AM 5-306 BASIC ELECTRONICS TRANSMISSION LINES JANUARY 2012 DISTRIBUTION RESTRICTION: Approved for Pubic Release. Distribution is unlimited. DEPARTMENT OF THE ARMY MILITARY AUXILIARY RADIO SYSTEM FORT HUACHUCA ARIZONA 85613-7070
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Table of Contents 1 TRANSMISSION LINES...1-1 1.1 Introduction:...1-1 2 WHEN ARE TRANSMISSION LINE EFFECTS SIGNIFICANT?...2-3 2.1 Example 1...2-3 2.2 Example #2...2-3 3 TRANSMISSION LINE MODELS...3-4 3.1 Characteristic Impedance - Z 0...3-6 3.2 The Propagation Constant:...3-7 4 SWR...4-8 4.1 Standing Waves...4-9 4.2 TABLE OF RETURN LOSS VS VOLTAGE STANDING WAVE RATIO... 4-12 Ver. 1.0 v
IMPROVEMENTS (Suggested corrections, or changes to this document, should be submitted through your State Director to the Regional Director. Any Changes will be made by the National documentation team. DISTRIBUTION Distribution is unlimited. VERSIONS The Versions are designated in the footer of each page if no version number is designated the version is considered to be 1.0 or the original issue. Documents may have pages with different versions designated; if so verify the versions on the Change Page at the beginning of each document. REFERENCES The following references apply to this manual: Allied Communications Publications (ACP): ACP - 167 - Glossary of Communications Electronics Terms US Army FM/TM Manuals 1. FM 6-02.52 Tactical Radio Operations 2. TM 5-811-3 - Electrical Design, Lightning and Static Electricity Protection Commercial References 1. Basic Electronics, Components, Devices and Circuits; ISBN 0-02-81860-X, By William P Hand and Gerald Williams Glencoe/McGraw Hill Publishing Co. 2. Standard Handbook for Electrical Engineers - McGraw Hill Publishing Co. CONTRIBUTORS This document has been produced by the Army MARS Technical Writing Team under the authority of Army MARS HQ, Ft Huachuca, AZ. The following individuals are subject matter experts who made significant contributions to this document. William P Hand vi Ver. 1.0
1 TRANSMISSION LINES 1.1 INTRODUCTION: You are no doubt familiar with transmission lines (sometimes abbreviated as tx lines). If you plug any electric device into a wall outlet, the wires that connect the wall outlet to the device is a transmission line. However, transmission lines behave very oddly at high frequencies. In traditional (low-frequency) circuit theory, wires connect devices, but have zero resistance. There is no phase delay across wires; and a short-circuited line always yields zero resistance. For high-frequency transmission lines, things behave quite differently. For instance, short-circuits can actually have an infinite impedance; open-circuits can behave like short-circuited wires. The impedance of some load (Z L =X L +jy L ) can be transformed at the terminals of the transmission line to an impedance much different than Z L. The goal of this tutorial is to understand transmission lines and the reasons for their odd effects. Let's start by examining a diagram as shown in Figure 4-1. A sinusoidal voltage source with associated impedance Z S is attached to a load Z L (which could be an antenna or some other device - in the circuit diagram we simply view it as an impedance called a load). The load and the source are connected via a transmission line of length L: Figure 4-1 Basic Transmission Line In traditional low-frequency circuit analysis, the transmission line would not matter. As a result, the current that flows in the circuit would simply be: However, in the high frequency case, the length L of the transmission line can significantly affect the results as shown in Figure 4-2. To determine the current that flows in the circuit, we would need to know what the input impedance is, Zin, viewed from the terminals of the transmission line: Ver. 1.0 1-1
The resultant current that flows will simply be: Figure 4-2 Length of Transmission Line Since antennas are often high-frequency devices, transmission line effects are often VERY important. That is, if the length L of the transmission line significantly alters Z in, then the current into the antenna from the source will be very small. Consequently, we will not be delivering power properly to the antenna. The same problems hold true in the receiving mode: a transmission line can skew impedance of the receiver sufficiently that almost no power is transferred from the antenna. Hence, a thorough understanding of antenna theory requires an understanding of transmission lines. A great antenna can be hooked up to a great receiver, but if it is done with a length of transmission line at high frequencies, the system will not work properly. Examples of common transmission lines, as shown in Figure 4-3, include the coaxial cable, the microstrip line which commonly feeds patch/microstrip antennas, and the two wire line: Figure 4-3 Examples of Transmission lines 1-2 Ver. 1.0
2 WHEN ARE TRANSMISSION LINE EFFECTS SIGNIFICANT? We know that at low frequencies, transmission lines don't affect power transfer in practical applications we use every day. However, at high frequencies, even short lengths of transmission lines will affect the power transfer. Why is there this difference? The answer is fundamentally important: NOTE It is not the length of the transmission line, or what frequency we operate at that determines if a transmission line will affect a circuit. What matters, is how long the transmission line is, measured in wavelengths at the frequency of interest. If a transmission line has a length greater than about 10% of a wavelength, then it will affect the circuit. 2.1 EXAMPLE 1 When you plug your vacuum cleaner into a wall outlet, the power chord (transmission line) that connects the power to the motor is, for example, 10 meters long. The power is supplied at 60 Hz. Should transmission line effects be taken into account. Answer: The wavelength at 60 Hz is 5,000 km (5 million meters). Hence, the transmission line in this case is 10/5,000,000 = 0.000002 wavelengths (2x10-6 wavelengths) long. As a result, the transmission line is very short relative to a wavelength, and therefore will not have much impact on the device. 2.2 EXAMPLE #2 Suppose a wireless device is transmitting at 4 GHz. Suppose also that a receiver is connect to a patch antenna via a microstrip transmission line that is 2.5 centimeters (cm) long. Should transmission line effects be taken into account? Answer: The wavelength at 4 GHz (4x10 9 Hz) is 7.5 cm. The transmission line is 2.5 cm long. Hence, the transmission line is 0.33 wavelengths long. Since this is a significant fraction of a wavelength (33%), the length of the line must be taken into account in analyzing the receiver/transmission line/antenna system. Ver. 1.0 2-3
3 TRANSMISSION LINE MODELS In this section, we'll form a model for transmission lines and then analyze the equations that govern their behavior. We'll then introduce the key property of characteristic impedance. To understand transmission lines, we'll set up an equivalent circuit to model and analyze them. To start, we'll take the basic symbol for a transmission line of length L and divide it into small segments as shown in Figure 4-4: Figure 4-4 Small Segment of Transmission Line Then we'll model each small segment with a small series resistance, series inductance, shunt conductance, and shunt capacitance as shown in Figure 4-5: Figure 4-5 Theoretic Transmission Line The parameters in the above figure are defined as follows: R' - resistance per unit length for the transmission line (Ohms/meter) L' - inductance per unit length for the tx line (Henries/meter) G' - conductance per unit length for the tx line (Siemans/meter) C' - capacitance per unit length for the tx line (Farads/meter) We will use this model to understand the transmission line. All transmission lines will be represented via the above circuit diagram. For instance, the model for coaxial cables will differ from microstrip transmission lines only by their parameters R', L', G' and C'. To get an idea of the parameters, R' would represent the dc resistance of one meter of the transmission line. The parameter G' represents the isolation between the two conductors of the transmission line. C' represents the capacitance between the two conductors that make up the tx 3-4 Ver. 1.0
line; L' represents the inductance for one meter of the tx line. These parameters can be derived for each transmission line. Assuming the +z-axis is towards the right of the screen, we can establish a relationship between the voltage and current at the left and right sides of the terminals for our small section of transmission line: Figure 4-6 Parameters of a Transmission Line Using ordinary circuit theory, the relationship between the voltage and current on the left and right side of the transmission line segment can be derived: Taking the limit as dz goes to zero, we end up with a set of differential equations that relates the voltage and current on an infinitesimal section of transmission line: These equations are known as the telegrapher s equations. Manipulation of these equations in phasor form, allow for second order wave equations to be made for both V and I: Ver. 1.0 3-5
The solution of the above wave-equations will reveal the complex nature of transmission lines. Using ordinary differential equations theory, the solutions for the above differential equations are given by: The solution is the sum of a forward traveling wave (in the +z direction) and a backward traveling wave (in the -z direction). In the above, V + is the amplitude of the forward traveling voltage wave, V + is the amplitude of the backward traveling voltage wave, I + is the amplitude of the forward traveling current wave, and I + is the amplitude of the backward traveling current wave. 3.1 CHARACTERISTIC IMPEDANCE - Z 0 We're now ready to introduce a fundamental parameter of every transmission line: its characteristic impedance. This is defined as the ratio of the magnitude of the forward traveling voltage wave to the magnitude of the forward traveling current wave: In terms of the transmission line per-length parameters, the characteristic impedance is given by: Z 0 will be extensively used in determining other transmission line parameters. This page will end with special cases of the characteristic Impedance. If R'=G'=0, then the conductors of the transmission line are perfectly conducting (so R'=0) and the dielectric medium that separates the conductors has zero conductivity (so that G'=0). In this case, the line is referred to as a Lossless Line. The characteristic impedance becomes: 3-6 Ver. 1.0
Another type of line of interest is the distortion less line. This type of line may contain loss (so that the voltage dies off somewhat as it propagates down the line), but the magnitude of the attenuation is frequency-independent, and the phase constant varies linearly with frequency. This is desirable; similar to filter theory, this would be considered linear phase - that is, signals that come out of the transmission line might be attenuated, but have the same shape. The criterion for this is: 3.2 THE PROPAGATION CONSTANT: The propagation constant shows up in the solution for the spatial variation of the voltage and current waves along the line (see above). The real part is given by α; this represents the rate of decay of the wave as it travels down the transmission line. The larger α is, the more "lossy" the line is, and the faster the wave decays. If α= 0, then the line is lossless, and the voltage and current waves do not die (shrink) as they travel down the line. The imaginary part of the propagation constant is given by β. This represents the rate at which the waves oscillate as a function of position on the line. In contrast, frequency represents the rate of change of oscillation as function of time. For a lossless line, β can be determined from the speed of propagation along the line (u). In general, β can be determined from: In the above, epsilon is the permittivity of the line, and mu (u) the permeability of the transmission line. Note that lambda (λ) in the above equation is the wavelength within the transmission line: it is not necessarily the wavelength of a wave of frequency f in free space. For an "air line", the speed u is equal to the speed of light c. Ver. 1.0 3-7
4 SWR We are now aware of the characteristic impedance of a transmission line, and that the tx line gives rise to forward and backward traveling voltage and current waves. We will use this information to determine the voltage reflection coefficient, which relates the amplitude of the forward traveling wave to the amplitude of the backward traveling wave. To begin, consider the transmission line with characteristic impedance Z 0 attached to a load with impedance Z L : Figure 4-7 Transmission Impedance At the terminals where the transmission line is connected to the load, the overall voltage must be given by: Recall the expressions for the voltage and current on the line (derived on the previous page): [1] If we plug this into equation [1] (note that z is fixed, because we are evaluating this at a specific point, the end of the transmission line), we obtain: 4-8 Ver. 1.0
The ratio of the reflected voltage amplitude to that of the forward voltage amplitude is the voltage reflection coefficient. This can be solved for via the above equation: The reflection coefficient is usually denoted by the symbol gamma. Note that the magnitude of the reflection coefficient does not depend on the length of the line, only the load impedance and the impedance of the transmission line. Also, note that if Z L =Z 0, then the line is "matched". In this case, there is no mismatch loss and all power is transferred to the load. At this point, you should begin to understand the importance of impedance matching: grossly mismatched impedances will lead to most of the power reflected away from the load. 4.1 STANDING WAVES Note The reflection coefficient can be a real, or a complex number. We'll now look at standing waves on the transmission line. Assuming the propagation constant is purely imaginary (lossless line), we can re-write the voltage and current waves as: If we plot the voltage along the transmission line, we observe a series of peaks and minimums, which repeat a full cycle every half-wavelength. If gamma equals 0.5 (purely real), then the magnitude of the voltage would appear as: Ver. 1.0 4-9
Figure 4-8 Voltage along the Transmission Line Similarly, if gamma equals zero (no mismatch loss) the magnitude of the voltage would appear as: Figure 4-9 Voltage When Gamma equals 0 Finally, if gamma has a magnitude of 1 (this occurs, for instance, if the load is entirely reactive while the transmission line has a Z 0 that is real), then the magnitude of the voltage would appear as: 4-10 Ver. 1.0
Figure 4-10 Voltage If Gamma is 1 One thing that becomes obvious is that the ratio of V max to V min becomes larger as the reflection coefficient increases. That is, if the ratio of V max to V min is one, then there are no standing waves, and the impedance of the line is perfectly matched to the load. If the ratio of V max to V min is infinite, then the magnitude of the reflection coefficient is 1, so that all power is reflected. Hence, this ratio, known as the Voltage Standing Wave Ratio (VSWR) or standing wave ratio is a measure of how well matched a transmission line is to a load. It is defined as: This parameter is commonly quoted in antenna spec sheets. It is typically given over a bandwidth, so that you have an idea of how much power is reflected by the antenna over a frequency range (or alternatively, how much power the antenna radiates). Ver. 1.0 4-11
4.2 TABLE OF RETURN LOSS VS VOLTAGE STANDING WAVE RATIO Return Loss (DB) VSWR Return Loss (DB) VSWR Return Loss (DB) VSWR Return Loss (DB) VSWR Return Loss (DB) VSWR 46.064 1.01 13.842 1.51 9.485 2.01 7.327 2.51 5.999 3.01 40.086 1.02 13.708 1.52 9.428 2.02 7.294 2.52 5.970 3.02 36.607 1.03 13.577 1.53 9.372 2.03 7.262 2.53 5.956 3.03 34.151 1.04 13.449 1.54 9.317 2.04 7.230 2.54 5.935 3.04 32.256 1.05 13.324 1.55 9.262 2.05 7.198 2.55 5.914 3.05 30.714 1.06 13.201 1.56 9.208 2.06 7.167 2.56 5.893 3.06 29.417 1.07 13.081 1.57 9.155 2.07 7.135 2.57 5.872 3.07 28.299 1.08 12.964 1.58 9.103 2.08 7.105 2.58 5.852 3.08 27.318 1.09 12.849 1.59 9.051 2.09 7.074 2.59 5.832 3.09 26.444 1.10 12.736 1.60 8.999 2.10 7.044 2.60 5.811 3.10 25.658 1.11 12.625 1.61 8.949 2.11 7.014 2.61 5.791 3.11 24.943 1.12 12.518 1.62 8.899 2.12 6.984 2.62 5.771 3.12 24.289 1.13 12.412 1.63 8.849 2.13 6.954 2.63 5.751 3.13 23.686 1.14 12.308 1.64 8.800 2.14 6.925 2.64 5.732 3.14 23.127 1.15 12.207 1.65 8.752 2.15 6.896 2.65 5.712 3.15 22.607 1.16 12.107 1.66 8.705 2.16 6.867 2.66 5.693 3.16 22.120 1.17 12.009 1.67 8.657 2.17 6.839 2.67 5.674 3.17 21.664 1.18 11.913 1.68 8.611 2.18 6.811 2.68 5.654 3.18 21.234 1.19 11.818 1.69 8.565 2.19 6.783 2.69 5.635 3.19 20.828 1.20 11.725 1.70 8.519 2.20 6.755 2.70 5.617 3.20 20.443 1.21 11.634 1.71 8.474 2.21 6.728 2.71 5.598 3.21 20.079 1.22 11.545 1.72 8.430 2.22 6.700 2.72 5.579 3.22 19.732 1.23 11.457 1.73 8.386 2.23 6.673 2.73 5.561 3.23 19.401 1.24 11.370 1.74 8.342 2.24 6.646 2.74 5.542 3.24 19.085 1.25 11.285 1.75 8.299 2.25 6.620 2.75 5.524 3.25 18.783 1.26 11.202 1.76 8.257 2.26 6.594 2.76 5.506 3.26 18.493 1.27 11.120 1.77 8.215 2.27 6.567 2.77 5.488 3.27 18.216 1.28 11.039 1.78 8.173 2.28 6.541 2.78 5.470 3.28 17.949 1.29 10.960 1.79 8.138 2.29 6.516 2.79 5.452 3.29 17.690 1.30 10.881 1.80 8.091 2.30 6.490 2.80 5.435 3.30 17.445 1.31 10.804 1.81 8.051 2.31 6.465 2.81 5.417 3.31 17.207 1.32 10.729 1.82 8.011 2.32 6.440 2.82 5.400 3.32 16.977 1.33 10.654 1.83 7.972 2.33 6.415 2.83 5.383 3.33 16.755 1.34 10.581 1.84 7.933 2.34 6.390 2.84 5.365 3.34 16.540 1.35 10.509 1.85 7.894 2.35 6.366 2.85 5.348 3.35 16.332 1.36 10.437 1.86 7.856 2.36 6.341 2.86 5.331 3.36 16.131 1.37 10.367 1.87 7.818 2.37 6.317 2.87 5.315 3.37 15.936 1.38 10.298 1.88 7.781 2.38 6.293 2.88 5.298 3.38 15.747 1.39 10.230 1.89 7.744 2.39 6.270 2.89 5.281 3.39 15.563 1.40 10.163 1.90 7.707 2.40 6.246 2.90 5.265 3.40 15.385 1.41 10.097 1.91 7.671 2.41 6.223 2.91 5.248 3.41 15.211 1.42 10.032 1.92 7.635 2.42 6.200 2.92 5.232 3.42 15.043 1.43 9.968 1.93 7.599 2.43 6.177 2.93 5.216 3.43 14.879 1.44 9.904 1.94 7.564 2.44 6.154 2.94 5.200 3.44 14.719 1.45 9.842 1.95 7.529 2.45 6.131 2.95 5.184 3.45 14.564 1.46 9.780 1.96 7.494 2.46 6.109 2.96 5.168 3.46 14.412 1.47 9.720 1.97 7.460 2.47 6.086 2.97 5.152 3.47 14.264 1.48 9.660 1.98 7.426 2.48 6.064 2.98 5.137 3.48 14.120 1.49 9.601 1.99 7.393 2.49 6.042 2.99 5.121 3.49 13.979 1.50 9.542 2.00 7.360 2.50 6.021 3.00 5.105 3.50 4-12 Ver. 1.0
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