Victor Taracila^{1}, Louis Vannatta^{1}, Robert Rainey^{1}, Miguel Navarro^{1}, Aleksey Zemskov^{1}, and Fraser Robb^{1}

In MR coils or MR table there are multiple cables. Their presence is not desirable because they distort the B1 field; therefore multiple RF traps are distributed along the cable length to minimize induced cable currents. Tuning and performance of the MR cable traps’, also called cable baluns, on the cable are always difficult to predict, because the cable shape and position may vary. A coaxial half-wave cavity resonator is shown to be a precise tool to test cable balun assembly. It facilitates precise measurement of a balun’s coupling to the cable, its loss, and coupling to neighboring baluns.

**Purpose**

**Theory**

A coaxial half-wave resonator is comprised of a central conductor (cylindrical rod or cable) and concentric circular conductive shield or cavity. The resonant central conductor is cut to the half wavelength. In this application, we utilize the first halfwave resonant mode, or optionally a quarterwave mode.

A resonator could be described by a series resistance $$$R_0$$$, inductance $$$L_0$$$ and capacitance $$$C_0$$$, which leads to other characteristics as resonant frequency $$$f_0=1/(2 \pi \sqrt{L_0 C_0})$$$ and quality factor $$$Q_0=2 \pi f_0L_0/R_0$$$ [1]. A resonator’s series impedance is equal to $$Z_0(f)=R_0 \left( 1+j \frac{f}{f_0} Q_0 \left( 1-\frac{f_0^2}{f^2}\right)\right).\hspace{70pt} (1)$$

To assess the resonator’s characteristics, we utilize scattering parameters and the S21 transmission measurement. Two electric or magnetic probes provide (electric monopoles (Figures 1 and 3) or magnetic dipoles (loops)) ports into the cavity resonator. The S21 transmission is found to be inversely proportional to the impedance (1) $$$S_{21}(f)\propto Z_0^{-1}(f)$$$. When the S21 is expressed in dB, the proportionally becomes an addition with a slight linear frequency dependency with the intercept $$$k_0$$$ and slope $$$k_1$$$

$$S_{21,\text{dB}}(f)=k_0+k_1 f-20 \lg \Bigl\lvert 1+j \frac{f}{f_0} Q_0 \left( 1-\frac{f_0^2}{f^2}\right)\Bigr\rvert.\hspace{70pt} (2)$$

In order to describe the balun’s coupling to the cable, we consider a balun series model with a resistance $$$R_1$$$ , self-inductance $$$L_1$$$, and capacitance $$$C_1$$$, interacting with the cable modeled as a mutual inductance, $$$M_1$$$. We may define some derivations as balun’s resonant frequency $$$f_1=1/(2 \pi \sqrt{L_1 C_1})$$$, quality factor $$$Q_1=2 \pi f_1L_1/R_1$$$ , and magnetic coupling coefficient between the balun and the cable $$$ k_{M_{1}} = M_1/ \sqrt{L_0 L_1}$$$ . Utilizing method [2] the cable impedance is modified by the addition of the balun, therefore following equation (2), the observed S21 transmission of the cable with a balun

$$\tilde{S}_{21,\text{dB}}(f)=k_0+k_1 f-20 \lg \Bigl\lvert 1+j \frac{f}{f_0} Q_0 \left( 1-\frac{f_0^2}{f^2}\right)+\frac{\frac{f^2}{f_0 f_1} k_{M_{1}}^2 Q_0 Q_1}{1+j \frac{f}{f_1} Q_1 \left( 1-\frac{f_1^2}{f^2}\right)}\Bigr\rvert.\hspace{70pt} (3)$$

Equation (3) can be generalized for multiple baluns in various positions on the cable.

**Apparatus**

**Conclusions**

1. D.M. Pozar. Microwave Engineering 3rd Edition, ch. 6.

2. L.D. Landau, L.P Pitaevskii, E.M. Lifshits, Electrodynamics of Continuous Media, 2nd Edition: Volume 8, § 62

Figure 1. Full scale model of half-wavelength cavity resonator: diameter 30 cm, overall length 150 cm.
Two monopole antennas may be seen on the top interior. The resonator rod or
cable is positioned on the inverted “W” shaped dielectric support.

Figure 2. Typical
S21 measurement of a half-wave resonator without (“red” and “green”) and with
balun (“black” and “magenta”). The equations (2) and (3) fit experimental data
with high accuracy.

Figure 3. HFSS simulations at 128MHz of the: top – empty Resonator,
middle-one balun, bottom – ten baluns. In the bottom left and right corners one
can see the excitation monopoles.