Transmission Lines and Waveguides |
65 |
thickness of the strip is t = 5 mm. Find the wavelength in the line at a frequency of 10 GHz.
Solution
From Figure 3.13 we see that w /h is about 1. Equation (3.90) gives A = 2.124. From (3.89) we solve w /h = 0.985 or w = 0.250 mm. To account for the effect of the strip thickness, we calculate D = 2h /t = 101.6 and solve from (3.93) to get Dwe = 0.009 mm. Therefore, the strip width should be 0.250 mm − 0.009 mm = 0.241 mm. A 5 mm thick and 0.241 mm wide strip corresponds to a 0.250 mm wide strip with t = 0. From (3.85)
we obtain |
ereff = 6.548. The wavelength at f = 10 GHz is l = |
c / X f √ereff |
C = 11.72 mm. (In practice, microstrip lines are dispersive and |
increases as the frequency increases. Therefore, l is slightly shorter.)
3.9 Wave and Signal Velocities
In a vacuum, radio waves propagate at the speed of light, c = 299,792,458 m/s. In a medium with parameters er and m r the velocity of propagation is
v = |
1 |
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= |
1 |
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c |
(3.96) |
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√ |
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√ |
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me |
m r er |
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The phase velocity
vp = |
v |
(3.97) |
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b |
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is that velocity with which the constant-phase points of a wave propagate. In case of a plane wave or a TEM wave propagating in a transmission line, the phase velocity is equal to the velocity of propagation in free space filled with the same medium, v p = v . Generally, the phase velocity of a wave propagating in a waveguide may be smaller or larger than v .
If the phase velocity and attenuation of a propagating wave do not depend on frequency, the waveform of a broadband signal does not distort as it propagates. However, if the phase velocity is frequency-dependent, the waveform will distort. This phenomenon is called dispersion (see Section 3.3).
The group velocity, v g , is that velocity with which the energy of a narrow-band signal (or signal experiencing no significant dispersion) propa-
66 Radio Engineering for Wireless Communication and Sensor Applications
gates. The group velocity of a plane wave or a TEM wave is equal to the velocity of propagation in free space filled with the same medium. For other wave modes, the group velocity is smaller than v . The group velocity can never exceed the speed of light. The group velocity is given by [2, 5]
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Sdv D |
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vg = |
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db |
−1 |
(3.98) |
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3.10 Transmission Line Model
A transmission line exhibits properties of capacitance, inductance, resistance, and conductance: The electric field between the conductors contains electric energy in the same way as a capacitor; the magnetic field produced by the currents contains magnetic energy as an inductor; the conductors have losses as a resistor; the leakage currents in the insulator produce losses as a resistor or a conductor. These properties cannot be separated because they are distributed along the line. Figure 3.14 shows the transmission line model, a short section of a transmission line with a length Dz having a series inductance L , a parallel capacitance C, a series resistance R , and a parallel conductance G, all being values per unit length.
In the transmission line model, voltages and currents are used instead of electric and magnetic fields to represent the propagating wave. This model best suits transmission lines carrying TEM wave modes. However, the transmission line model may also be applied for other transmission lines and waveguides by defining the voltage and current properly and by restricting the analysis for such a narrow band that dispersion may be neglected.
The voltage and current on a transmission line depend on the position and time, V (z , t ) and I (z , t ). We can derive for voltage and current the so-called telegrapher equations:
Figure 3.14 Transmission line model.
Transmission Lines and Waveguides |
67 |
∂2V (z , t ) |
= LC |
∂2V (z , t ) |
+ (RC + |
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∂z 2 |
∂t 2 |
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∂2I (z , t ) |
= LC |
∂2I (z , t ) |
+ (RC + |
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∂z 2 |
∂t 2 |
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LG )
LG )
∂V (z , t )
∂ + RGV (z , t ) t
(3.99)
∂I (z , t )
∂ + RGI (z , t ) t
(3.100)
The voltage and current of a sinusoidal signal are
V (z , t ) = V (z ) e jvt
I (z , t ) = I (z ) e jvt
For sinusoidal signals, the telegrapher equations simplify to
d 2V (z ) − g2V (z ) = 0 dz 2
d 2I (z ) − g2I (z ) = 0 dz 2
where
g = √(R + jvL ) (G + jvC ) = a + jb
(3.101)
(3.102)
(3.103)
(3.104)
(3.105)
The solutions of the telegrapher equations are of the form
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V (z ) = V +e −gz + V −e +gz |
(3.106) |
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I (z ) = |
V + |
e |
−gz − |
V − |
e +gz = I +e −gz − I −e +gz |
(3.107) |
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Z 0 |
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Z 0 |
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where |
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Z 0 = √ |
R + jvL |
(3.108) |
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G + jv C |
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68 Radio Engineering for Wireless Communication and Sensor Applications
is the complex characteristic impedance of the transmission line. In (3.106) and (3.107) V + and I + are the complex amplitudes for a wave propagating into the positive z direction and V − and I − are those for a wave propagating into the negative z direction.
References
[1]Chatterjee, R., Elements of Microwave Engineering, Chichester, England: Ellis Horwood, 1986.
[2]Collin, R. E., Foundations for Microwave Engineering, 2nd ed., New York: IEEE Press, 2001.
[3]Collin, R. E., Field Theory of Guided Waves, New York: IEEE Press, 1991.
[4]Gardiol, F. E., Introduction to Microwaves, Dedham, MA: Artech House, 1984.
[5]Pozar, D. M., Microwave Engineering, 2nd ed., New York: John Wiley & Sons, 1998.
[6]Ramo, S., J. Whinnery, and T. van Duzer, Fields and Waves in Communication Electronics, New York: John Wiley & Sons, 1965.
[7]Liao, S. Y., Microwave Circuit Analysis and Amplifier Design, Englewood Cliffs, NJ: Prentice Hall, 1987.
[8]Bahl, I. J., and D. K. Trivedi, ‘‘A Designer’s Guide to Microstrip Line,’’ Microwaves, May 1977, pp. 174–182.