Methodology for Analysis of LMR Antenna Systems

Methodology for Analysis of LMR Antenna Systems Steve Ellingson June 30, 2010 Contents 1 Introduction 2 2 System Model 2 2.1 Receive System Model................................... 2 2.2 Calculation of Transceiver Input-Referred Signal-to-Noise Ratio............ 5 2.3 Transmit System Model.................................. 5 3 Example: UHF-Band Quarter-Wave Monopole 7 Bradley Dept. of Electrical & Computer Engineering, 302 Whittemore Hall, Virginia Polytechnic Institute & State University, Blacksburg VA 24061 USA. E-mail: ellingson@vt.edu 1

1 Introduction Land mobile radio (LMR) refers to wireless communications, typically in the frequency range 30 3000 MHz, in which at least one end of the link is mobile; typically a handheld or vehicle-mounted transceiver (radio). In the most general terms, the mobile station consists of (1) an antenna, (2) a transceiver, and (3) an interface consisting of cable and perhaps other devices connecting the antenna to the transceiver. The primary performance considerations in such a system are different for receiving and transmitting. When receiving, the primary performance consideration is sensitivity; that is, the relationship between signal-to-noise ratio and incident signal power. When transmitting, the primary performance considerations are efficiency; that is, the fraction of power available from the transceiver that is successfully radiated by the antenna; and voltage standing wave ratio (VSWR), which is a measure of extent to which power sourced by the transceiver is reflected back into the transceiver. This report describes a simple methodology for quantifying these considerations in terms of the characteristics of the antenna, the antenna-transceiver interface, the transceiver, and the relevant properties of the radio frequency environment. 2 System Model Different but related models are used for the receive and transmit cases. 2.1 Receive System Model The system model for receive is shown in Figure 1. Figure 1: Receive system model. Note that it is assumed that the input impedance of the transceiver, R R, is equal to Z 0, the real-valued reference impedance used for the analysis. Z 0 will typically be 50 Ω. The antenna is modeled as a voltage source v A in series with its self-impedance Z A. v A is the voltage across the antenna terminals when open-circuited. Given an incident electric field E, v A is given by v A = E l e (1) where l e is the vector effective length (VEL) of the antenna. Assuming E is co-polarized with l e and we are not concerned with the phase of v A, we have the simplified relationship v A = E l e (2) where E and l e are the magnitudes of E and l e respectively. The effective length l e can in principle be determined experimentally as ratio of v A to an applied co-polarized electric field having magnitude 2

E. However, it is usually more convenient to compute l e from the transmit case by applying a test current to the antenna terminals, and then determining the resulting current distribution over the antenna; l e is then the ratio of the integral of the co-polarized component of the current distribution over the antenna, to the test current. is The available power P A that can be delivered by the antenna into a reflectionless matched load P A = v A 2 4R A (3) where R A is the real part of Z A and it is assumed that v A is measured in RMS volts. Thus the available power may be written in terms of E: P A = E 2 l 2 e 4R A. (4) It is important to account for noise from the radio frequency environment, as it is possible in LMR systems for this to be a significant or dominant component of the system noise. If the signal received by the antenna is expressed in terms of an antenna temperature T A, the corresponding available power is given by the well-known expression P A = kt A B. (5) where k is Boltzmann s Constant (1.38 10 23 J/K) and B is the bandwidth of interest. T A is property of the radio frequency environment; depending on frequency, time, and location. However, the minimum possible value can be estimated. In the absence of man-made noise and unusual solar or atmospheric activity, T A in our frequency range of interest is strongly dominated by the Galactic background noise plus a 2.7 K contribution from the cosmic microwave background (CMB). The spectrum of the Galactic background is well-understood, and so the antenna temperature can be approximately lower-bounded by ( ) 2.52 f T A 9000 + 2.7 K (6) 38 MHz where f is frequency [1]. Thus, we see that the lower bound on T A is approximately 9000 K at 38 MHz, dropping to just a few K at 1 GHz. It should be noted that this minimum level actually fluctuates on the order of a few db due to rotation of the Earth with respect to the spatially-varying Galactic brightness temperature distribution. Man-made noise can be modeled in a similar way; in fact, it is useful to know that the man-made radio noise spectrum in the LMR bands also exhibits a power-law spectral dependence with approximately the same exponent, with the primary difference being simply that the leading coefficient is greater [2, 3]. We now consider P R, the power delivered to the transceiver. This is given by { P R = Re v + ( ) R i + } R (7) where v + R is the voltage wave traveling into the transceiver, i+ R is the corresponding current wave, and we continue to use RMS quantities. Since i + R = v+ R /R R, we have P R = v + 2 1 R (8) R R Now consider a situation in which the antenna is terminated directly into a load impedance equal to R R, and let v + A be defined as the voltage wave traveling into the load in this case. Then R R v + A = v A. (9) Z A + R R 3

We use this to manipulate our expression for P R as follows: P R = v + R v + A and using Equation 3, we obtain 2 v + A 2 1 R R = v + R P R = P A v + A v A 2 R R Z A + R R v + 2 R v + A 2 2 1 R R, (10) 4R A R R Z A + R R 2. (11) The factor v + R /v+ 2 A can be conveniently expressed in terms of the s-parameters 1 of the antennatransceiver interface stage (i.e., cable and other devices, as defined in Figure 1). To accomplish this, we take R R to be the real-valued reference impedance Z 0 with respect to which the s-parameters and associated reflection coefficients are calculated. We then have: 2 s 21 P R = P A 4R A Z 0 1 s 11 Γ A Z A + Z 0 2. (12) where Γ A is the voltage reflection coefficient looking into the antenna from a termination of impedance Z 0 ; i.e., Γ A = Z A Z 0 Z A + Z 0 (13) Equation 12 can be used with Equations 4 and 5 to compute the contributions of a deliberate signal and environmental noise, respectively, to the total power delivered to the transceiver. A common situation is that the antenna-transceiver interface will consist simply of a coaxial cable which connects the antenna output to the transceiver input. In this case, the associated s-parameters for a cable having characteristic impedance equal to the standard impedance Z 0 are: s 12 = s 21 = e γl, and (14) s 11 = s 22 = 0 ; where (15) f γ = α + j 2πf f 0 v f c, (16) where l is length, α is the attenuation constant, f 0 is the frequency at which α is determined, and v f is the velocity factor. The s-parameters for more complicated interfaces including those consisting of combinations of cables, filters, amplifiers, couplers, and so on can be determined using well-known theory and/or measurement techniques [4]. At this point the only noise we have accounted for is the noise contributed by the environment. Additional noise sources include thermal noise associated with ohmic loss in the antenna-transceiver interface, and noise contributed by the transceiver itself. 2 The former can be computed using expressions reported by [5], which can be summarized in our case as follows: ( 1 Γ A 2) s 21 2 N T = kt p B 1, (17) 1 s 11 Γ A where T p is the ambient physical temperature of the antenna-transceiver interface. Note that for the commonly-assumed special case Γ A = 0, this reduces to the well-known expression for the outputreferred noise power produced by a lossy stage in an RF system cascade. 1 A recommended reference on s-parameters which is particularly well-suited to the analysis described here is [4]. 2 This is the noise normally quantified by the transceiver s noise figure. 4

Finally, we consider the self-noise of the transceiver. This can be inferred from the specified sensitivity of the transceiver as follows. For transceivers operating in analog FM mode, it is common to specify sensitivity as the power at the input which produces an audio output equal to a given SINAD; for example, the TIA-603 specification is 116 dbm for 12 db SINAD [6]. It can be shown that for the 12.5 khz bandwidth variant of analog FM, 12 db audio SINAD requires an RF signalto-noise ratio of 6.5 db (see Appendix A of [3] for a derivation). Thus, the noise power at the transceiver input must be 116 6.5 = 122.5 dbm (in 12.5 khz) at the threshold of sensitivity in this case. This corresponds to a noise power spectral density of about 4.5 10 20 W/Hz. Similar calculations can be made of other transceivers using either specified or measured sensitivity. 2.2 Calculation of Transceiver Input-Referred Signal-to-Noise Ratio As stated in Section 1, our primary consideration when receiving is signal-to-noise ratio referenced to the input of the transceiver, which can be calculated in terms of the receive system model defined in the previous section as where S N = S N E + N T + N R (18) S is the power of the desired signal which is delivered to the transceiver, which is given by Equations 12 and 4. Summarizing: S = E 2 le 2 s 21 4R A 1 s 11 Γ A 2 4R A Z 0 Z A + Z 0 2. (19) N E is the power associated with environmental noise which is delivered to the transceiver, which is given by Equations 12 and 5. Summarizing: N E = kt A B s 21 1 s 11 Γ A 2 4R A Z 0 Z A + Z 0 2, (20) where T A is approximately lower-bounded by Equation 6 and can be determined for other environments as described in [2] or [3]. N T is the power associated with thermal noise due to ohmic losses in the antenna-transceiver interface and which is delivered to the transceiver, which is given by Equation 17. N R is the input-referred noise power generated by the transceiver itself. As explained above, this is about 4.5 10 20 W/Hz times B for a transceiver operating at the TIA-603-specified sensitivity for B = 12.5 khz analog FM, and can be similarly computed for actual receivers or other modes. 2.3 Transmit System Model The system model for transmit is shown in Figure 2. Note that the model is the same as for receive, except that the antenna is now modeled as a simple load having impedance Z A, the transceiver is modeled as a source of power P T incident on the 2 port of the antenna-transceiver interface from an impedance Z 0, and the desired direction of power flow is obviously from right to left. Note that the meanings of all parameters in the transmit model retain the same meanings as in the receive model, including the definitions of the s-parameters for the antenna-transceiver interface. As stated in Section 1, the two parameters of interest in the transmit case are efficiency and VSWR. The VSWR for the transceiver s input to the system is given by VSWR = 1 + Γ 2 1 Γ 2, (21) 5

Figure 2: Transmit system model. The transceiver is assumed to be providing power P T incident on the antenna-transceiver interface. Note that the definitions of the 1 and 2 ports for s-parameter definitions is unchanged from the receive system model. where Γ 2, the voltage reflection coefficient looking into the input of the antenna-transceiver interface from the output of the transceiver, is given by Γ 2 = s 22 + s 12s 21 Γ A 1 s 11 Γ A. (22) For transmit efficiency, many definitions are possible. Here, we consider two. First, we define the total transmit power efficiency ǫ T as the ratio of the total power successfully radiated by the antenna; to P T, the power that would be delivered by the transceiver into a reflectionless matched load. The derivation is very similar to that used to obtain Equation 12, and the result is: ǫ T = s 12 1 s 11 Γ A 2 ( 1 Γ A 2). (23) Second, we define the antenna pattern-modified transmit power efficiency ǫ TA as ǫ T times the directive gain D H of the antenna in the direction of the horizon; i.e., for zero elevation. Note that ǫ TA = ǫ T for an isotropic antenna, but in practice ǫ TA will be typically be greater than ǫ T by a few db, corresponding to the directivity of an LMR mobile antenna in the direction of the horizon. It is also useful to note that ǫ TA is equal to the effective isotropic radiated power (EIRP) in the direction of the horizon, divided by the power that the transceiver can deliver into a reflectionless matched load. The benefit of computing ǫ TA in addition to ǫ T is that the former is more closely related to link range, and thus is closer to facilitating a bottom-line, apples-to-apples comparison between systems using the same transceivers but different antennas and antenna-transceiver interfaces. However, when there is uncertainty about D H, or in situations where horizon directivity is not necessarily the most relevant concern (e.g., in urban or mountainous environments), then ǫ T is probably a better metric of transmit efficiency. Thus, both ǫ T and ǫ TA are useful. D H can be computed in a number of ways. Since we require the effective length l e for the receive analysis, it is convenient to recycle that result to obtain an expression for D H. To do this, we note 6

that the effective aperture A e of an antenna can be expressed in terms of l e, as follows: A e = η l2 e 4 R A Z A 2 (24) where η is the impedance of free space ( 377 Ω). In terms of D H, we have: A e = D H λ 2 where λ is wavelength. Combining these equations we find: D H = π ηr A Z A 2 4π, (25) ( ) 2 le. (26) λ Of course, D H can also be obtained from an analysis or measurement of the antenna pattern. 3 Example: UHF-Band Quarter-Wave Monopole In this section we present an example of a simple but commonly-considered antenna system to demonstrate the methodology described in the previous sections of this report. The antenna is a monopole of height h = 15.7 cm and radius a = 1 mm, over an infinite perfectly-conducting ground plane, designed to have a quarter-wavelength resonance at 453 MHz. The antenna is connected to the transceiver through RG-58 coaxial cable having length 5.18 m (17 ft). The transceiver is assumed to be operating in an analog FM mode with bandwidth B = 12.5 khz. The remaining details of the system model are as follows: The antenna self-impedance Z A is obtained by first calculating the theoretical impedance of a dipole having length 2h and radius a, using a method described in [7]. Invoking image theory, the actual value of Z A is then taken to be one-half this value. The antenna effective length l e is calculated from the ideal sinusoidal current distribution for the corresponding image dipole of length 2h: I(z) = I(0) sin (2π (h z)/λ) sin (2πh/λ) (27) for h z h, using the method described in Section 2.1. The cable is modeled as described in Equations 14 16, with α = 0.00553 m 1 at f 0 = 10 MHz, and v f = 0.66 (representing RG-58 coaxial cable). The lower-bound environmental noise model described in Equation 6 is used. The assumed ambient physical temperature is T p = 293 K. The assumed input-referred noise power spectral density of the transceiver s receiver is 4.5 10 20 W/Hz, corresponding to the TIA-603-specified sensitivity, as suggested in Section 2.1. The co-polarized incident electric field magnitude E was set to achieve S/N = 12.5 db at the input of the transceiver at 453 MHz, under the conditions of the system model and parameters given above. This is 6 db greater than the minimum 6.5 db determined in Section 2.1 to be required to achieve the TIA-603 sensitivity specification. The resulting value is found to be E = 7.2 µv/m. This value is used as the incident signal strength, independent of frequency. The results are shown in Figures 3 (S/N delivered to receiver), 4 (transmit VSWR) and 5 (transmit efficiency). Also, Table 1 shows a numerical summary of performance. Further observations about this antenna system are summarized as follows: 7

Receive Co-pol. E required to exceed TIA-603 sensitivity by 6 db Bandwidth over which S/N is greater than 3 db below peak Transmit Bandwidth over which transmit VSWR < 2:1 Max. radiated power relative to power available from transmitter (ǫ T ) Max. horizon EIRP relative to power available from transmitter (ǫ TA ) Bandwidth over which ǫ TA is within 3 db of peak 7.2 µv/m RMS 161 MHz 95 MHz 1.9 db +3.3 db 65 MHz Table 1: Quarter-wavelength monopole example: Summary. Note that tuning range for acceptable sensitivity is greater than the tuning range for reasonable transmit VSWR, and much greater than the tuning range over which ǫ TA is close to its nominal value. Said simply, the useful bandwidth for receiving is much greater than the useful bandwidth for transmitting. Note in Figure 5 that the difference between the two curves is equal to D H, the directive gain in the direction of the horizon. The peak difference is about 5 db, corresponding to the nominal D H of an ideal quarter-wavelength monopole. As the frequency is varied from this nominal value, both D H and the transmit efficiency (ǫ T ) are reduced. Figure 6 shows the desired signal power and the various noise components contributing to the S/N result shown in Figure 3, individually. Note that the total noise is strongly dominated by the transceiver s self-noise, and that the environmental noise and antenna system thermal noise are relatively weak. (The dip in the thermal noise around resonance is because under this condition thermal noise from the cable can efficiently be radiated out the antenna as well as into the receiver.) Said differently, the system S/N could be improved dramatically by improving the sensitivity of the transceiver itself. In fairness, it should be noted that there are significant technical barriers to doing this in practice; large among them is the tradeoff between receiver sensitivity and intermodulation performance. 8

15 10 5 S/N [db] 0-5 -10 100 200 300 400 500 600 700 800 900 Frequency [MHz] Figure 3: Quarter-wavelength monopole example: S/N delivered to receiver. 9

6 5 4 VSWR 3 2 1 100 200 300 400 500 600 700 800 900 Frequency [MHz] Figure 4: Quarter-wavelength monopole example: Transmit VSWR. 10

10 ε T [db] ε TA [dbi] 5 [db] 0-5 -10 100 200 300 400 500 600 700 800 900 Frequency [MHz] Figure 5: Quarter-wavelength monopole example: Transmit efficiency ǫ T and antenna patternmodified transmit efficiency ǫ TA. 11

-110-120 S N E N T N R -130 [dbm] -140-150 -160-170 100 200 300 400 500 600 700 800 900 Frequency [MHz] Figure 6: Quarter-wavelength monopole example: Signal power and the various noise components contributing to the S/N result shown in Figure 3, shown individually. 12

References [1] S. W. Ellingson, Antennas for the Next Generation of Low Frequency Radio Telescopes, IEEE Trans. Ant. & Prop., Vol. 53, No. 8, August 2005, pp. 2480 9. [2] International Telecommunication Union, Radio Noise, P.372-8, 2003. [3] S. M. Shajedul Hasan, New Concepts in Front End Design for Receivers with Large, Multiband Tuning Ranges, Ph.D. Dissertation, Virginia Polytechnic Inst. & State U., 2009. [4] Agilent Technologies, S-Parameter Design, Application Note AN 154, 2006. [5] C. K. S. Miller, W. C. Daywitt, & M. G. Arthur, Noise Standards, Measurements, and Receiver Noise Definitions, Proc. IEEE, Vol. 55, No. 6, June 1967, pp. 865 77. [6] Telecommunications Industry Association, TIA Standard: Land Mobile FM or PM Communications Equipment Measurement and Performance Standards, TIA-603-C, December 2004. [7] S. J. Orfandis, Electromagnetic Waves and Antennas, [online] www.ece.rutgers.edu/ orfanidi/ewa. Section 22.3 ( Self and Mutual Impedance ), including associated Matlab source code, of Chapter 22 ( Coupled Antennas ), Apr 28, 2010. 13