Radio Science

Direct measurement of the radiation resistance of a dipole antenna in the whistler/lower hybrid wave regime



[1] An electrically short dipole antenna radiating high-frequency waves is instrumented in such a way as to directly measure its radiation impedance when immersed in a laboratory plasma. By measuring both current and voltage at the radiating end, transmission line effects are largely eliminated, and extremely accurate results can be obtained. Initial results are presented for wave frequencies ranging from lower-hybrid to whistler, for density varying from 2 × 1018 m−3 down to 2 × 1015 m−3. The initial experiments are performed for ω/ωce ≪ 1. At the highest density, the impedance is approximately real, varying from 400 Ω at 30 MHz down to 100 Ω at 180 MHz, while at the lowest density, the impedance is almost entirely capacitive, at roughly 1.6 pF.

1. Introduction

[2] In a free-space broadcasting arrangement, such as a ham radio or a commercial broadcast station, a high-frequency antenna is located at the end of a transmission line. The antenna and the transmission line connection are carefully constructed so as to present a real impedance, matched to the transmission line impedance. Thus, there is little or no reflected wave energy back into the transmission line, and the bulk of the energy is broadcast as radio waves. In contrast, when an antenna is located in a plasma, it does not radiate into free space but is surrounded by a dielectric medium with tensor components to the dielectric. In addition, the load is generally not constant, and the impedance of the antenna system cannot be matched to its feeder line. In particular, in the laboratory case, the transmitting antenna is located at the end of a probe shaft and is immersed in the plasma; a feeder transmission line is threaded through the shaft, connecting to a driver at the outside and the antenna at the plasma end; the plasma is anisotropic and linearly dispersive. There have been numerous attempts to directly measure the antenna impedance in both space plasma and laboratory plasma settings. Many of the measurements rely on a circuit model of the antenna and transmission line feed even if a network analyzer is used. The model itself can be a large source of error. In this work we have constructed a specialized dipole in which the current and voltage is measured at the probe tips eliminating the necessity for modeling.

[3] Whistler waves [Helliwell, 2006] have long been observed in space [Kintner et al., 1995; Parks, 2003] and were in fact first observed in the early days of telephony [Preece, 1894; Fuchs, 1958]. They were a puzzle until they were explained as a wave in the plasma surrounding the earth, triggered by high-altitude lightning [Shawhan, 1979; Sonwalker and Inan, 1989]. They have since been observed in the magnetospheres of other planets [Gurnett et al., 1979; Xin et al., 2006a, 2006b] and may be excited in the solar corona [Le Quéau and Roux, 1986]. The first laboratory observation of whistlers was by Gallet et al. [1960]. Subsequent to this, highly detailed laboratory observations of whistlers were made by Stenzel [1976a] and Sugai et al. [1978]. Whistlers near the earth have relatively low frequencies (f ∼ 1 kHz) and are audible after detection by a suitable receiver. In space around the earth, the plasma density and magnetic field are low (n = 103 cm−3; B < 10−6 T). The dispersion of the waves is complicated, and the group and phase velocities are, in general, not in the same direction. However, for cold plasma, the frequency is related to the wavelength by

equation image

where ωpe, ωce are the electron plasma and cyclotron frequencies, respectively, and θ is the propagation direction with respect to the local magnetic field. For parallel propagation equation and ωωce and ωωpe, equation (1) reduces to

equation image

where n is the plasma density in units of 1012 cm−3, B is in units of gauss, f in units of MHz, and the wavelength in units of centimeter. For the above mentioned conditions in space above the equator at L = 3 the parallel wavelength is about 30 km at 10 kHz. However, the same physics is readily accessible in laboratory plasmas: much shorter wavelengths can be achieved by scaling to higher plasma densities, magnetic field, and higher-frequency waves. The upper frequency limit that whistler waves exist is the electron cyclotron frequency (fce = 2.8 × 106B Hz, if B is given in gauss).

[4] There has been a great deal of theoretical work on the impedance of antennas in magnetoplasmas. For example the impedance of a short dipole antenna (antenna length shorter than the wavelength of radiated wave) in a magnetoplasma was worked out analytically by Balmain [1964] in 1964, and this formula is still used. This and other works assumed an RF current profile along the dipole, generally triangular with no current on the tips. [Wang and Bell, 1972a] derived an analytic expression for the impedance in a cold, collisionless plasma and then a finite temperature plasma [Wang and Bell, 1972b]. Chevalier et al. [2008] have calculated the impedance of a dipole probe, in the lower hybrid, and whistler regime with no restriction on the current profile across the antenna for magnetospheric conditions. They conclude that for a plasma frequency up to twenty times what is generally found at L = 2 the cold plasma theory of Wang and Bell works well. The radiation pattern of dipole probe in a magnetoplasma was measured by Stenzel [1976b], but the antenna impedance was not evaluated. Recent experiments done at National Research Laboratory (NRL) measured the impedance of spherical antennas [Blackwell et al., 2005] as well as dipoles [Blackwell et al., 2007]. The experiments were done at low densities (n < 1015 m−3) and low magnetic fields (4 ≤ B ≤ 50 G). The experiments were done at frequencies close to the plasma frequency, and data in the whistler wave regime were not emphasized; however, experiment seemed to agree well with theory. Another NRL experiment in the whistler regime [Amatucci et al., 2005] used a loop antenna to launch the waves. In the NRL whistler regime experiment, the data trend follows theory, but there are frequency regimes where there is an order on magnitude difference.

2. Experimental Setup, Radiated Whistler Wave Pattern

[5] These experiments were performed in the Large Plasma Device at UCLA [Gekelman et al., 1991]. The plasma is 18 m long and 60 cm in diameter with axial magnetic fields of up to 2.5 kG. The plasma is formed with a DC discharge between a barium oxide-coated cathode on one end and an anode located 50 cm from the cathode. Plasmas are typically generated from inert gases (He, Ne, Ar, and Xn). The machine has over seventy gate valves through which probes and antennas may be inserted into the plasma. Vacuum interlocks with cryopumps and small turbomolecular pumps are used to pump down the probes and antennas before they are inserted into the machine. Computer-activated probe drives controlled by a custom data acquisition system move the probes in lines or planes orthogonal to the background magnetic field. The plasma is driven by an electrical arc discharge at one end of the machine for (typically) 15 ms and is pulsed at 1 Hz. Once the drive stops, the plasma density decays exponentially by a factor of 10 every 50 to 100 ms. The plasma profile is approximately flat over its diameter of 60 cm during this period, with nonuniformities in density being a few percent or less.

[6] As an initial test, we verified the launch of a whistler wave from the antenna. This measurement was performed 4 ms after the Helium discharge was terminated. At this time the plasma density had dropped to 5.0 × 1017 m−3, the electrons were cold (Te ≈ 0.5 eV), and the plasma very quiescent. The background magnetic field was B0z = 0.6 kG. The corresponding plasma frequency is 2.0 GHz, the electron cyclotron frequency is 1.7 GHz and the lower hybrid frequency about 30 MHz. For this test, whistler waves were launched using a tone burst at 170 MHz, well below ωce/2, and the parallel wavelength for zero degree propagation is 14 cm. The launching antenna length is about 5 cm tip-to-tip, or about 0.03 of the free-space wavelength; this is excited with tone burst having peak voltage between 0.05 and 0.2 V. The receiving probe is an antenna with each of three orthogonal one-turn loops having a diameter of 1 cm. A schematic diagram of the experimental setup is shown in Figure 1, and Figure 2 shows the whistler wave radiation pattern on two planes (25 × 25 cm) located 1 and 2 m from the dipole radiator. The waves were detected using a three-axis orthogonal magnetic loop detector, each loop 1 cm in diameter, sensitive to equation image(i = x, y, z). The radiating dipole had its tips perpendicular to the machine axis. The antenna voltage and detected magnetic field was digitized with a sampling rate of 5 GHz.

Figure 1.

A schematic view of the experiment. The Large Plasma Device plasma is produced by a DC discharge at one end of the chamber and streams down the axial field. The vacuum chamber has more than 400 diagnostic ports. The launching dipole is inserted in one of them about 3 m from the plasma source, and measurements of the radiated wave pattern for this experiment (see Figure 2) are taken “downstream” of the launcher, in this case, at planes located at 1 and 2 m from the antenna.

Figure 2.

The measured magnetic field vectors of whistler waves at two axial distances from the dipole launch antenna. The planes are both transverse to the background (B0Z = 600 G) magnetic field. Each plane is 25 cm on a side, and data were acquired at spatial locations 0.6 cm apart. The wave frequency is 170 MHz.

3. Antenna Impedance

[7] There are at least three schemes for measuring (or inferring) the impedance of an antenna, and in particular the antenna as it is immersed in the plasma. In the most common, as discussed in many textbooks [Maxwell, 1990; Van Valkenburg, 1993], forward and reflected powers are measured at some point in the system. Assuming only a single transmission line is present in the signal path and knowing its exact characteristics as a function of frequency, the antenna impedance can be inferred as follows:

equation image

Here Z0 is the transmission line impedance and Γ(d) is a generalized reflection coefficient, Γ(d) ≡ ΓRe−2γd, where ΓR is the reflection coefficient at the antenna end of the transmission line (ZL is the load impedance).

equation image

Note that in the general case (which is necessary to reproduce the observed behavior, as discussed below), both conductor and dielectric losses in the transmission line must be taken into account. Thus:

equation image

The quantities R, L, G, and C, are the value per unit length of the series resistance, series inductance, shunt conductance (=1/shunt resistance) and shunt capacitance that characterize the transmission line [Jordan, 1968]. Measurement of the load impedance requires determination of 4 unknowns for the transmission line in question. In fact, the situation is worse because R and G can be weak functions of frequency in the regime of interest. Obviously, when there are two transmission lines in series with the load, determination of these parameters is complicated not only by the fact that there are twice as many characteristic quantities to be accounted for (R, L, G, C) but also by the reflections at the junction between the lines.

4. Measurement of Probe Impedance: First Antenna Design

[8] The first dipole probe used to launch and measure the radiation resistance is shown schematically in Figure 3. The transmission line feeding the dipole is composed of the outer conductors of two semirigid coaxial cables, insulated from each other. In a cup at the outside end of the probe, these cables are each wrapped once through a small ferrite toroid, then the outer conductors are connected to ground. The RF source is connected to another winding on the toroid in a balun configuration, i.e., in such a way that a balanced emf is generated on the transmission line in the shaft, launching the propagating mode that is eventually radiated/reflected by the antenna. At the plasma end, a 50 Ω resistor is connected in series with each arm of the dipole antenna, and the coax inner conductors are connected to these resistors to monitor the current through them. The 50 Ω impedance of the coax is in parallel with the resistor if it is appropriately terminated, so when the coax finally penetrates the vacuum boundary at the outside end of the probe, the voltage on the center conductor referred to ground is Iant × 25 Ω for each leg (with the associated phase delay). The probe, except for the dipole arms, is housed inside a shaft which has two components. One is a 1.4 m long 3/8″ diameter stainless steel tube, while the second is a 5 mm diameter ceramic extension supporting the last 25 cm of transmission line as it penetrates the plasma. Thus, there may be two or more sections of different impedance transmission lines in series. For simplicity, these were modeled as a single coax, which may have led to some errors described below.

Figure 3.

Schematic diagram of the dipole launch antenna that measured the current to the legs of the dipole. The antenna current is measured with the two 50 Ω surface mount resistors.

[9] This first measurement of the antenna impedance was performed with the idea that measuring the instantaneous current at the antenna end would be advantageous and would help avoid some of the pitfalls in estimating the large number of unknowns. However, determination of the impedance requires both voltage and current measurements and, consequently, the voltage needed to be computed from the transmission line equations. Measurements of a set of known resistive loads over a range of 0 to 5100 Ω, together with 5 and 10 nF capacitors, were used to back out the transmission line characteristics. The resistors were attached to the antenna leads by twisting their leads onto the antenna conductors. The equivalent circuit is shown in Figure 4.

Figure 4.

Equivalent circuit of transmission line. The balun in the end cap was a small ferrite toroid.

[10] When computing Zend, the best fits to the known impedances included 20 nH of inductance in series with it, together with a stray capacitance of about 2 pF. Note that a total of 50 Ω of resistance also needed to be taken into account, representing the two current-sense impedances, so the load as measured was Zload = [Zknown + 50 + jωLend] ∥ equation image, where the operator Z1Z2 is intended to mean that the two impedances are in parallel. Inferred characteristics of the transmission line are shown in Table 1.

Table 1. Inferred Transmission Line Characteristicsa
Transmission line characteristics  
  • a

    A series of known resistances and capacitances were measured and used to fit these numbers.

   Capacitance per length C058.3pF/m
   Inductance per length L0390nH/m
   Impedance Z081.7Ω
Lumped elements  
   Stray capacitance at antenna Cstray1.75pF
   Stray inductance in series with known impedance Lstray20nH
   Series inductance at driver Lseries2900nH
   Series resistance at driver Rseries200Ω
Transmission lines losses  
   Conduction losses R2.5Ω/m
   Insulator losses G0.0027mhos/m

[11] The above transmission line model performed quite well when used to predict the measured response as a function of frequency for various loads. Even so, this model proved to be inadequate for measurement of the plasma-loaded impedance. As an example, Figure 5a shows the computed versus measured amplitude response when a 5 pF capacitor is attached to the antenna leads. Despite the excellent agreement between the measurement and the predictions around the frequencies of interest (170 MHz in this case) with the various defined load impedances, the open circuit measurement diverges sharply from the prediction.

Figure 5.

Predicted and measured ratio equation image close to launch frequency. The measured current is taken from the current monitoring resistors at the probe tips and the applied voltage is measured at the outside end of the probe shaft. Despite excellent agreement with a series of known impedances, for example, (a) 5.6 pF, (b) the open circuit result is poorly predicted. This transmission line technique was abandoned for the more accurate measurement described in the text.

5. Measurement of Probe Impedance: Second Antenna Design

[12] The uncertainties in the transmission line approach led to the design of a second antenna in which both the current and voltage at the dipole tips were measured. The reconfigured antenna is shown in Figure 6. In this version, the transmission line conductors are actually a pair of semirigid coaxial cables with shields soldered together. The conductors are then insulated with Teflon before they are threaded into the shaft.

Figure 6.

Antenna in which the current and voltage are both measured at the probe tips. Resistors marked a, identical in function to those in Figure 3, are used to measure the antenna current; for example, Ileft = equation image. The voltage is divided by 25 Ω because resistor a (50 Ω) is in parallel with the coaxial cable terminated by 50 Ω. Resistors b and c are used to measure the voltage between the tips and in a voltage divider configuration wherein V1 = equation image. The differential voltage and current signals that exit the vacuum boundary are connected to balun transformers to perform the subtraction, with the result that Zmeas = equation image. The outer conductors of the paired coax cables comprise the transmission line, with an emf being applied as shown in Figure 7.

[13] The transformer coupling for the driver was also simplified for this antenna. Rather than coupling through a ferrite core, which led to a large leakage inductance in the first antenna version, an air core coupling arrangement was constructed by simply threading a loop into the shaft in proximity with the conductors making up the transmission line. This arrangement is illustrated in Figure 7a. The loop is composed of one turn of coax with the shield split at the far end to allow flux to escape. The RF driver is connected to one end of this loop, with the other terminated by 50 Ω. The actual drive signal that couples to the transmission line is made approximately differential by using a ferrite toroid to create a large impedance to common-mode signals, as shown (i.e., to create a balun). Figure 7b is a simple circuit diagram illustrating this coupling.

Figure 7.

(a) A transformer consisting of a one turn coaxial loop, 30 cm long, is placed inside the shaft in such a way that it couples flux to the transmission line. The outer conductor is removed to allow flux from the drive loop to couple to the transmission line. The ferrite core minimizes the common mode voltage and allows for balanced drive of the differential transmission line. A 50 Ω resistor terminates the flux loop coax. Note that the shield of both legs of the primary are connected together, and to ground, at the driver end. (b) Schematic detail of coupling scheme.

[14] The currents I1 and I2 form a differential signal proportional to the antenna current. This was made single ended using a commercial balun connected outside the vacuum. The same is true for voltages V1 and V2. The final current and voltage signals went to a network analyzer (HP E5100A), which recorded their ratio as a function of time. The driver frequency was set within the network analyzer and was constant for a given plasma discharge and changed from shot to shot. For a given value of the magnetic field, the drive frequency was varied in steps of 5 MHz from 25 to 180 MHz. The ratio of the plasma frequency to the cyclotron frequency was varied by changing the background magnetic field. The antenna tips were field aligned. A measurement of the real and imaginary parts of the impedance is shown in Figures 8 and 9. The discharge is terminated at t = 10 ms, and during the plasma afterglow, the density decreases from n = 2 × 1018 m−3 (during the discharge) to approximately 2 × 1015 m−3 changing the ratio of equation image. The real part of the impedance at late times is largest at close to 25 MHz which corresponds to the lower hybrid resonance.

equation image

The approximation is true if fpifci, which is true in this case. Here fpi is the ion plasma frequency. The lower hybrid frequency at 100 ms after the main discharge is approximately 18 MHz; lower hybrid waves occur above the lower hybrid frequency. Debye length in these experiments is quite short, varying from about 10 to 70 μm as the density decays and is small compared to every dimension including the radius of arms of the launching antenna. Computationally, with the small amplitude of the launched wave, we expect to be in the quasilinear regime. The Gendrin angle θ = cos-1 2ω/ωce varies from approximately 89° at the lowest frequency to about 83°. As the density decreases, the plasma becomes more like a vacuum and the antenna load becomes more capacitive. This is reflected in reactive part of Z which is negative and larger at low frequencies. The small periodic oscillations versus frequency at late times most likely are an artifact due to a slight mistermination of the coax. During the discharge and at high frequencies, the real part and reactive parts of the antenna impedance is very low (several Ohms) indicating the antenna is well coupled to the plasma and is a good radiator. Near the lower hybrid frequency, the opposite is true; this makes dipoles poor sources for lower hybrid waves. In fact, specialized antennas have to be constructed to effectively radiate them [Stenzel and Gekelman, 1975; Bernabei et al., 1975; Bellan and Porkolab, 1975]. Data were acquired with the dipole aligned with the background magnetic filed as well as perpendicular to it, and in both cases, the impedance was similar; the behavior is also similar at B = 400 G.

Figure 8.

(a) Real and (b) imaginary parts of the dipole impedance at a magnetic field of 1000 G. As the plasma decays the ratio of equation image changes from 4.54 ≤ equation image ≤ 0.454. The dipole antenna is aligned with the magnetic field in this case.

Figure 9.

(a) Real and (b) imaginary parts of the dipole impedance at a magnetic field of 1000 G. These traces are sections of the contour plots shown in Figure 8, taken at 0, 75, and 150 ms. Corresponding plasma densities are 2 × 1018 m−3, 3 × 1016 m−3, and 3 × 1015 m−3. The oscillation in Figure 9a for the lowest density case is almost certainly an artifact of the measurement, as we expect this impedance to approach zero. In Figure 9b, the dashed line shows the impedance of a 1.6 pF capacitor for comparison.

6. Conclusions

[15] A novel dipole antenna was constructed which allows its radiation impedance to be measured as a function of frequency in the whistler wave regime. The design allows one to circumvent the complications of calculating the transmission line properties of the antenna and modeling the plasma dielectric. Measurements show the antenna impedance is high close to the lower hybrid frequency, but it is an effective radiator at higher frequencies and densities. In an upcoming publication, these results will be compared to a theoretical calculation by Chevalier et al. [2008] in this parameter regime.


[16] We acknowledge support by the ONR under the MURI grant N000140710789. This work was done on the Large Plasma Device at UCLA. The device is part of the Basic Plasma Science Facility funded by the Department of Energy and the National Science Foundation. We would also like to thank Marvin Drandell, Zoltan Lucky, and Mio Nakamoto for their expert technical assistance.