Antenna Pattern Measurement: Theory and Equations

Michael D. Foegelle

The second installment on antenna pattern measurement describes the calculations involved in determining properties such as TRP, EIRP, directivity, and efficiency.

Illustration by TAISHA PAYTON

This is the second article in a two-part series on antenna pattern measurement. This installment presents the theory and equations governing a variety of antenna properties and includes a complete description of a site calibration for pattern-measurement testing.

Range Calibration

With a two-axis positioner setup, it is quite straightforward to perform general pattern measurements and determine a variety of relative data such as 3-dB beam width, front-to-back ratio, and directivity. However, before accurate measurements of values such as total radiated power (TRP), effective isotropic radiated power (EIRP), or antenna gain can be made, it is necessary to perform a reference calibration to correct for the various factors affecting these tests. The factors include components such as range-length loss, gain of the receive antenna, cable losses, and so forth.

Normally, this calibration is done using a reference antenna (typically either a dipole or standard-gain horn) with known gain characteristics. The reference antenna is mounted at the center of the positioner as the antenna under test (AUT) and adjusted to be at bore-sight level with the receive antenna. The reference calibration is repeated for each polarization of the receive antenna, with the reference antenna polarized parallel to the corresponding receive element. Figure 1 shows a typical range-calibration setup and calls out various components that are included in the measurement.

Typically, a signal generator or the output of a network analyzer is connected to the reference antenna by one or more cables, possibly through a power amplifier. The receive antenna is connected to a receiver or the input of a network analyzer through one or more additional cables, possibly through a preamplifier.

The power at the transmit antenna input port, P_t, is given in terms of the signal generator output, P_SG, by

(1)

where g_a is the gain of the amplifier, and cl₁ and cl₂ are the cable losses of the corresponding transmit cables.

The power at the receiver, P_RX, is given in terms of the power at the receive antenna output port, P_r, by

(2)

where g_pa is the gain of the preamplifier, and cl₃ and cl₄ are the cable losses of the corresponding receive cables.

If any of the components are missing, the corresponding gain or loss for that variable in the equation should be 1. In terms of decibels, these formulas become

P_t = P_SG + g_a – cl₁ – cl₂(3)

and

P_RX = P_r + g_pa – cl₃ – cl₄, (4)

and the gain or loss of missing components would be 0 dB.

The Friis transmission equation governs the interaction between two antennas in the far field:

(5)

where P_r is the power measured at the receive antenna output port; P_t is the power measured at the transmit antenna input port; G_t is the gain of the transmit antenna; G_r is the gain of the receive antenna; l is the wavelength; and r is the separation between the two antennas (the range length).^{1, 2}

The exact definition of P_t is often a source of some confusion and is somewhat dependent on what terms are included in the definition of gain. If the antenna is perfectly matched to the source cable, then all power applied to the antenna is radiated (or absorbed by losses in the antenna). However, in the more common case of a mismatch between the source impedance and the antenna impedance, a portion of the energy is reflected back to the source so that the net power transmitted is the difference between the applied forward incident power and the power reflected back to the source:

P_net = P_inc – P_refl. (6)

If a theoretical gain value is used in the Friis equation, then P_net should be used for P_t because the theoretical formula typically would not be able to account for the voltage standing wave ratio (VSWR) caused by the impedance mismatch. This requires either using a bidirectional coupler and power meter configuration at the transmit antenna to determine P_net directly, or measuring the VSWR of the antenna and performing additional calculations to predict the net power from the forward power.

If measured gain values are used, it is important to know how those gain values were determined and whether they already contain a contribution from the VSWR. Because any calibration technique is inherently governed by this same formula, the resulting gain will be different depending on whether VSWR effects have been accounted for separately. If not, the gain will be changed simply by the ratio of net power to forward power:

(7)

An impedance mismatch is just as likely to happen with the receive antenna, leading to similar measurement issues, but it would not be as easy to observe directly because, in this case, the reflected energy would be reradiated. There is no good way to measure the forward and reflected received energy. However, the VSWR of the receive antenna can be used to determine this effect. Fortunately, the gain of the receive antenna does not need to be known exactly (other than to double-check the calibration result against theoretical predictions) because it will be measured as part of the range calibration process.

As indicated in Figure 1, other factors are typically involved in the measurement, unless power meters and directional couplers are used right at the antennas to measure the net transmitted and received power. These factors include cable losses and the gain of any power amplifiers or preamplifiers.

Figure 1. Some typical components of a range-calibration setup.

To minimize the uncertainty of resulting measurements, it is usually desirable to perform the range calibration with all cables in place and use the same configuration for both calibration and pattern measurements. Should any component be changed or damaged, the entire calibration must be redone. It is possible to perform individual calibrations on various system components, but each additional measurement increases the total measurement uncertainty involved. Therefore, it is preferable to calibrate the system as a whole whenever possible.

TRP

To determine exactly how to apply the range calibration, it is important to make a comparison between the desired measurement quantities and what will actually be measured by the test system. The primary quantity of interest is the TRP, which can be obtained by integrating the time-averaged power density of the radiated signal across the entire spherical surface enclosing the AUT.

The time-averaged power density of a radiating signal is given by the real part of the Poynting vector:

(8)

where r is the time-averaged power density, E is the peak electric field strength, H is the peak magnetic field strength, E_rms is the root-mean-square (rms) electric field strength, and h is the impedance of free space (120p).^{3, 4}

The factor of 1/2 in the definition of the power density originates from the time averaging of the power across a complete period. Although most reference materials and numerical analysis tools refer to wave magnitudes by their peak values,

(9)

most measurement instrumentation reports rms values,

(10)

Therefore, when determining the power density from the rms electric field, the factor of 1/2 has already been accounted for. The difference between rms and peak field values can result in an immediate 3-dB error in reported measurement results if it is not treated correctly.

The TRP is given by integrating the power density across the surface of the reference sphere:

(11)

where TRP is the total radiated power, r is the time-averaged power density, r is the radius of the sphere (the range length), q is the elevation angle, and f is the azimuth angle.

The electric field generated at a point in the far field as a function of the transmitted power is given by

(12)

where E is the electric field generated at the distance r from the transmit antenna, P_t is the power measured at the transmit antenna input port, G_t (q, f) is the angle-dependent gain of the transmit antenna, and r is the distance from the transmit antenna to the test point (the range length).¹

Combining the equation for the power density with that of the electric field gives

(13)

Combining this result with the equation for TRP gives

(14)

Received Power

Unfortunately, the receiver used to perform the test cannot measure power density directly; instead, it measures received power (again, neglecting cable losses, etc.). A related quantity to the TRP would then be the total received power, given by integrating the received power across all of the measurement points of the AUT. The total power received is

(15)

where TP_r is the total power received and P_r is the power measured at the receive antenna output port.

The received power is given by the Friis transmission equation described earlier, so in terms of the transmit power and the angle-dependent gain, the equation becomes

(16)

Because the desired value is TRP, the required correction factor is simply the ratio of TRP to the total power received:

(17)

which, when simplified, becomes

(18)

This constant makes sense because the factor is related to the range length and the gain of the receive antenna, both of which are exactly what needs to be calibrated out of the system. Going back to the Friis equation, the reference measurement performed with the reference antenna results in a site reference constant given by

(19)

where C is the ratio of received power to transmitted power. Substituting this into the previous equation gives a correction factor of

(20)

The required site-calibration constant is now represented in terms of the gain of the reference antenna and a single-path loss measurement for each polarization. The ratio C could contain contributions from other terms, such as cable loss and so forth, as long as those contributions are present in both the reference calibration and the pattern measurements.

Accounting for VSWR

The treatment of the transmit antenna VSWR is an important part of both the range calibration and the measurement of various antenna properties. In general, VSWR is a measurement of the mismatch between two transmission lines. It provides a measurement of the amount of signal being reflected back from the mismatch, which is directly related to the amount of energy that is transmitted.

For many antennas, the VSWR represents the largest component of the antenna efficiency (the rest results from ohmic losses in the antenna itself). To determine the contribution from VSWR, it is necessary to calculate the ratio of the net power to the forward power.

VSWR is defined as the ratio of maximum to minimum voltage on the transmission line and is given by

(21)

where V_max is the maximum voltage on the transmission line (feed cable), V_min is the minimum voltage on the transmission line, V_inc is the magnitude of the incident wave, and V_refl is the magnitude of the reflected wave.⁵

The reflection coefficient r (not to be confused with the power density described previously) is the ratio of reflected to incident waves and is given by

(22)

or, in terms of impedance,

(23)

where V₊ is the incident wave (magnitude and phase), V_– is the reflected wave (magnitude and phase), Z_o is the characteristic impedance of the transmission line (magnitude and phase), and Z_L is the impedance of the load line (magnitude and phase).

If the load impedance is equal to the characteristic impedance of the transmission line, the reflection coefficient would be zero because there is no mismatch in this case. In addition, unlike VSWR, the reflection coefficient has both magnitude and phase. The magnitude of the reflection coefficient is then

(24)

The transmission coefficient t is defined as the ratio of transmitted to incident waves and is given by

(25)

or, in terms of impedance,

(26)

where V_L is the wave transmitted through the mismatch to the load side (magnitude and phase).

By definition, t – r = 1. However, the transmission coefficient is not very useful for determining the net transmitted power from the VSWR because it also requires some knowledge of the impedance of the load. Although the necessary information could be determined from the reflection coefficient, it is considerably easier to determine the ratio of the reflected power to the incident power, and then use that to determine the net transmitted power:

(27)

so that

P_net = P_inc – P_refl
	= P_inc – P_inc• \|r\|²	(28)
	= P_inc (1 – \|r\|²).

This results in a VSWR correction factor given in dB by

C_VSWR = 10log₁₀ (1 – |r|²)

(29)

The VSWR component covered here is not the only antenna VSWR term related to antenna measurements. If an antenna is not in a free-space environment, energy reflected back from other objects will affect the VSWR measurement. However, this term is a measure of the antenna's interaction with its environment rather than a measurement of an inherent property of the antenna.

Care should be taken when measuring VSWR to be used for range calibrations to ensure that the measurement represents a true free-space VSWR. A simple way to do this is to alter the orientation and location of the reference antenna when measuring VSWR. If no variation is seen in the resulting VSWR measurements, then the environment probably does not have a significant effect.

Gain, Directivity, Efficiency, and EIRP

Once the range has been calibrated, a number of antenna properties can be determined from the pattern measurement. The first property of interest is EIRP. EIRP is the power required for a theoretical isotropic radiator (one that radiates the same power in all directions) to generate the same field level in all directions as the maximum field seen from the AUT. Starting from the definition of TRP, EIRP is given by

(30)

where r_max is the maximum time-averaged power density found over the surface of the measurement sphere.

Assuming that the maximum power density can be defined using the bore-sight gain of the AUT,

(31)

Combine this with the equation for EIRP to get

(32)

EIRP is simply the transmitted power increased by the AUT gain, which brings some clarity to the definition of gain. Gain (over isotropic) is defined as the increase in received signal from the AUT over that which would be received from an isotropic radiator with the same source power. Therefore, to create an isotropic radiator that generates the same field level as the maximum seen from the AUT, the source power must be increased by the gain.

Rearranging the equation for EIRP gives the definition of gain:

(33)

where P_t is often referred to as the antenna-port input power (APIP). Again, there is the question of whether this term should refer to the incident power or whether it should refer to the net power. This decision affects the calculation of the efficiency of the antenna.

The ratio of EIRP to TRP is defined as the directivity of the antenna:

(34)

where D_t(q, f) is the relative magnitude of the AUT pattern at any angle with respect to the maximum.² For an isotropic radiator, D_t(q, f) = 1, so that D = 1. For any real antenna, D_t(q, f)< 1 for much of the surface, resulting in D > 1. Directivity is the only term related to the antenna gain, which is solely a relative term. Range calibration does not show up in this equation.

As with the TRP measurement, the measurement system is only capable of measuring received power, so instead of EIRP, the corresponding value calculated would be the effective isotropic received power:

(35)

where P_{r max} is the maximum received power from the pattern measurement.

Assuming again that the maximum received power is the bore-sight transmission response, the same site-reference constant, C, can be used:

(36)

It is apparent that the same range calibration holds in this case as well. Therefore, the directivity can also be represented directly in terms of measured quantities as

(37)

The efficiency of the AUT is defined as the ratio of TRP to APIP. The choice of defining APIP as incident power or net power determines whether VSWR is part of the efficiency term. If the net power definition is used, the efficiency only represents the ohmic losses of the antenna and not the mismatch effects:

(38)

Comparing this to the definition of gain and directivity makes it clear that gain is given by the product of directivity and efficiency:

(39)

If the AUT has no losses or mismatch, the directivity and gain should be equivalent.

Other Antenna Properties

There are plenty of other properties that can be determined from an antenna pattern, such as front-to-back ratio, average radiated power, average gain, and beam widths. The calculation of most of these properties is straightforward, usually using simple formulas. The most important part of many of the calculations is the data search algorithms used to find values like the maximum point, minimum point, and –3-dB points.

Some of these antenna properties have little or no meaning for some antennas. In addition, the orientation of the AUT can affect the result of an automated calculation without additional input from the user to indicate the desired alignment information. For example, the meaning of E- and H-plane beam widths is commonly understood. However, if an AUT is randomly oriented for the pattern test, or has an unusual pattern, there is no simple way to determine automatically what constitutes each plane.

CTIA Requirements

The Cellular Telecommunications and Internet Association (CTIA) has developed some very specific antenna-property requirements in addition to the EIRP and TRP measurements.⁴ One of these is the near-horizon partial radiated power, which is used to determine the power radiated in a small band (typically ±22.5° or ±45°) along the azimuth axis. This requirement is intended to determine how a cellular phone will interact with the network of cellular base stations arranged around it along the horizon during normal operation. The orientation of the AUT will have a great effect on this result, so the standard calls out precise positioning requirements for the phone.

Because a cellular phone has both transmit and receive modes, the CTIA standard also contains receive property requirements, including total isotropic sensitivity (TIS) and near-horizon partial isotropic sensitivity (NHPIS), in addition to the radiated pattern requirements. These values are calculated from the received power pattern instead of the transmitted power pattern. Because the CTIA standard is still in draft form and subject to change, the details of these calculations are not covered in this article.

Conclusion

The techniques for complex pattern measurement are rather straightforward, but the calculations involved in determining certain antenna properties can be much more complicated. Nonetheless, with appropriate care and understanding of the associated quantities, it is not difficult to obtain excellent results.

The information provided in this article can help even the novice RF or EMC engineer to determine a variety of antenna properties.

References

1. "Antenna Calculations," ETS-Lindgren Antenna Catalog (Cedar Park, TX: ETS-Lindgren, 2002), 71.

2. CA Balanis, Antenna Theory, Analysis and Design (New York: Harper & Row, 1982), 29, 65.

3. JD Jackson, Classical Electrodynamics, 2nd ed. (New York: Wiley, 1975), 347.

4. "Method of Measurement for Radiated RF Power and Receiver Performance, Draft Revision 1.2" (Washington, DC: Cellular Telecommunications and Internet Association, 2001).

5. BC Wadell, Transmission Line Design Handbook (Boston: Artech House, 1991), 497.

Michael D. Foegelle, PhD, is senior principal design engineer at ETS-Lindgren (Cedar Park, TX). He can be reached at 512-531-6444 or michael.foegelle@emctest.com.