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APPENDIX B
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Transmission Line Calculations
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Velocity of Propagation
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Velocity of propagation of electrical energy along a transmission line of length D meters is given by vp L C vp inductance of line, in henrys capacitance of line, in farads velocity of propagation, in meters per second D _____ ___
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Characteristic Impedance of a Transmission Line
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A schematic diagram of a section transmission line follows
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RS Z1 X 2 L RS Z1 X 2 L
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RS is ac series resistance caused by wires, in ohms XL is a series inductive reactance caused by twisted wires RP is insulation leakage resistance between conductors XC is capacitive reactance caused by capacitance between conductors It can be shown that ZO
Z1Z2
_____
Copyright 2005 by The McGraw-Hill Companies, Inc Click here for terms of use
Appendix B
where Z is the impedance of two sets of RS in series with XL Also, it can be shown that if RS 0 RP for the lossless transmission line ZO
L __
Therefore, as series inductance is added by loading the pair, the characteristic impedance of the line is increased Given the following transmission line,
Transmission line characteristic ZO impedance ZS ac voltage source VS Sending end L ZR
Receiving end
if ZS and ZR do not equal ZO, voltage and current re ections will result along the transmission line length L The re ection coef cient K is de ned as the magnitude of the re ected voltage wave to the incident voltage wave V RMS voltage of the re ected wave, re ected from the mismatch impedance V RMS voltage of the incident wave going toward the mismatch impedance K |V | _____ |V | 0 K in terms
Example: If V 0 V, there is no re ected wave and K of the impedance mismatch is represented as at the sending end KS ZS ZO ________ ZS ZO
at the receiving end KR
ZR ZO ________ ZR ZO
Transmission Line Calculations
Example: For a perfect impedance match, ZO KS ZR ZS
0 KR, or 0
meaning no re ections If V is the incident voltage wave traveling down the line and V is the re ected wave traveling up the line caused by the mismatch of the receiving end impedance, VMAX occurs when they reinforce each other to cause a voltage maximum |VMAX| |VMIN| V and V de ned as |V | |V | |V | |V | (B-1) (B-2)
cancel each other out The voltage standing-wave ratio is
VSWR
|VMAX| ________ |VMIN|
Substituting in Equations B-1 and B-2, VSWR |V | |V | _____________ |V | |V | (B-3)
and multiplying Equation B-3 by 1/|V |, |V | 1 _____ |V | _________ |V | 1 _____ |V |
VSWR
and K
|V | _____ |V |
so VSWR 1 K ______ 1 K
K can be KR or KS Return loss is a measure of the incident wave power to the re ected wave power Mathematically, RL |P | 10 log ______ |P |
and |P | Substituting, |V | ______ RL ZO 10 log ______ 2 |V | ______ ZO RL Since K |V |/|V |, RL 1 20 log __ K |V | 10 log ______ 2 |V |
2 2 2 2
Appendix B
|V | ______ ZO
|P |
|V | ______ ZO
|V | 20 log _____ |V |
Return loss is the difference expressed in dB between the incident and re ected waves along a transmission line
APPENDIX C
Coaxial Cable System Powering
A section of cable plant is shown; the resistances of cable sections and the distances between ampli ers are noted
10 1A CR1 2200 ft 2A 3A 4A 4A CR3 CR2 CR4 9 7 1 8 P1 2300 ft 2100 ft 2000 ft 3A CR5 2200 ft 2A CR6 2100 ft 1A CR7 2200 ft
8A 60 V CC 60 Hz regulated power supply 12-ampere full load
532 V
484 V
45 V
432 V
Power block signal pass
Cable section CR1 CR2 CR3 CR4 CR5 CR6 CR7
Length (in 1000 ft) 22 23 2 21 22 21 22
Resistance (in ohms)* 18 18 16 17 18 17 18
Current (in amps) 1 2 3 4 3 2 1
Voltage drop (in volts) 18 36 48 68 48 34 18
*08 ohms per 1000 ft
Resistance
current
Copyright 2005 by The McGraw-Hill Companies, Inc Click here for terms of use
Appendix C
Cable loop resistance is 08 ohm per thousand feet The ampli ers draw 1 ampere each and the minimum allowable ampli er voltage is 40 volts The power supply delivers 8 amps Now the voltage at each ampli er location can be calculated and labeled on the diagram Amp 1: 60 V Amp 2: 532 V Amp 3: 484 V Amp 4: 45 V 68 V 48 V 34 V 18 V 532 V 484 V 45 V 432 V Amp 7: No voltage drop Amp 8: 60 V Amp 9: 552 V Amp 10: 516 V 48 552 V 516 V 49 V 36 V 18 V
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