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Antenna Analysis and Optimisation This series is about understanding how antennas work and designing matching circuits for them. This first article will cover antenna fundamentals, reactance, Smith charts and some related topics. Next month, a follow-up article will go into using antenna analysis software. Part 1 by Roderick Wall, VK3YC R adio Amateurs (hams) frequently build and install antennas. We know that the Voltage Standing Wave Ratio (VSWR) needs to be as close as possible to 1:1 for good performance (to achieve efficient power transfer to the antenna). To help us achieve this, we can use an antenna analyser hardware device. Software is also available to aid in this endeavour. Traditional Smith charts can help us understand how to adjust antennas or design matching circuits. We will also demonstrate how some common antenna types work. Amateur radio clubs often have antenna analysers for members to use. They can usually measure complex impedance and indicate the sign of the antenna image. Those that only measure |Z| magnitude and do not show the image are not as useful. When an antenna analyser is connected to a set of antenna terminals, it ‘sees’ your antenna as being made up of what looks like three components: • An inductor (L) with inductive reactance gives a positive imaginary +jW impedance component. • A capacitor (C) with capacitive reactance gives a negative imaginary -jW impedance component. • A resistor (R) that dissipates some of the power as heat and radiates the rest, with a real resistance (W) value. Editor’s note: j is being used as the engineering substitute for the complex value i, where i = √-1. Like resistance, the SI unit for impedance is ohms (W). An antenna can be considered a complex resistive-­ inductive-capacitive (RLC) network – see Fig.1. Antennas can have impedances like: • 50W of real resistance with a capacitive reactance of 25W, written as (50 – j25)W – see Fig.2. • 25W of real resistance with 50W of inductive reactance, written as (25 + j50)W – see Fig.3. • 50W of real resistance with no reactance; the antenna is resonant. Written as (50 + j0)W – see Fig.4. Some antenna analysers use X rather than j to represent reactance. The three antenna states The above are three possible antenna conditions that an antenna analyser will display. Real resistance will always be there, while reactance can either be inductive (+j), capacitive (-j) or absent (j0). At some frequencies, it may have inductive (+j) reactance; at other frequencies, it could have capacitive (-j) reactance. At a specific frequency, both reactances will be equal in magnitude, but opposite in influence and cancel each other out (j0). The antenna is said to be resonant at the specific frequency that the impedance is purely resistive. Real resistance The real resistance is where power is dissipated. The power dissipated in radiation resistance (Rr) is radiated as electromagnetic waves, while the power dissipated in loss resistance (Rl) is lost as heat. For an antenna to be efficient, the radiation resistance should be as high as possible compared to the loss resistance. However, an antenna analyser is only able to measure the total resistance. It is not easy to measure each resistance separately and indicate antenna efficiency, where efficiency = Rr / (Rr + Rl). However, refer to the later section on an experimental method to derive the loss resistance, Rl. Reactance (1) Modelling the complex impedance of an antenna as three passive components in series. (2) An antenna with a -jW complex impedance component has capacitive reactance. (3) An antenna with a +jW complex impedance component has inductive reactance. (4) An antenna with a j0W complex impedance component is purely resistive. Reactance is the imaginary part of electrical impedance. Antennas can have both inductive and capacitive reactance. These reactances are opposing, so the presence of both will mean that they partially (or possibly wholly) cancel. The antenna analyser will only display the net resultant reactance as inductive, capacitive or no reactance. Australia's electronics magazine siliconchip.com.au 40 Silicon Chip When the reactance is j0W, the antenna is said to be resonant. Often, it is not practical for the reactance to reach j0W at the desired frequency; with a low reactance value, close to zero, we may still say that the antenna is resonant. Ideal inductors and capacitors do not dissipate power; they store energy and then return it. However, real inductors and capacitors are not ideal components and will have some resistive loss, even if it is small. The antenna reactance stores power in the antenna’s near field and gives it back. Inductance is measured in henries (H), while capacitance is measured in farads (F). The amount of reactance a capacitor or inductor has depends on the value (in henries or farads) and the frequency. The formulas are Xc = 1 ÷ (2πfC) and Xl = 2πfL. When the frequency increases, a capacitor’s reactance decreases while an inductor’s reactance increases. If frequency decreases, the opposite happens. An antenna analyser can determine the resonant frequency of tuned circuits and antennas. Simulating an antenna Discrete components can be used to make the equivalent circuits shown in Figs.2-4 for a given fixed frequency. However, the radiation resistance (Rr) will be low compared to the loss resistance (Rl), and the circuit’s efficiency as an antenna will be very low. Electromagnetic waves will not travel far; most of the power will be dissipated as heat. These equivalent circuits can be helpful as calibration standards to check the accuracy of antenna analysers at specific frequencies. For example, a 1% 50W non-inductive resistor is equivalent to the Fig.4 circuit. An antenna analyser should give a reading of (50 + j0)W and a VSWR of 1:1 if the resistor leads are short, with no inductive reactance, and the antenna analyser impedance is 50W. How complex impedance determines VSWR Let’s start by using Cartesian coordinates to draw a real resistance line, as shown in Fig.5, with 0W at one end and infinite ohms at the other. The 50W system impedance is in the middle. We can place +j inductive reactance above the real resistance line and -j capacitive reactance below it. siliconchip.com.au Fig.5: by plotting the complex impedance on a Cartesian plane with an X-axis that ranges from zero to infinite ohms, we obtain circles of constant VSWR, with the ideal 1:1 VSWR in the centre. Next, we can draw constant VSWR circles to indicate various VSWR values. Larger circles indicate a higher VSWR than smaller circles. A VSWR of 1:1 is a dot at the 50W point in the middle of the real resistance line, where the VSWR circle has collapsed into a dot. A 100W resistor would have a VSWR of 2:1, as would a 25W resistor. To achieve a VSWR of 1:1, the resistance has to match the system impedance, which is 50W in this case (it may be 75W in some situations). For a VSWR of 1:1, the antenna must also be resonant, ie, having no reactance (j0W). If your antenna has a worse VSWR, it may be possible to adjust it to get closer to 1:1 or use LC matching circuits, which use an inductor (L) and a capacitor (C) to improve the VSWR. T and Pi matching circuits can also be used; how to design matching circuits will be discussed later. The Smith chart The Smith chart was invented by Philip H. Smith (1905-1987). It is a graphical aid or nomogram designed Australia's electronics magazine for engineers specialising in radio-­ frequency engineering to assist in solving problems with transmission lines and matching circuits. The Smith chart shows complex impedance, real resistance and imaginary reactance for a single frequency or a range of frequencies. Fig.6 shows a modern version of the original Smith chart, published in Electronics magazine, January 1939, under the title “Transmission Line Calculator By P. H. Smith, Radio Development Department Bell Telephone Laboratories”. It has been rotated on its side, as that is how we usually see Smith charts these days. The beauty of Smith charts is that they make it easy to plot impedance changes and impedance matching on paper. Software for plotting Smith charts is also available, which can be more accurate than drawing on paper and can usually perform component calculations – so no maths is required! Smith charts are often displayed on modern RF test instruments, including antenna analysers. February 2025  41 Fig.6: a blank Smith chart, which is similar to Fig.5 except that lines of constant reactive impedance are curved rather than straight. The Smith chart is similar to Fig.5, except instead of having straight vertical lines for real resistance and straight reactance lines, the Smith chart has constant circles and constant curves. Smith charts also have constant VSWR circles. The Smith chart shown in Fig.6 is a normalised version, with 1.0 at the centre. That means it can be used with any system impedance (50W, 75W etc). To convert those values to ohms, you multiply by the system impedance. To convert back to a normalised chart, you divide by the system impedance. 42 Silicon Chip Fig.7 is a simplified version of the Smith chart with some highlighted lines and points. The impedance values on it are shown for a 50W system; the red dot in the middle represents (50 + j0)W. Several of the constant-­ resistance circles are highlighted in green and labelled with their values. For example, any point on the constant resistance circle that goes through 50W has a real resistance component of 50W. A mauve dot has been placed on the constant real-­ resistance 25W line (it is at [25 + j50] W). The left-most point on the Smith Australia's electronics magazine chart represents 0W, while the rightmost point represents infinite ohms. The 50W circle is called the unity resistance circle or Z-matching circle. It is the road home to where VSWR is 1:1, in the middle of the Smith chart. The blue lines and values in Fig.7 show the inductive imaginary +j portion of the complex impedance. The mauve dot mentioned earlier is on the +j50W line, hence its value of (25 + j50)W. The equivalent circuit of an antenna that falls at this point was shown in Fig.3. It comprises a series resistor and inductor. siliconchip.com.au Fig.7: this simplified Smith chart show lines of equal inductive reactance (blue), capacitive reactance (red) and resistance (green). The red lines are the imaginary -j (capacitive) part of the complex impedance. The yellow dot is at (50 – j25)W, and its equivalent circuit, with a series capacitor and resistor, was shown in Fig.2. If you analyse an antenna and find it is above or below the horizontal line at the centre, you generally want to try to get it onto that horizontal line, ie, make it resonant. But remember that the VSWR will only be 1:1 at the system impedance, 50W in the example shown in Fig.7. If the real resistance is higher or lower siliconchip.com.au than that, you ideally want to make changes to move it to the 1:1 VSWR point in the middle for maximum efficiency. Wavelength vs frequency In a vacuum, electromagnetic waves travel at the speed of light, c ≈ 3 × 108m/s (light is a type of electromagnetic wave). For most practical purposes, air is sufficiently close to a pure vacuum that you can use the same figure. A signal’s frequency and wavelength can therefore be determined using the following formulas: Australia's electronics magazine ƒ = c ÷ λ or λ = c ÷ ƒ The Greek letter lambda (λ) is the wavelength in metres, while c is the speed of light (in m/s). The wavelength is the distance travelled by one cycle of an electromagnetic wave, while the frequency ( ƒ) is in cycles per second (Hz). Antenna impedance vs wavelength How you adjust an antenna to obtain a VSWR of 1:1 depends on the type of antenna. The following may give some ideas. February 2025  43 ground systems and objects around them will have different complex impedance curves than the one shown. Dipole antennas Fig.8: a plot of resistance (cyan) and reactance (red) versus length as a fraction of the wavelength for a lossless Marconi vertical antenna with a perfect ground plane. Fig.9: how a monopolar (“Marconi”) antenna with a ground plane (left) can be reconfigured into a dipole (right). The Fig.8 plot is for a Marconi vertical antenna with a perfect ground that has no losses. It shows the antenna drive point real resistance (Rd, cyan) and the imaginary reactance (Xd, red) as functions of the length of the driven vertical element. Positive values above the X-axis are for the real resistance (Rd) and inductive reactance (+j), while values beneath it indicate capacitive reactance (-j). The antenna’s resonant points are when reactance is j0, ie, where the red Xd curve crosses the X-axis. The horizontal scale shows the length of the driven vertical element as a multiple of the wavelength (λ). This graph was made using data from the EZNEC antenna simulator. The antenna is resonant at points (b) 1/4 wavelength, (d) 1/2 wavelength, (f) 3/4 wavelength and (g) one wavelength. Points (c) and (e) are if the driven element is cut in length so that the real resistance is 50W, to match a 50W system impedance. Point (b) indicates 44 Silicon Chip that if you were cutting the 1/4-wave driven element for resonance at a fixed frequency and the reactance is capacitive, you need to make the driven element longer to make it resonant. Likewise, if the reactance is inductive, the driven element must be shorter. For a 1/2-wave driven element, if it is inductive, make it longer, or shorter if it is capacitive. Point (d) shows that the real resistance for a 1/2-wavelength resonant antenna is 1889W. You may find that 1889W is too high for an antenna tuner to cope with. In that case, you may want to reduce or increase the driven element length to reduce the real resistance, to allow the tuner to work. If it is a multi-band antenna, you may need to select a length that is not a 1/2-wavelength or multiple of it on the other bands. The Marconi vertical antenna with a perfect ground and no losses used in Fig.8 is a reasonable reference, but practical antennas with different Australia's electronics magazine So, how does the plot in Fig.8 relate to a 1/2-wave dipole antenna? A dipole antenna can be built by combining two Marconi antennas, as shown in Fig.9. Section (a) on the left is a Marconi 1/4wave vertical antenna. Replacing the no-loss ground with a conductive (eg, tin) sheet gives us (b). Adding another 1/4-wave Marconi vertical antenna on the other side of the tin sheet results in the configuration shown in (c). Because the field lines between the top and bottom elements line up and match each other, the tin sheet can be removed, giving (d). The complex impedance for each 1/4wave resonant vertical antenna is (36 + j0)W. Connect them in series doubles the antenna impedance to (72 + j0)W, as in (e). To make the vertical polarised 1/2-wave dipole a horizontal polarised dipole, we just need to lay it horizontally. A 1/2-wave dipole can be broken into two 1/4-wave lengths called elements. The elements are set at 180° from each other and fed in the middle. This type of antenna is called a centre-feed 1/2wave dipole. Its impedance is (72 + j0) W in free space. When placed near the ground, the complex impedance will be different. In Fig.9, we showed how two 1/4wave Marconi antennas can be made into a 1/2-wave centre-feed dipole. The same can be done with two 5/8-wavelength vertical antennas, converting them into a centre-feed Extended Double Zepp antenna. Reflected power & transmission line losses Fig.10 shows an antenna matching circuit at the transmitter end of the transmission line and not at the antenna end (as would be the case when the antenna tuner is part of the transceiver). Because the VSWR at the antenna is not 1:1, power is reflected back from the antenna towards the matching circuit. The matching circuit reflects and adds the reflected power to the forward power from the transmitter. Thus, the forward power supplied to the antenna is now higher than the power supplied to it just from the transmitter. This must happen if the matching siliconchip.com.au Fig.10: when the antenna matching circuit is at the transmitter end, some power is reflected back at the antenna end and circulates to ensure the antenna receives the full transmitter power. circuit at the transmitter end is doing its job, delivering the full transmitter power to the antenna when the antenna VSWR is not 1:1. The reflected power circulates from end to end of the transmission line. Essentially, the matching circuit boosts the power level on the transmission line until all the power from the transmitter reaches the antenna. The ‘extra’ power on the transmission line does not come out of thin air, it is simply recirculating power from the transmitter that has not yet reached the antenna. The level of the reflected circulating power depends on the antenna VSWR. If antenna VSWR is 1:1, there is no circulating power and you do not need a matching circuit. In this example, an SWR meter inserted at either end of the transmission line will indicate a standing wave, while an SWR meter between the transmitter and matching circuit will indicate a VSWR of 1:1. Losses will increase because of the extra distance the reflected power travels. Because the forward power from the matching circuit to the antenna is now higher than the power from the transmitter, forward transmission line losses will also increase. The ARRL Antenna Book presents detailed graphs of increased line losses as a function of VSWR for a variety of real lines. Some transceivers have an Antenna Tuning Unit (ATU) built inside them. This allows the ATU at the transmitter output to be tuned at any frequency within the band. And make the transmitter output VSWR to be close to 1:1 across the band. It also allows the antenna VSWR to be higher than 1:1 as in Fig.10. If the VSWR at the antenna is not 1:1 and there is no matching circuit at the transmitter end, the reflected energy is dissipated in the transmitter’s output resistance. Some transceivers measure siliconchip.com.au VSWR to determine the reflected power and reduce the transmitted power to protect the transceiver if VSWR is high. A low-loss balanced transmission line can reduce losses under dry conditions, as shown in Fig.11; losses can increase in wet conditions with such a configuration. An added advantage is that the antenna tuning unit (ATU) does not need to be mounted in the air at the antenna connection terminals. An experimental method to derive Rl Point (b) in Fig.8 for a 1/4-wavelength resonant vertical antenna shows that the feed point impedance is (36 + j0)W when the ground is perfect and has no loss resistance (Rl). In chapter 7 of the ARRL book “Antenna Physics: An Introduction” by Robert J. Zavrel, Jr W7SX, he describes a non-­ambiguous method to determine Rl for a given location for a 1/4-wavelength vertical antenna. To do this, install a 1/4-wavelength resonant vertical over the ground plane. The base feed point impedance should show as little reactance as possible (ideally, it is a pure real resistance, but some small reactance value is acceptable). We are only interested in the real resistance of the impedance that dissipates power. In this case, the value for radiation resistance (Rr) will be very close to 36W. Therefore, the loss resistance (Rl) will simply be Rl = Rfeed – 36W As you add radials, change their lengths and so on, the feed point complex impedance should change accordingly. These can form the basis of an approximation for general use. However, the multiple differentials involved will vary by antenna location. As you experiment, simply use the following equation to approximately determine efficiency: Efficiency = Rr ÷ Rr + Rl Ground properties can also affect radiation resistance (Rr). For example, if the ground under the vertical/ radial system has very low conductivity and dielectric constant and is gradually made more lossy, it will begin to approach the characteristics of free space. In this case, the antenna radiates electromagnetic waves into the ground, which are lost as heat. Consequently, an accurate differentiation between radiated and absorbed power is nearly impossible. A radiated resistance (Rr) calculation accounts for all radiated power, even that which goes under the surface and can never reach the receiver. Next month That covers all the basic theory we need to analyse and tune antennas. Next month, we are using software to SC make antenna analysis easier. Fig.11: this configuration can reduce losses in the line between the transmitter and antenna when the matching network is at the transmitter end. Australia's electronics magazine February 2025  45