Lab Manual Electrical Networks 2015

Introduction to the Transmission Lines Circuit Board

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Figure 1-24. The SIGNAL GENERATOR.

The LOAD section, shown in Figure 1-25, consists of a network of resistors,inductors, and capacitors that can be configured in various ways, through the settingof toggle switches. A BNC connector located at the LOAD-section input permitsconnection of this input to the common via the desired load configuration. Forexample, to connect the LOAD-section input to the common via resistor R1 in serieswith inductor L2, switches S1 and S8 are set to the I (ON) position, while all the otherswitches are set to the O (OFF) position.


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Figure 1-25. The LOADs.

Thevenin's Theorem

Thevenin's theorem is named after the French engineer M.L. Thevenin. Thevenin'stheorem allows any electrical linear circuit seen at two terminals to be representedby a Thevenin equivalent circuit. The Thevenin equivalent circuit consists of avoltage source, ETH, and an impedance in series with this source, ZTH. Figure 1-26shows how a simple circuit is thevenized.

Voltage ETH is equal to the open-circuit voltage, VOC, measured across the twoterminals of the circuit to thevenize.

Impedance ZTH is the impedance seen at the two terminals when the voltagesource of the circuit to thevenize is replaced by a short circuit.


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Figure 1-26. Thevenizing a simple circuit.

Thevenizing the STEP GENERATOR and the SIGNAL GENERATOR of theTRANSMISSION LINES Circuit Board

The STEP GENERATOR of the TRANSMISSION LINES circuit board can berepresented by its Thevenin equivalent. To determine the Thevenin voltage of theThevenin equivalent, the STEP GENERATOR output voltage is measured with noload connected to the generator output (that is, with the load impedance in the open-circuit condition, equal to 4 ), as Figure 1-27 shows. The measured voltagecorresponds to the Thevenin voltage, ETH.


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Figure 1-27. Thevenizing the STEP GENERATOR.

Assuming that the Thevenin impedance of the STEP GENERATOR Theveninequivalent is purely resistive, this impedance can then be determined by connectinga resistive load, whose resistance can be varied, to the output of the STEPGENERATOR, as Figure 1-28 shows.

Figure 1-28. Voltage divider rule.


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According to the voltage divider rule, the voltage across this load, VL, is

where VL = Voltage across the load (V);ZL = Load impedance ();

ZTH = Thevenin impedance ();ETH = Thevenin voltage (V).

When the load is adjusted so that the voltage across it, VL, is equal to ETH/2 (seeFigure 1-29), the equation for calculating VL becomes:

Figure 1-29. ZTH = ZL when VL = ETH/2.

Rewriting and simplifying this equation for solving ZTH, gives:

Consequently, by adjusting the resistance of the load so that VL = ETH/2, and thenmeasuring this resistance, the value of ZTH can be determined.

A method identical to that just described can be used to determine the Theveninequivalent circuit of the SIGNAL GENERATOR of the TRANSMISSION LINES circuitboard.

As will be seen in detail in Unit 2, a transmission line acts as a load when it isconnected to a voltage source. This causes the instantaneous applied voltage to beattenuated by a specific amount determined by the voltage divider rule.


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Procedure Summary

In this procedure section, you will determine the Thevenin equivalents of the STEPGENERATOR and SIGNAL GENERATOR on your circuit board.

PROCEDURE

Determining the Thevenin Equivalent at the STEP GENERATOR 50- BNCOutput

G 1. Make sure the TRANSMISSION LINES circuit board is properly installedinto the Base Unit. Turn on the Base Unit and verify that the LED's next toeach control knob on this unit are both on, confirming that the circuit boardis properly powered.

G 2. Referring to Figure 1-30, connect the STEP GENERATOR 50- BNC outputto the BNC connector at the LOAD-section input, using a short coaxialcable.

Then, connect the STEP GENERATOR 100- BNC output to the triggerinput of the oscilloscope, using a coaxial cable.

Finally, using an oscilloscope probe, connect channel 1 of the oscilloscopeto the probe turret just next to the BNC connector at the LOAD-section input.Make sure to connect the ground conductor of the probe to the associated(nearby) common (L) turret.

G 3. In the LOAD section, set all the toggle switches to the O (OFF) position.Then, connect the input of the LOAD section to the common via resistor R1(500- potentiometer) by setting the appropriate switches in this section tothe I (ON) position. (That is, set both switches S1 and S10 to the I position.The other switches must all be left to the O position).

Turn the knob of resistor R1 fully clockwise. This sets the impedance of theSTEP GENERATOR output load to 500 approximately.


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Figure 1-30. STEP GENERATOR 50- BNC output connected to oscilloscope channel 1 and to theLOAD-section input.

G 4. Make the following settings on the oscilloscope:

Channel 1Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NormalSensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.5 V/divInput Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DC

Time Base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 s/divTrigger

Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ExternalLevel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.3 VInput Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 M or more

Note: Throughout this course, the oscilloscope settings for the timebase and channel sensitivity are given as a starting point forguidance and may be modified as necessary to obtain the maximumpossible measurement accuracy.

G 5. Observe the STEP GENERATOR output signal on the oscilloscope screen.Is this signal a rectangular pulse having a period of 20 s approximately, asFigure 1-31 shows?


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G Yes G No

Figure 1-31. STEP GENERATOR output signal.

G 6. In the LOAD section, slowly turn the knob of resistor R1 fullycounterclockwise, which will cause the impedance of the STEPGENERATOR output load to decrease from 500 to 0 approximately.

While doing this, observe what happens to the pulses in the STEPGENERATOR output signal on the oscilloscope screen.

Which of the following statements best describes your observation?

a. The voltage of the pulses stays constant as the load impedance isdecreased.

b. The voltage of the pulses decreases as the load impedance isdecreased.

c. The voltage of the pulses is half the maximum voltage when the loadimpedance is minimum (0 ).

d. The pulses become absent from the displayed signal when the loadimpedance becomes maximum.

G 7. In the LOAD section, set all the toggle switches to the O (OFF) position.This places the impedance of the load at the STEP GENERATOR 50-output in the open-circuit condition (4 ).


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Measure the voltage (height of the rising edge) of the pulses on theoscilloscope screen. This is the Thevenin voltage, ETH, at the STEPGENERATOR 50- BNC output.

ETH = V

G 8. Connect the input of the LOAD section to the common via resistor R1(500- potentiometer) by setting the appropriate switches in this section tothe I (ON) position.

Adjust the knob of resistor R1 until the voltage of the pulses on theoscilloscope screen is equal to half the Thevenin voltage measured in theprevious step.

Record below this voltage, ETH/2.

ETH/2 = V

G 9. Using an ohmmeter, measure the resistance between the LOAD-sectioninput and the common (current resistance setting of resistor R1). Since R1has been adjusted to create a voltage drop of ETH/2 at the STEPGENERATOR 50- BNC output, the R1-resistance setting corresponds tothe Thevenin impedance, ZTH, at this output:

Disconnect the end of the coaxial cable connected to theBNC connector at the LOAD-section input. (Leave the other endconnected to the STEP GENERATOR 50- BNC output.)

Disconnect the oscilloscope probe from the probe turret at the LOAD-section input.

Hold the tip of one of the ohmmeter probes on the probe turret at theLOAD-section input, while touching the nearby common (L) turret withthe other ohmmeter probe.

Record below the measured resistance, ZTH.

ZTH =

G 10. Reconnect the coaxial cable coming from the STEP GENERATOR 50-BNC output to the BNC connector at the LOAD-section input.

G 11. Connect the input of the LOAD section to the common via resistor R4(100- resistor) by setting the appropriate switches in this section to the I(ON) position.

Reconnect channel 1 of the oscilloscope to the probe turret at the LOAD-section input.


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G 12. Measure the voltage of the pulses on the oscilloscope screen. This is thevoltage across the 100- load currently connected to the STEPGENERATOR output, VL.

VL = V

G 13. Using the Thevenin voltage, ETH, and the Thevenin impedance, ZTH,measured in steps 7 and 9 of this exercise, use the voltage divider rule tocalculate the theoretical voltage present across a 100- load, ZL, connectedto the STEP GENERATOR output:

Is your calculation result approximately equal to the practical voltage, VL,measured in the previous step?

G Yes G No

G 14. Disconnect the circuit by removing all the connecting cables and probes.

Determining the Thevenin Equivalent at the SIGNAL GENERATOR 50- BNCOutput

G 15. Now, determine the Thevenin equivalent at the SIGNAL GENERATOR 50-BNC output:

Referring to Figure 1-32, connect the SIGNAL GENERATOR 50- BNCoutput to the BNC connector at the LOAD-section input, using a shortcoaxial cable.

Then, connect the SIGNAL GENERATOR 100- BNC output to thetrigger input of the oscilloscope, using a coaxial cable.

Finally, using an oscilloscope probe, connect channel 1 of theoscilloscope to the probe turret just next to the BNC connector at theLOAD-section input. Make sure to connect the ground conductor of theprobe to the associated (nearby) common (L) turret.


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Figure 1-32. SIGNAL GENERATOR 50- output connected to oscilloscope channel 1 and to theLOAD-section input.

G 16. In the LOAD section, set all the toggle switches to the O (OFF) position.This places the impedance of the load at the SIGNAL GENERATOR 50-output in the open-circuit condition (4 ).


Channel 1Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NormalSensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 V/divInput Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AC

Time Base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.1 s/divTrigger


G 18. Adjust the frequency of the SIGNAL GENERATOR output signal to 3 MHzapproximately. To do so, adjust the FREQUENCY knob of this generator


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until the period, T, of the sinusoidal signal displayed on the oscilloscope is0.33 s, approximately, as Figure 1-33 shows. To make sure theFREQUENCY knob is properly adjusted, you can verify that the voltage atthe REFERENCE OUTPUT of the SIGNAL GENERATOR is 3.0 V, using aDC voltmeter.

Figure 1-33. SIGNAL GENERATOR output signal frequency set to 3 MHz approximately.

Measure the peak (positive) amplitude of the sinusoidal voltage on theoscilloscope screen. This is the Thevenin voltage, ETH, at the SIGNALGENERATOR 50- BNC output.

ETH = VPK

G 19. Connect the input of the LOAD section to the common via resistor R1(500- potentiometer) by setting the appropriate switches in this section tothe I (ON) position.

Adjust the knob of resistor R1 until the peak (positive) amplitude of thesinusoidal voltage on the oscilloscope screen is equal to half the Theveninvoltage measured in the previous step. Record below this voltage, ETH/2.

ETH/2 = VPK

G 20. Using an ohmmeter, measure the resistance between the LOAD-sectioninput and the common (current resistance setting of resistor R1). Since R1has been adjusted to create a voltage drop of ETH/2 at the SIGNALGENERATOR 50-BNC output, the R1 resistance setting corresponds to theThevenin impedance, ZTH, at this output:


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Disconnect the end of the coaxial cable connected to theBNC connector at the LOAD-section input.

Disconnect the oscilloscope probe from the probe turret at the LOAD-section input.

Hold the tip of one of the ohmmeter probes on the probe turret at theLOAD-section input, while touching the nearby common (L) turret withthe other ohmmeter probe.

Record below the measured resistance, ZTH.

ZTH =

G 21. Turn off the Base Unit and remove all the connecting cables and probes.

CONCLUSION

The TRANSMISSION LINES circuit board has five sections: theTRANSMISSION LINES, the AUXILIARY POWER INPUT, the STEPGENERATOR, the SIGNAL GENERATOR, and the LOADs.

The Thevenin equivalent of the STEP GENERATOR or SIGNAL GENERATORcan be determined at any of their BNC outputs. To do so, the generator outputvoltage is measured with no load connected to the generator output. Themeasured voltage corresponds to the Thevenin voltage, ETH.

Then, a variable load is connected to the generator output, and the load isadjusted until the voltage across it is equal to half ETH. In this condition, the loadimpedance, which corresponds to the Thevenin impedance ZTH, can bemeasured with an ohmmeter. ZTH does not necessarily correspond exactly to thenominal resistance of the BNC output where it is measured.

Once the Thevenin equivalent at a generator output is known and a load isconnected to this output, the voltage applied to the load is determined by thevoltage divider rule.

REVIEW QUESTIONS

1. The STEP GENERATOR on your circuit board produces a rectangular signal

a. whose frequency can be determined through measurement of the voltageat the REFERENCE OUTPUT.

b. available at five BNC connectors, each connector corresponding to adifferent line input impedance.

c. having a frequency of 50 kHz.d. that occurs every 20 ms.

2. The TRANSMISSION LINES on your circuit board


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a. have four probe turrets and their associated shield turrets that divide the lineinto four segments of equal length [6 meters (19.7 feet) each].

b. can be connected end-to-end to obtain a line of 48 meters (158 feet).c. each consists of a 75- coaxial cable of the RG-174 type.d. each has a length of 48 meters (158 feet).

3. According to Thevenin's theorem,

a. the Thevenin impedance ZTH is the impedance seen at the two terminals ofthe circuit to thevenize, when the voltage source of this circuit is replaced byan open circuit.

b. the Thevenin voltage ETH is determined by measuring the short-circuitvoltage at the two terminals of the circuit to thevenize.

c. the Thevenin equivalent circuit consists of a voltage source, ETH, and animpedance in parallel with this source, ZTH.

d. any electrical linear circuit seen at two terminals can be represented by aThevenin equivalent circuit.

4. What is the Thevenin voltage, ETH, of the STEP GENERATOR if an open-circuitvoltage VOC of 1.0 V is measured at the 5- BNC output of this generator?

a. 5.0 Vb. 0.2 Vc. 0.5 Vd. 1.0 V

5. If the Thevenin equivalent at a BNC output of the STEP GENERATOR isETH = 2 V and ZTH = 75 , what will be the voltage present across a 50- loadconnected to this output?

a. 0.8 Vb. 4.0 Vc. 2.0 Vd. 1.2 V

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Exercise 1-2

Velocity of Propagation

EXERCISE OBJECTIVE

Upon completion of this unit, you will know how to measure the velocity ofpropagation of a signal in a transmission line, using the step response method.Based on the measurements, you will know how to determine the relative permittivityof the dielectric material used to construct this line.

DISCUSSION


A radio signal travels in free space at the velocity of light (approximately3.0 @ 108 m/s, or 9.8 @ 108 ft/s). In a transmission line, a signal will travel at a relativelylower speed. This is due mainly to the presence of the dielectric material used toconstruct the line. In fact, the velocity of propagation of a signal in a transmissionline, vP, is dependent upon the distributed inductance and capacitance of the line, L'and C' (see Figure 1-34). The equation for calculating vP is:

where vP = Velocity of propagation (m/s or ft/s);L' = Distributed inductance, in henrys per unit length (H/m or H/ft);C' = Distributed capacitance, in farads per unit length (F/m or F/ft).


1-36

Figure 1-34. Equivalent circuit of a two-conductor transmission line.

Step (Transient) Response Method

The velocity of propagation of a signal in a transmission line can be measured byusing the step response method. This methods requires that a step generator anda high-impedance oscilloscope probe be both connected to the sending end of theline, using a bridging connection, as Figure 1-35 shows. The receiving end of the lineis left unconnected [impedance of the load in the open-circuit condition (4 )].


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Figure 1-35. Measuring the velocity of propagation of a signal by using the step response method.

The signal propagation through the line is described below (refer to Figure 1-35).

At time t = 0, the step generator launches a fast-rising, positive-going voltage, VI,into the line. The rising edge of VI is called a step, or transient. This step isincident because it comes from the generator and is going to travel down the linetoward a possibly reflecting load.

Incident step VI propagates at a certain velocity, vP, along the line. It arrives at thereceiving end of the line after a certain transit time, T. There its level hasdecreased by a certain amount due to the resistance of the line.

Since the impedance of the load at the receiving end of the line is in the open-circuit condition (4 ), it does not match the characteristic impedance of the line.This impedance mismatch causes the incident step to be reflected back towardthe generator. The reflected step, VR, gets back to the step generator after atime equal to twice the transit time, 2T. 2T is synonymous with round-trip time,or back-and-forth trip time.

The signal at the sending end of the line, as a function of time, is the step responsesignal. As Figure 1-36 shows, this signal is the algebraic sum of the incident stepVI and reflected step VR. Step VR is superimposed on step VI, and is separated bya time 2T from the rising edge of VI.


1-38

Figure 1-36. Voltage at the sending end of the open-circuit line (step response signal).

By measuring time 2T on the oscilloscope screen, the velocity of propagation of asignal in a transmission line, vP, can be determined, using the formula below.

where vP = Velocity of propagation (m/s or ft/s);l = Length of the line (m or ft);

2T = Round-trip time, i.e. time taken for the launched step to travelfrom the generator to the line receiving end and back again tothe generator (s).

Transmission lines that are lossy, and whose series losses are predominant, willappear as a simple RC network (resistor-capacitor network) for a short time followingthe launching of a voltage step, as Figure 1-37 shows. This is due to the high-frequency components contained in the voltage step.


1-39

Figure 1-37. Lossy line with predominant series losses.

The time constant, , of the RC network (not to be confused with the transit time T)is determined by constants Rs and C, which are themselves derived from thedistributed series resistance, R's, series inductance, L', and parallel capacitance, C',of the line. Consequently, the time constant of the RC network is independent of thelength of the line.

In that case, the incident and reflected steps observed at the sending end of the linewill first rise to a certain level, and then increase exponentially at a rate determinedby the time constant of the RC network, as Figure 1-38 shows. This does not preventthe measurement of time 2T on the oscilloscope screen for calculation of the velocityof propagation. However, it is clear that lossy lines cause a degradation in the risetime of voltage steps.


1-40

Figure 1-38. Incident and reflected steps at the sending end of a lossy line with predominant serieslosses.

Velocity Factor

The velocity of propagation of a signal in a transmission line is usually expressed asa percentage of the velocity of light in free space. This percentage is called thevelocity factor, vF. For example, a transmission line with a vF of 66% will transmitsignals at about 66% of the velocity of light.

where vF = Velocity factor (%);vP = Velocity of propagation in the transmission line (m/s or ft/s);c = Velocity of light in free space (about 3.0 @ 108 m/s, or

9.8 @ 108 ft/s).


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In the case of coaxial cables, the velocity factor varies from about 66 to around 85%,as indicated in Table 1-1.

TYPE OF COAXIAL CABLE VELOCITY FACTOR, vF (%)

RG-8 66

RG-58 66

RG-174 66

RG-400 70

RG-11 75

RG-316 79

LMR-195 83

RG-8X 84

LMR-400 85

Table 1-1. Velocity factor of various types of coaxial cables.

TRANSMISSION LINES A and B of the circuit board are RG-174 coaxial cables.Consequently, they have a theoretical velocity factor, vF, of 66%.

Relative Permittivity (Dielectric Constant)

The velocity of propagation of a signal in a transmission line is determined mainly bythe permittivity of the dielectric material used to construct the line. Permittivity is ameasure of the ability of the dielectric material to maintain a difference in electricalcharge over a given distance.

The permittivity of a particular dielectric material is normally expressed in relation tothat of vacuum. This ratio is called relative permittivity, or dielectric constant. Whenthe velocity of propagation in a transmission line is known, the relative permittivityof the dielectric material used to construct that line, r, can be determined by usingthe equation below.

where r = Relative permittivity (dielectric constant);c = Velocity of light in free space (3.0 @ 108 m/s, or 9.8 @ 108 ft/s);

vP = Velocity of propagation (m/s or ft/m).

The formula for calculating relative permittivity indicates that a higher velocity ofpropagation indicates a lower relative permittivity, since the velocity of light is aconstant value.

Table 1-2 lists the relative dielectric constants of various materials.


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MATERIAL RELATIVE PERMITTIVITY, r

VELOCITY FACTOR,vF (%)

Vacuum 1.00000 100

Air 1.0006 99.97

Teflon 2.10 69.0

Polyethylene 2.27 66.4

Polystyrene 2.50 63.2

Polyvinyl chloride (PVC) 3.30 55.0

Nylon 4.90 45.2

Table 1-2. Relative dielectric constant of various materials.

Procedure Summary

In this procedure section, you will measure the velocity of propagation of voltagesteps in the transmission lines of the circuit board. Based on the measured velocity,you will determine the relative permittivity of the dielectric material used to constructthese lines.

PROCEDURE

Measuring the Velocity of Propagation


G 2. Referring to Figure 1-39, connect the STEP GENERATOR 50- BNC outputto the BNC connector at the sending end of TRANSMISSION LINE A.Leave the BNC connector at the receiving end of TRANSMISSION LINE Aunconnected (open-circuit).


Finally, using an oscilloscope probe, connect channel 1 of the oscilloscopeto the 0-meter (0-foot) probe turret at the sending end of TRANSMISSIONLINE A. Make sure to connect the ground conductor of the probe to theassociated 0-meter (0-foot) shield turret.

Note: When connecting an oscilloscope probe to one of the fiveprobe turrets of a transmission line, always connect the groundconductor of the probe to the associated (nearest) coaxial-shieldturret. This will minimize noise in the observed signal due to theparasitic inductance introduced by undesired ground return paths.


1-43

Figure 1-39. Measuring the velocity of propagation of voltage steps through TRANSMISSIONLINE A.


Channel 1Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NormalSensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.2 V/divInput Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DC


Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ExternalLevel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.3 VInput Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 M or more

Note: Throughout this course, the oscilloscope settings for thetime base and channel sensitivity are given as a starting point forguidance and may be modified as necessary to obtain themaximum possible measurement accuracy.


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G 4. On the oscilloscope screen, observe the step response signal at the sendingend of TRANSMISSION LINE A. This signal corresponds to the stepresponse of TRANSMISSION LINE A. Does the reflected step appearsuperimposed on the incident step, a certain time interval separating thesetwo steps, as Figure 1-40 shows?

G Yes G No

Figure 1-40. Incident and reflected steps at the sending end of TRANSMISSION LINE A.

G 5. Observe that the incident and reflected steps first rise to a certain level, andthen increase exponentially, as the voltage across a capacitor chargingthrough a series resistor. Does this indicate that TRANSMISSION LINE Ahave predominant series losses?

G Yes G No


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G 6. When the incident step arrives at the receiving end of TRANSMISSIONLINE A, it is reflected back toward the sending end because

a. TRANSMISSION LINE A is not terminated by a load impedance equalto the Thevenin impedance of the STEP GENERATOR.

b. TRANSMISSION LINE A is not terminated by a load impedance equalto its characteristic impedance.

c. the receiving end of TRANSMISSION LINE A is open-circuit, causingthe characteristic impedance of the line to be infinite.

d. the Thevenin impedance of the STEP GENERATOR is not equal to thecharacteristic impedance of TRANSMISSION LINE A.

G 7. Decrease the oscilloscope time base to 0.05 s/div.

On the oscilloscope, measure the round-trip time, 2T, separating the risingedge of the incident step from the rising edge of the reflected step, asFigure 1-41 shows. This is the time required for the step launched by thestep generator to travel to the receiving end of TRANSMISSION LINE A andthen back to the step generator.

2T = @ 10!9 s

Figure 1-41. Measuring time 2T.


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G 8. Based on the round-trip time, 2T, measured in the previous step, and on aline length, l, of 24 meters (78.7 feet), calculate the velocity of propagation,vP, through the line.

vP = @ 108 m/s or @ 108 ft/s

G 9. Express the velocity of propagation, vP, obtained in the previous step as apercentage of the velocity of light, or velocity factor, vF, using the formulabelow.

Your result should be near the theoretical value of 66% for a RG-174 coaxialcable (type of cable used for TRANSMISSION LINES A and B of your circuitboard).

where c = velocity of light in free space (3.0 @ 108 m/s, or 9.84 @ 108 ft/s)

vF = %

Determining the Relative Permittivity (Dielectric Constant)

G 10. Based on the velocity of propagation vP obtained in step 8, determine therelative permittivity, r of the dielectric material used to construct the RG-174coaxial cables used for TRANSMISSION LINES A and B.

The result should be quite near the theoretical value of 2.25 for polyethylene(dielectric material used to construct the RG-174 coaxial cables used forTRANSMISSION LINES A and B).

where c = velocity of light in free space (3.0 @ 108 m/s, or 9.84 @ 108 ft/s)

r =

Effects that a Change in Line Length Has on the Round-Trip Time (2T)

G 11. As Figure 1-42 shows, increase the length of the line from 24 to 48 meters(78.7 to 157.4 feet) through end-to-end connection of TRANSMISSIONLINEs A and B. To do so, connect the BNC connector at the receiving endof TRANSMISSION LINE A to the BNC connector at the sending end ofTRANSMISSION LINE B, using a short coaxial cable. Leave the


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BNC connector at the receiving end of TRANSMISSION LINE Bunconnected (open-circuit).

Figure 1-42. Increasing the length of the line from 24 to 48 meters (78.7 to 157.4 feet).

G 12. Set the oscilloscope time base to 0.2 s/div. Observe that the round-triptime, 2T, separating the rising edge of the incident step from the rising edgeof the reflected step has doubled, as Figure 1-43 shows.


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Figure 1-43. The round-trip time, 2T, separating the rising edges of the incident and reflected stepshas doubled.

Time 2T has doubled because the

a. velocity of propagation has decreased by a factor of two.b. length of the line has doubled.c. relative permittivity has doubled.d. characteristic impedance of the line has doubled.

G 13. On the oscilloscope screen, observe that the incident and reflected stepsfirst rise to a certain level, and then increase exponentially as they did withthe shorter 24-meter (78.7-foot) long line.

These steps increase at the same rate as they did with the shorter length.This occurs because the time constant of the series RC network temporarilypresented by the line is determined by the

a. characteristic impedance, which is a constant.b. total series resistance and parallel capacitance of the entire line.c. series resistance, parallel capacitance, and series inductance of the line

per unit length.d. velocity factor, which is a constant.



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CONCLUSION

The velocity of propagation of a signal in a transmission line can be measuredby using the step response method: a fast-rising (transient) step is launched intothe line. The time required for this step to travel from the generator to thereceiving end of the line and then back to the generator is measured. Thistime, 2T, permits calculation of the velocity of propagation. 2T is synonymouswith round-trip time, or back-and-forth trip time.

The velocity of propagation of a signal in a transmission line is only a percentageof the velocity of light in free space. The velocity of propagation in a transmissionline, when expressed as a percentage of the velocity of light in free space, iscalled the velocity factor.

The velocity of propagation in a transmission line is determined mainly by therelative permittivity (dielectric constant) of the dielectric material used toconstruct that line. The lower the relative permittivity is, the higher the velocityof propagation will be.

REVIEW QUESTIONS

1. In a transmission line, a signal travels at a velocity

a. that is null if the impedance of the load at the receiving end of the line is inthe open-circuit condition (4 ).

b. that is directly proportional to the relative permittivity of the dielectric materialused to construct the line.

c. that usually increases as the diameter of the line conductors is decreased.d. relatively less than 3.0 @ 108 m/s, or 9.8 @ 108 ft/s.

2. The permittivity of the dielectric material used to construct a transmission line

a. is a measure of the ability of the material to maintain a difference inpropagation velocity over a given distance.

b. is called dielectric constant, or relative permittivity, when expressed inrelation to the permittivity of vacuum.

c. is usually expressed as a percentage of the velocity of light in free space.d. does not determine the velocity factor of that line.

3. The velocity of propagation of a signal in a transmission line can be determinedby using

a. a high-impedance oscilloscope probe connected to the sending end of theline and a step generator connected to the receiving end of the line.

b. a simple formula, if the time required for a voltage step to travel to thereceiving end of the line and back to the generator is known.

c. the step response method, provided that the load impedance perfectlymatches the characteristic impedance of the line.

d. a step generator and a high-impedance oscilloscope connected to thereceiving end of the line.


1-50

4. When the step response method is used, the signal observed on theoscilloscope at the sending end of the line consists of

a. a reflected step superimposed on an incident step, the rising edge of theincident step being of higher voltage than that of the reflected step due toattenuation.

b. an incident step superimposed on a reflected step, the rising edge of theincident step being of higher voltage than that of the reflected step due toattenuation.

c. a reflected step superimposed on an incident step, the time separating thesesteps being directly proportional to the velocity of propagation.

d. several incident steps, the time separating two successive incident stepsbeing determined by the length of the line.

5. When a voltage step is launched into a lossy line whose series losses arepredominant,

a. the high-frequency components contained in the voltage steps make the linetemporarily appear as a simple RC network.

b. the incident and reflected steps will first rise to a certain level and thendecrease exponentially.

c. it is not possible to measure the time separating the incident and reflectedsteps.

d. the line will appear as a simple LC network from the perspective of the load.

1-51

Exercise 1-3

Transient Behavior of a LineUnder Resistive Load Impedances

EXERCISE OBJECTIVE

Upon completion of this unit, you will know how a transmission line terminated byvarious types of loads behaves when voltage steps are launched into the line. Youwill also know about two methods of determining the characteristic impedance of aline.

DISCUSSION

Determining the Nature of the Load Impedance by Using the Step ResponseMethod

When a transmission line is terminated by a load of unknown impedance, the stepresponse method can be used to determine the nature of this impedance (whetherpurely resistive or complex). The measurements are performed by using the stepresponse method.

A step generator and a high-impedance oscilloscope probe are connected to thesending end of the line, using a bridging connection, as Figure 1-44 shows.

At time t = 0, the step generator produces an incident step, VI, that is launchedinto the line.

The incident step travels down the line until it reaches the receiving end of theline at the transit time T. If the load impedance does not perfectly match thecharacteristic impedance of the line, the incident step experiences a change inimpedance as it quits the line and encounters the load. This causes part of theenergy contained in the incident step to be reflected back toward the generatorinstead of being absorbed by the load. Consequently, the step response signalobserved at the sending end of the line is the algebraic sum of the incident step,VI, and reflected step, VR.


1-52

Figure 1-44. Determining the nature of the load impedance by using the step response method.

The step response signal can have several different shapes, this shape beingdetermined by the nature of the load impedance ZL. When ZL is purely resistive, thereflected voltage has the same shape as the incident voltage, as Figure 1-45 shows.

Figure 1-45. ZL is purely resistive.

When ZL is both resistive and inductive, the reflected voltage in the step responsesignal has the same shape as the voltage across a capacitor discharging through aseries resistor. Thus, this voltage decreases exponentially until it stabilizes to acertain level, as Figure 1-46 shows.


1-53

Figure 1-46. ZL is both resistive and inductive.

When ZL is both resistive and capacitive, the reflected voltage in the step responsesignal has the same shape as the voltage across a capacitor charging through aseries resistor. Thus, this voltage increases exponentially until it stabilizes to acertain level, as Figure 1-47 shows.

Figure 1-47. ZL is both resistive and capacitive.


1-54

Purely Resistive Load Impedance

When the load impedance is purely resistive, the voltage step reflected from themismatched impedance at the receiving end of the line has the same shape as theincident step. The magnitude (voltage) and polarity of this voltage are determined bythe relation between the load impedance ZL, and the characteristic impedance, Z0,as indicated by the equation below:

where VR = Voltage of the reflected step at the receiving end of the line at thetransit time T (V);

ZL = Load impedance ();Z0 = Characteristic impedance ();VI = Voltage of the incident step at the receiving end of the line (V).

The equation indicates that

when ZL is greater than Z0, the voltage of the reflected step, VR, is of positivepolarity. Consequently, the reflected voltage adds up to the incident step whenit gets back to the sending end of the line, as Figure 1-48 shows.

when ZL is lower than Z0, the voltage of the reflected step, VR, is of negativepolarity. Consequently, the reflected voltage subtracts from the incident stepwhen it reaches the sending end of the line, as Figure 1-48 shows.

when ZL is equal to Z0, the voltage of the incident step is perfectly absorbed bythe load. Consequently, there is no reflected voltage in the step response signal,as Figure 1-48 shows.

Figure 1-48. Behavior of a line terminated by a purely resistive load.


1-55

In part (b) of Figure 1-48, observe that the reflected voltage is approximately equalto voltage VI when the load impedance, ZL, is infinite (4 ).

The reflected voltage, VR, is equal to VI when ZL is infinite because

Determining the Characteristic Impedance

The principles just discussed suggest that the step response method can be usedto measure the characteristic impedance of a line. To do this, a purely resistiveload, ZL, whose resistance can be varied, is connected to the receiving end of theline, as Figure 1-49 shows.

Figure 1-49. Measuring the characteristic impedance by means of a variable resistor connectedto the receiving end of the line.

The resistance of the load is adjusted until no reflected voltage appears in the stepresponse signal, as Figure 1-49 shows. In this condition, ZL is equal to Z0. The loadcan then be disconnected from the line, and its resistance value be measured todetermine ZL.

When the receiving end of the line is not accessible for connection to a variable-resistance load, there is another way of determining the characteristic impedance ofthe line. This method consists in measuring the voltage of the rising edge, Vre, of theincident step in the step response signal. The method can be applied regardless ofthe nature of the load impedance (see Figure 1-50):

capacitive; inductive; purely resistive.


1-56

Figure 1-50. Measuring the voltage of the rising edge (Vre) of the incident step in order todetermine the characteristic impedance.

Voltage Vre is determined by the impedance seen by the step generator immediatelyafter it launches the voltage step into the line, as Figure 1-50 shows. This impedanceis the input impedance of the line, that is, the characteristic impedance of the line.

Consequently, the voltage of the rising edge, Vre, is

where Vre = Voltage of the rising edge of the incident step (V);ETH = Thevenin voltage of the step generator (V);ZTH = Thevenin impedance of the step generator ();Z0 = Characteristic impedance of the line ().

Rewriting and simplifying the above equation for solving Z0 gives:

Procedure Summary

In this procedure section, you will observe the step response of a transmission lineunder various purely resistive load impedances. You will then measure thecharacteristic impedance of this line, using two different methods.


1-57

PROCEDURE

Step Response of a Transmission Line Under Various (Purely Resistive)Impedances


G 2. As Figure 1-51 shows, connect the STEP GENERATOR 50- BNC outputto the BNC connector at the sending end of TRANSMISSION LINE A, usinga coaxial cable. Then, connect the BNC connector at the receiving end ofTRANSMISSION LINE A to the BNC connector at the input of the LOADsection, using a coaxial cable.


Finally, using an oscilloscope probe, connect channel 1 of the oscilloscopeto the 0-meter (0-foot) probe turret at the sending end of TRANSMISSIONLINE A. Make sure to connect the ground conductor of the probe to theassociated shield turret.



1-58

Figure 1-51. Step response of a transmission line under various (purely resistive) loadimpedances.

G 3. In the LOAD section of the circuit board, make sure all the toggle switchesare set to the O (off) position. Then, connect the LOAD-section input to thecommon via resistor R1 (500- potentiometer) by setting the appropriateswitches in this section to the I (ON) position.

Turn the knob of resistor R1 fully clockwise. This sets the impedance of theload at the receiving end of TRANSMISSION LINE A to around 500 .






1-59


G 5. On the oscilloscope screen, observe the step response signal at the sendingend of the transmission line. Since the impedance of the load connected toTRANSMISSION LINE A (about 500 ) is greater than the characteristicimpedance of this line (50 ), the reflected voltage adds up to the incidentvoltage, as Figure 1-52 shows. Is this your observation?

G Yes G No

Figure 1-52. Step response signal with R1 set to 500 approximately.

G 6. Slowly turn the knob of resistor R1 fully counterclockwise, which will causethe impedance of the load connected to TRANSMISSION LINE A todecrease from about 500 to 0 .

While doing this, observe what happens to the step response signal on theoscilloscope screen. As the load impedance is decreased,

a. the reflected voltage, which initially subtracts from the incident voltage,increases, becomes equal to the incident voltage, and then adds up tothe incident voltage.

b. the incident voltage, which is initially lower than the reflected voltage,increases, becomes equal to the reflected voltage, and then adds up tothe reflected voltage.


1-60

c. the reflected voltage, which initially adds up to the incident voltage,decreases, becomes equal to the incident voltage, and then subtractsfrom the incident voltage.

d. the incident voltage, which is initially higher than the reflected voltage,decreases, becomes equal to the reflected voltage, and then subtractsfrom the reflected voltage.

Determining the Characteristic Impedance by Using a Purely Resistive Load

G 7. Adjust the knob of resistor R1 until no reflected voltage appears in the stepresponse signal. If a small notch (discontinuity) remains in the reflectedvoltage, adjust R1 in order to reduce this notch to a minimum, asFigure 1-53 shows.

Figure 1-53. Adjust resistor R1 to reduce the reflected voltage to a minimum.

G 8. Using an ohmmeter, measure the resistance between the LOAD-sectioninput and the common (current resistance setting of resistor R1). Since R1has been adjusted to reduce the reflected voltage to a minimum, its currentresistance should approximately correspond to the characteristicimpedance, Z0:



1-61

Hold the tip of one of the ohmmeter probes on the probe turret just nextto the BNC connector at the LOAD-section input, while touching thenearby common (L) turret with the other ohmmeter probe.

Record below the measured resistance, Z0. Note that Z0 could differfairly from the manufacturer's value of 50 , since the obtained value isdependent upon the accuracy of measurement and on theR1-adjustment.

Z0 =

G 9. Leave the connections as they are, with the BNC connector at the LOAD-section input unconnected, and proceed with the exercise.

Determining the Characteristic Impedance Based on the Rising Edge of theIncident Step

G 10. Determine the Thevenin voltage, ETH, at the STEP GENERATOR 50- BNCoutput by using the following steps:

Connect the STEP GENERATOR 50- BNC output directly to theLOAD-section input. To do so, disconnect the end of the coaxial cableconnected to the BNC connector at the sending end of TRANSMISSIONLINE A, and connect it to the BNC connector at the LOAD-section input.Set all the toggle switches in this section to the O (OFF) position. Thissets the impedance of the load at the STEP GENERATOR 50- outputto the open-circuit condition (4 ).

Disconnect the oscilloscope probe from TRANSMISSION LINE A, andconnect it to the probe turret just next to the BNC connector at theLOAD-section input. Connect the ground conductor of the oscilloscopeprobe to the nearby common (L) turret.

On the oscilloscope screen, measure the voltage of the incident step inthe step response signal. Record below the measured voltage, ETH.

ETH = V

G 11. Determine the Thevenin impedance, ZTH, at the STEP GENERATOR 50-BNC output by using the following steps:

In the LOAD section, set the toggle switches in such a way as toconnect the input of this section to the common, via resistor R1 (500-potentiometer). Adjust the knob of R1 until the voltage of the incidentvoltage on the oscilloscope screen is equal to half the Thevenin voltagemeasured in the previous step.



1-62

Measure the resistance setting of resistor R1 by connecting anohmmeter across the probe turret and associated common (L) turret justnext to the BNC connector at the LOAD-section input. Record below themeasured resistance, ZTH.

ZTH =

G 12. Reconnect the coaxial cable coming from the STEP GENERATOR 50-BNC output to the BNC connector at the sending end of TRANSMISSIONLINE A. Reconnect the coaxial cable coming from the receiving end ofTRANSMISSION LINE A to the BNC connector at the LOAD-input section.

In the LOAD section, turn the knob of resistor R1 fully clockwise.

Using an oscilloscope probe, connect channel 1 of the oscilloscope to the0-meter (0-foot) probe turret at the sending end of TRANSMISSION LINE A.

G 13. Set the oscilloscope time base to 0.2 s/div.

G 14. On the oscilloscope screen, measure the voltage (height) of the fast-risingedge, Vre, of the incident step, as Figure 1-54 shows. As oscillatory noisemight appear in the upper section of the rising edge, measure theapproximate voltage of this edge.

Record below the measured voltage, Vre.

Vre = V


1-63

Figure 1-54. Measuring the voltage (height) of the fast-rising edge, Vre, of the incident step.

G 15. Based on the fast-rising edge voltage Vre, and on the STEP GENERATORThevenin equivalent measured in the previous steps, calculate thecharacteristic impedance, Z0, of TRANSMISSION LINE A.

Z0 =

The result should be quite near the characteristic impedance of 50 specified for the RG-174 coaxial cables used as TRANSMISSION LINESA and B. However, since your result is dependent upon the accuracy of themeasurements made on the oscilloscope and on the rounding accuracyused for the calculations, this value may differ fairly from the manufacturer'svalue of 50 .



1-64

CONCLUSION

The step response method can be used to determine the nature of the loadimpedance terminating a line. The shape of the step response signal indicateswhether the load impedance is purely resistive or complex.

When the load impedance is purely resistive, the reflected voltage has thesame shape as the incident voltage. It adds up to or subtracts from theincident voltage, depending on the relation between the load impedance andthe characteristic impedance of the line.

When the load impedance is both resistive and inductive, the reflectedvoltage decreases exponentially until it stabilizes to a certain level, thereforehaving the same shape as the voltage across a capacitor dischargingthrough a series resistor.

When the load impedance is both resistive and capacitive, the reflectedvoltage increases exponentially until it stabilizes to a certain level, thereforehaving the same shape as the voltage across a capacitor charging througha series resistor.

The characteristic impedance of a line can be determined by connecting avariable resistance load to the receiving end of the line and adjusting theresistance until no reflected voltage appears in the step response signal. In thiscondition, the load resistance is equal to the characteristic impedance.

When the receiving end of the line is not accessible, the characteristicimpedance can be determined by measuring the voltage of the rising-edge of theincident voltage in the step response signal. This voltage and the Theveninequivalent of the step generator are then used to calculate the characteristicimpedance.

REVIEW QUESTIONS

1. When the load impedance is both resistive and inductive, the reflected voltagein the step response signal

a. decreases more and more with time until it stabilizes to a certain level.b. increases exponentially until it stabilizes to a certain level.c. suddenly increases and then remains at a constant level.d. has the same shape as the incident voltage.


1-65

2. When the load impedance is purely resistive and lower than the characteristicimpedance of the line, the voltage of the reflected step is

a. of negative polarity, so that it subtracts from the voltage of the incident stepwhen it gets back to the sending end of the line.

b. equal to the voltage of the incident step, so that it cancels out this step whenit gets back to the sending end of the line.

c. of positive or negative polarity, depending on the extent of the mismatchbetween the load and line impedances.

d. of positive polarity, so that it adds up to the incident step when it gets backto the sending end of the line.

3. What is the voltage and polarity of the step reflected at the receiving end of theline if the impedance of the load is perfectly equal to the load impedance?

a. The voltage of the reflected step is half the voltage of the incident step andis of positive polarity.

b. The voltage of the reflected step is twice the voltage of the incident step andis of negative polarity.

c. The reflected step has the same voltage and polarity as the incident step.d. There is no reflected step.

4. What is the voltage and polarity of the step reflected at the receiving end of theline if the impedance of the load is in the short-circuit condition (0 )?

a. The voltage of the reflected voltage is equal to the voltage of the incidentstep and is of positive polarity.

b. The voltage of the reflected step is equal to the voltage of the incident stepand is of negative polarity.

c. The voltage of the reflected step is twice the voltage of the incident step andis of negative polarity.

d. There is no reflected step.

5. What is the characteristic impedance of a line if the voltage of the rising edge ofthe incident step in the step response signal is 2.5 V? Assume the Theveninimpedance and Thevenin voltage of the step generator to be 75 and 5 V,respectively.

a. 100 b. 50 c. 75 d. 125

1-67

Exercise 1-4

Attenuation and Distortion

EXERCISE OBJECTIVE

Upon completion of this unit, you will know what attenuation and distortion are, andhow they can affect the shape of the transmitted signal. You will be able to explainwhat causes attenuation and distortion. You will know about a method of evaluatingsignal quality in high-speed transmission systems.

DISCUSSION

Attenuation

In transmission lines that are lossy, the transmitted signals lose some energy as theytravel down the line. This occurs because the energy gradually dissipates in eachseries resistance, R'S, and parallel resistance, R'P, per unit length of the line.

The energy lost in each R'S is by heating of the conductors (I2R losses). The energy

lost in each R'P is by heating of the dielectric material used to construct theconductors (shunt or dielectric losses), as Figure 1-55 shows.

Figure 1-55. Signals lose some energy in each R'S and R'P.

The energy losses cause the level of the transmitted signal to gradually decrease asthe signal travels down the line, as Figure 1-56 shows. The decrease in signal levelover distance is called attenuation. Attenuation increases as the distance from thetransmission point increases.


1-68

Figure 1-56. Attenuation of a rectangular signal and a sinusoidal signal.

Attenuation is normally expressed in decibels (dB). The formula for calculating theattenuation in signal power at a distance D from the sending end of a line is asfollows:

where A = Attenuation in signal power (dB);log = Base-10 logarithm;PD = Signal power at a distance D from the sending end of the

line (W);PS = Signal power at the sending end of the line (W).


1-69

Table 1-3 indicates the attenuation, A, for different PD/PS ratios. Each time the ratiodecreases by a factor of 2, the signal power is attenuated by 3 dB.

PD/PS RATIO POWER ATTENUATION (dB)

1 0

0.5 -3

0.25 -6

0.125 -9

Table 1-3. Power attenuation for different PD/PS ratios.

For example, the attenuation in signal power at a distance D from the sending endof the line, if the ratio PD/PS is 0.75, will be -1.25 dB.

When voltage measurements, which are most common, are performed instead ofpower measurements, the formula for calculating the attenuation in signal power ata distance D from the sending end of the line becomes:

where A = Attenuation in signal power (dB);log = Base-10 logarithm;VD = Signal voltage at a distance D from the sending end of the

line (V);VS = Signal voltage at the sending end of the line (V).

For example, the attenuation in signal power at a distance D from the sending endof the line, if the ratio VD/VS is 0.75, will be -2.5 dB.

Line manufacturers usually provide graphs that indicate the attenuation per unitlength, , of a line as a function of signal frequency. They must do this because athigher frequencies, the attenuation per unit length, instead of being constant,increases with frequency due, among other things, to a phenomenon known as skineffect.

The skin effect is illustrated on Figure 1-57. At direct current (DC) or low frequency,the current density is quite uniform across the conductor. At higher frequencies, thecurrent density tends to concentrate near the surface (hence the term "skin") of theconductor, thereby increasing the resistance to current flow and, in turn, theattenuation per unit length, .


1-70

Figure 1-57. The skin effect.

Frequency Components of a Signal

A pure sinusoidal wave is composed of a single frequency component, called afundamental. However, periodic signals usually consist of a superposition of severalfrequency components.

These components are waves that are all sinusoidal in shape, but are of differentfrequencies and amplitudes. They include a fundamental, or first harmonic, at thefrequency of the signal, and several higher-order harmonics whose frequencies aremultiples of the fundamental frequency.

Figure 1-58 shows the time-domain and frequency-domain representations ofdifferent signals. A periodic signal can be expanded as an infinite sum of sines andcosines of different amplitudes and frequencies, called a Fourier series.

When observing the frequency components of a signal on a spectrum analyzer, wesee that the frequency of the fundamental is the reciprocal of the signal period, T.The magnitude of the harmonics decreases as the order, or number, of the harmonicincreases.


1-71

Figure 1-58. Time-domain and frequency-domain representations of different signals.

The frequency spectrum differs from one type of signal to another, as Figure 1-58shows. For example, a rectangular signal consists of a set of odd harmonics, whilea sawtooth signal consists of both even and odd harmonics.

Distortion

In a transmission line, the velocity of propagation of the fundamental and harmonicsthat compose a transmitted signal is determined mainly by the relative permittivityof the line dielectric material.

In lines that are lossless or that have very low losses, relative permittivity staysapproximately constant with frequency. Consequently, the fundamental andharmonics of the transmitted signal all propagate at the same velocity along theline. As a result, the signal at the receiving end of the line is a faithful reproductionof the transmitted signal, as the left-hand section of Figure 1-59 shows. Thesignal is said to be distortionless.

In lines that are lossy, however, relative permittivity varies with frequency.Consequently, the fundamental and harmonics of the transmitted signalpropagate at differing velocities. This phenomenon is known as dispersion.Dispersion causes distortion: the signal at the receiving end of the line has a


1-72

shape that is quite different than that of the transmitted signal, as the right-handsection of Figure 1-59 shows.

If, additionally, the fundamental and its harmonics are of relatively highfrequencies, they will be attenuated differently since, as earlier mentioned, theattenuation per unit length is frequency dependent at higher frequencies. This willtend to aggravate distortion in the received signal.

Figure 1-59. An undistorted signal versus a distorted and attenuated signal.

Thus, the change in shape of the transmitted signal in the figure occurs because therise time and fall time of the transients in the transmitted signal are longer in thereceived signal.

Attenuation and distortion can be significant problems in today's high-speedtransmission systems, due to the high-frequency signals inherently involved in thesesystems. Figure 1-60, for example, shows how distortion affects the transmission ofnon-return-to-zero (NRZ) data. A comparison of the transmitted data (A) shows thatthe recovered data (C) does not truly reproduce the original data, which mayintroduce errors.


1-73

Figure 1-60. Attenuation and distortion affect the recovery of the original NRZ data.

A popular method of evaluating signal quality in digital transmission systems is theeye-pattern method. This method requires that a pseudo-random binary signal(PRBS) be applied to the vertical input of an oscilloscope. The oscilloscopehorizontal sweep is triggered by a signal of the same frequency as the binary signal.The time base is adjusted so as to see about one period of the PRBS, as Figure 1-61shows. In this way, the oscilloscope display is a pattern that resembles an eye, dueto the superposition of the transitions and constant bit levels that occur randomly onsuccessive periods of the signal. The width of the eye opening indicates the degreeof distortion. The narrower the eye opening, the greater the signal distortion and,therefore, the lower the probability of error-free data recovery.


1-74

Figure 1-61. The eye pattern.

Procedure Summary

In this procedure section, you will measure the attenuation of the STEPGENERATOR output signal at the receiving end of a 48-meter (157.4-foot) line. Youwill then answer theoretical questions about distortion.

Note: Since the lines used on the TRANSMISSION LINES circuit boardare not long enough to permit observation of the effects of dispersion,the procedural section on distortion will consist of theoretical questions.

PROCEDURE

Attenuation



1-75

G 2. Referring to Figure 1-62, connect the STEP GENERATOR 50- output tothe sending end of TRANSMISSION LINE A, using a coaxial cable. Connectthe receiving end of TRANSMISSION LINE A to the sending end ofTRANSMISSION LINE B, using a coaxial cable. Finally, connect thereceiving end of TRANSMISSION LINE B to the input of the LOAD section,using a coaxial cable.

Figure 1-62. Observing the attenuation of the STEP GENERATOR output voltage along the line.

Now, connect the STEP GENERATOR 100- output to the trigger input ofthe oscilloscope, using a coaxial cable.

Connect channel 1 of the oscilloscope to the 0-meter (0-foot) probe turretat the sending end of TRANSMISSION LINE A, using an oscilloscopeprobe.

Finally, using another oscilloscope probe, connect channel 2 of theoscilloscope to the 24-meter (78.7-foot) probe turret at the receiving end ofTRANSMISSION LINE B.


1-76


G 3. In the LOAD section of the circuit board, make sure all the toggle switchesare set to the O (OFF) position. Then, connect the LOAD-section input tothe common via resistor R3 (50- resistor) by setting the appropriateswitches in this section to the I (ON) position.




Time Base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 s/divTrigger



G 5. On the oscilloscope screen, observe the signal transmitted by the STEPGENERATOR at the sending end and receiving end of the 48-meter(157.4-foot) line formed by TRANSMISSION LINEs A and B beingconnected end-to-end.


1-77

Figure 1-63. Measuring the attenuation at D = 48 meters (157.4 feet).

Note that the voltage of the pulses in the STEP GENERATOR signal islower at the receiving end of the line, as Figure 1-63 shows. This occursbecause

a. of the energy lost in each series inductance and parallel capacitanceper unit length of the line.

b. the pulses at the sending end of the line lost some energy as theytraveled down the line.

c. of a phenomenon known as dispersion.d. the transmitted signal lost energy in the 50- characteristic impedance

of the line.

G 6. Measure the voltage (height) of the pulses at the sending end of the line, VS.

VS = V

G 7. Measure the voltage (height) of the pulses at the receiving end of theline, VD.

VD = V

G 8. Using the measured voltages, VS and VD, calculate the attenuation in pulsepower at the receiving end of the 48-meter (157.4-foot) line by using theformula below.


1-78

where A = Attenuation in pulse power (dB);log = Base-10 logarithm;VD = Signal voltage at a distance D from the sending end of the

line (V);VS = Signal voltage at the sending end of the line (V).

A = dB

Distortion

G 9. Which of the following signals best correspond to the frequency-domainrepresentation shown in Figure 1-64?

a. A pure sinusoidal signal having a period of 5 s.b. A rectangular signal having a period of 5 s.c. A sawtooth signal having a period of 0.5 s.d. A rectangular signal having a period of 0.5 s.

Figure 1-64. Frequency-domain representation of a signal.

G 10. If you look at Figure 1-65, which of the following statements could explainwhy the received signal has a shape that is quite different than that of thetransmitted signal?

a. The harmonics of higher order in the transmitted signal were lessattenuated than those of lower order as they traveled down the lossyline.

b. The fundamental and harmonics of the transmitted signal propagatedat the same velocity along the lossy line.

c. The harmonics of higher order in the transmitted signal were moreattenuated than those of lower order as they traveled down the losslessline.

d. The fundamental and harmonics of the transmitted signal propagatedat differing velocities along the lossy line.


1-79

Figure 1-65. The received signal is attenuated and distorted.

G 11. If you look at Figure 1-66, which eye pattern corresponds to the bestprobability of recovering the transmitted pseudo-random NRZ data withouterror?

a. Eye pattern Ab. Eye pattern Bc. Eye pattern Cd. Eye pattern D


1-80

Figure 1-66. Eye patterns.


CONCLUSION

Attenuation is a decrease in the level of a transmitted signal as it travels alonga line. Attenuation occurs in lines that are lossy. It is due to the dissipation ofpart of the signal energy in the distributed series and parallel resistances of theline.

Distortion is a change in the shape of the transmitted signal that also occurs inlines that are lossy. Distortion is caused mainly by dispersion, a phenomenon bywhich the fundamental and harmonics that compose a transmitted signalpropagate at differing velocities. Distortion can also be caused by the high-frequency signal components being attenuated differently, since attenuation isfrequency dependent.

Distortion of the high-frequency components in the transmitted signals increasethe rise time and fall time of the signal transients, causing the signals to berounded.


1-81

In a high-speed transmission system, a popular method of evaluating signalquality is the eye-pattern method. This method provides an eye-pattern displaythat resembles an eye. The width of the eye opening indicates the degree ofdistortion.

REVIEW QUESTIONS

1. The attenuation of a transmitted signal in a lossy line

a. is due to the dissipation of part of the signal energy by heating of the load.b. is a decrease in the signal level as the signal travels down the line.c. decreases as the distance from the transmission point increases.d. decreases as the signal frequency is increased.

2. The skin effect causes the

a. current density across the line conductors to remain uniform if the signalfrequency is increased.

b. resistance to current flow of the line conductors to decrease as the signalfrequency is increased.

c. current density to concentrate near the surface of the line conductors at lowsignal frequencies.

d. attenuation per unit length to increase as the signal frequency is increased.

3. A periodic signal usually consists of a superposition of several frequencycomponents that

a. include, among others, harmonics whose frequencies are odd and/or evenmultiples of the fundamental frequency.

b. include, among others, a fundamental at twice the frequency of the signal.c. are all of the same frequency.d. are all rectangular in shape.

4. The signal at the receiving end of a lossy line is distorted

a. when the low-frequency components of the transmitted signal areattenuated differently, due to the skin effect.

b. mainly because the frequency components of the transmitted signalpropagate at differing velocities.

c. when it does not truly reproduce the signal applied to the line load.d. when it has the same shape as the transmitted signal.


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5. The eye-pattern method of evaluating signal quality

a. provides an eye-pattern display, the width of the eye opening beinginversely proportional to the degree of distortion.

b. requires that a pseudo-random audio signal be applied to the vertical inputof an oscilloscope.

c. provides a display of the frequency components of the signal as a functionof time.

d. is used with low-speed data transmission systems.

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Unit Test

1. The harmonics of a rectangular signal having a period of 0.5 s are

a. even multiples of 5 MHz.b. odd multiples of 2 MHz.c. even multiples of 2 MHz.d. odd multiples of 5 MHz.

2. The velocity of propagation of a signal in a transmission line can be determinedby using

a. a high-impedance oscilloscope probe connected to the sending end of theline and a step generator connected to the receiving end of the line.

b. a simple formula, if the time required for a voltage step to travel to thereceiving end of the line and back to the sending end is known.

c. the step response method, provided that the load impedance perfectlymatches the characteristic impedance of the line.

d. a step generator and a high-impedance oscilloscope probe connected to thereceiving end of the line.

3. When a voltage step is launched into a lossy line whose series losses arepredominant,

a. the high frequency components contained in the voltage steps make the linetemporarily appear as a simple RC network.

b. the incident and reflected steps will first rise to a certain level and thendecrease gradually.

c. it is not possible to measure the time separating the incident and reflectedsteps.

d. the line will appear as a simple LC network from the perspective of the load.

4. Theoretically speaking, the characteristic impedance, Z0, of a transmission linecorresponds to the input impedance, ZIN, of a

a. line whose load impedance is in the short-circuit condition (0 ).b. line terminated with a purely resistive load.c. particular length of line.d. line of infinite length.

5. According to Thevenin's theorem,

a. the Thevenin impedance ZTH is the impedance seen at the two terminals ofthe circuit to thevenize, when the voltage source of this circuit is replaced byan open circuit.

b. the Thevenin voltage ETH is determined by measuring the short-circuitvoltage at the two terminals of the circuit to thevenize.

c. the Thevenin equivalent circuit consists of a voltage source, ETH, and animpedance in parallel with this source, ZTH.

d. any electrical linear circuit seen at two terminals can be represented by aThevenin equivalent circuit.

6. The eye-pattern method of evaluating signal quality

Unit Test (cont'd)

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a. provides an eye-pattern display, the width of the eye opening indicating thedegree of distortion.

b. requires that a pseudo-random audio signal be applied to the vertical inputof an oscilloscope.

c. provides a display of the frequency components of the signal as a functionof time.

d. is used with low-speed data transmission systems.

7. In a transmission line, a signal travels at a velocity that

a. is directly proportional to the relative permittivity of the dielectric materialused to construct the line.

b. is null if the impedance of the load at the receiving end is in the open-circuitcondition (4 ).

c. usually increases as the diameter of the line conductors is decreased.d. is relatively less than 3.0 @ 108 m/s, or 9.8 @ 108 ft/s.

8. The skin effect causes the

a. current density across the line conductors to remain uniform if the signalfrequency is increased.

b. resistance to current flow of the line conductors to decrease as the signalfrequency is increased.

c. current density to concentrate near the surface of the line conductors at lowsignal frequencies.

d. attenuation per unit length to increase as the signal frequency is increased.

9. The step response method can be used to determine the characteristicimpedance of a transmission line

a. by connecting a purely resistive load to the receiving end of the line andadjusting the load resistance until no reflected voltage appears in the stepresponse signal.

b. by connecting a resistive load having half the input impedance of the lineand measuring the voltage of the rising edge of the incident step in the stepresponse signal.

c. through measurement of the voltage of the rising edge of the reflected stepin the step response signal.

d. provided that the receiving end of the line can be made accessible forconnection to a specific load.

Unit Test (cont'd)

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10. When the load impedance is purely resistive and lower than the characteristicimpedance of the line, the voltage of the reflected step is

a. of negative polarity, so that it subtracts from the voltage of the incident stepwhen it gets back to the sending end of the line.

b. equal to the voltage of the incident step, so that it cancels out this step whenit gets back to the sending end of the line.

c. of positive or negative polarity, depending on the extent of the mismatchbetween the load and line impedances.

d. of positive polarity, so that it adds up to the incident step when it gets backto the sending end of the line.

2-1

Unit 2

Transmission Line Measurements Under Transient(Step Testing) Conditions

UNIT OBJECTIVES

Upon completion of this unit, you will be able to measure the distributed inductanceand distributed capacitance of a line, and then use the measured values todetermine the characteristic impedance and velocity of propagation through the line.You will be familiar with the concept of voltage reflection coefficient. You will knowhow to represent the creation of reflections and the distribution of the voltage alonga line, using a graph called the voltage reflection diagram. You will know how a lineterminated by complex load impedances behaves when voltage steps are launchedinto the line. Finally, you will learn how to use time-domain reflectometry (TDR) tolocate and identify discontinuities (faults) introduced by your instructor along thetransmission lines of the circuit board.

UNIT FUNDAMENTALS

Determining Characteristic Impedance and Velocity of Propagation

The characteristic impedance, Z0, of a line is constant, regardless of how long theline may be. This occurs because Z0 is determined by the geometrical and physicalcharacteristics of the line. Thus, a relationship exists between the value of Z0 and thevalue of its distributed parameters, L', C', R'S, and R'p.

When the losses in a line are low and the frequency of the carried signals is relativelyhigh, Z0 is considered to be purely resistive. In this condition,

Moreover, the velocity of propagation of signals through the line, vP, is

Consequently, Z0 and vP can easily be calculated when L' and C' are known:

L' and C' are normally specified by the line's manufacturer. If L' and C' are unknown, they can be measured by creating an impedance

mismatch at both ends of the line in order to obtain a step response signal withmeasurable time constant, as Figure 2-1 shows. The measured time constantspermit calculation of L' and C'.


2-2

Figure 2-1. Examples of step response signals obtained for the measurement of L' and C'.

Voltage Reflection Coefficient

Whenever the signal carried by a transmission line encounters a change inimpedance, or discontinuity, a reflection is created. The reflection causes part ofthe signal energy to be reflected in a direction opposite to the direction of travel ofthe signal.

The ratio of the reflected voltage to incident voltage at a discontinuity is called thevoltage reflection coefficient. This coefficient is usually represented by , theGreek capital letter "gamma". It is determined by the relationship between thecharacteristic impedance, Z0, and the new impedance encountered by the signal atthe discontinuity, as Figure 2-2 shows.

Figure 2-2. Discontinuity (impedance change) at the load.

When neither the impedance of the load, ZL, and of the step generator, ZTH, matchthe characteristic impedance, multiple successive reflections occur on the line untilthe voltage on the line stabilizes to a final value.


2-3

In that case, a convenient way of representing the creation of the multiple reflectionsand the resulting distribution of the voltage along the line as a function of time is byusing a voltage reflection diagram, as Figure 2-3 shows.

The voltage reflection diagram, on the left-hand side of the figure, is a verticallyoriented graph. It indicates each instant when a sudden change (voltage step)occurs in the line voltage.

The voltage reflection diagram can be used conveniently to graph the voltage asa function of time at any arbitrary point, z, along the line, as shown on the right-hand side of the figure.

Figure 2-3. Voltage reflection diagram.

Transient Behavior of a Line Terminated by Complex Load Impedances

When the load connected to the end of a transmission line does not match thecharacteristic impedance of the line, and the load impedance is not purely resistive,the voltage reflected back toward the step generator varies with time, so that it doesnot have the same shape as the incident voltage.

For example, Figure 2-4 shows the step response signal of a line terminated by aninductive load impedance, and that of a line terminated by a capacitive loadimpedance:

when the load impedance is mostly inductive, the reflected voltage decreasesexponentially over time;


2-4

when the load impedance is mostly capacitive, the reflected voltage increasesexponentially over time.

In both cases, the time constant of the exponentially-decreasing or increasingvoltage can be measured in order to determine the inductance or capacitance of theload, as Figure 2-4 shows.

Figure 2-4. Examples of step response signals with inductive and capacitive load impedances.

Appendix B provides a summary of the step response signal of a line terminated byinductive and capacitive loads.

Time-Domain Reflectometry (TDR)

Time-domain reflectometry (TDR) is a technique used to detect, locate, andidentify the nature of discontinuities (impedance changes) along a transmission line.Discontinuities can be due, for example, to broken conductors, loose connectors,shorted conductors, or sheath faults.

TDR requires the use of a device called a time-domain reflectometer, as Figure 2-5shows. Conventional time-domain reflectometers consist of a built-in step generatorand an oscilloscope. They operate by using the same technique as bats or radars:the echo technique.

The step generator launches a pulse into the line under test.

When the pulse encounters a discontinuity along the line, part of the pulse energyis reflected back to the step generator for display on an oscilloscope.

In the displayed signal, the discontinuities encountered by the pulse appear in theform of transients.

By measuring, on the oscilloscope screen, the time between the rising edge ofthe launched pulse and the transient caused by a discontinuity, the location(distance) of the discontinuity can be determined. Moreover, the shape and


2-5

magnitude of the transient indicate the nature and severity of the discontinuityand, therefore, give clues to the probable cause(s) of the fault.

Figure 2-5. Using time-domain reflectometry (TDR) to detect and locate discontinuities along aline.

EQUIPMENT REQUIRED

In order to complete the exercises in this unit, you will need the following:

FACET Base Unit TRANSMISSION LINES circuit board Oscilloscope, dual trace, 40 MHz, (Lab-Volt P/N 797 or equivalent)

2-7

Exercise 2-1

Determining Characteristic Impedance and Velocityof Propagation by Measuring the DistributedCapacitance and Inductance of a Line

EXERCISE OBJECTIVES

Upon completion of this exercise, you will know how to measure the distributedcapacitance and distributed inductance of a line. You will use the measured valuesto determine the characteristic impedance of the line and the velocity of propagationof the signals along the line.

DISCUSSION

Relationship Between the Characteristic Impedance of a Line and itsDistributed Parameters

The characteristic impedance, Z0, of a line is an intrinsic property of the line.Because of this, Z0 is determined by the geometrical and physical characteristics ofthe linenot by the length of the line. Z0, therefore, is of constant value, whether theline is short, long, or infinite.

The physical characteristics that determine Z0, which are the diameter of theconductors, the relative spacing between the conductors, and the insulating materialused, also determine the value of the distributed parameters of the line.Consequently, a relationship exists between the value of Z0 and the value of thedistributed parameters.

Figure 2-6. Distributed parameters of a line.

Determining Characteristic Impedance and Velocityof Propagation by Measuring the DistributedCapacitance and Inductance of a Line

2-8

When R'S is negligible and R'P is relatively high (G' is relatively low), and thefrequency of the carried signals, f, is relatively high (such that 2fL' o R'S and2fC' o R'P), the reactive component of Z0 can be disregarded and Z0 be consideredas purely resistive. In this condition, it can be demonstrated that

where Z0 = Characteristic impedance ();L' = Distributed inductance, in henrys per unit lengt

Lab Manual Electrical Networks 2015

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