWER STANDING-WAVE RATIO (pswr). Therefore:

This ratio is useful because the instruments used to detect standing waves react to the square of the voltage. Since power is proportional to the square of the voltage, the ratio of the square of the maximum and minimum voltages is called the power standing-wave ratio. In a sense, the name is misleading because the power along a transmission line does not vary.

The ratio of maximum to minimum current along a transmission line is called CURRENT STANDING-WAVE RATIO (iswr). Therefore:

This ratio is the same as that for voltages. It can be used where measurements are made with loops that sample the magnetic field along a line. It gives the same results as vswr measurements.

Q28. At what point on an open-circuited rf line do voltage peaks occur?

Q29. What is the square of the voltage standing-wave ratio called?

Q30. What does vswr measure?

3-47

**SUMMARY**

This chapter has presented information on the characteristics of transmission lines. The information that follows summarizes the important points of this chapter.

A

3-48

A

3-49

A transmission line is either electrically

3-50

Introduction to Matter, Energy, and Direct Current, Introduction to Alternating Current and Transformers, Introduction to Circuit Protection, Control, and Measurement, Introduction to Electrical Conductors, Wiring Techniques, and Schematic Reading, Introduction to Generators and Motors, Introduction to Electronic Emission, Tubes, and Power Supplies, Introduction to Solid-State Devices and Power Supplies, Introduction to Amplifiers, Introduction to Wave-Generation and Wave-Shaping Circuits, Introduction to Wave Propagation, Transmission Lines, and Antennas, Microwave Principles, Modulation Principles, Introduction to Number Systems and Logic Circuits, Introduction to Microelectronics, Principles of Synchros, Servos, and Gyros, Introduction to Test Equipment, Radio-Frequency Communications Principles, Radar Principles, The Technician's Handbook, Master Glossary, Test Methods and Practices, Introduction to Digital Computers, Magnetic Recording, Introduction to Fiber Optics