Those interested in understanding some of the RF concepts behind common interconnections in the MRI hardware setup.

This talk will cover the following topics:

- Distributed-circuit analysis and wave equation of a terminated transmission line (TL)

- Type of TLs commonly used in MRI

- Scattering parameters and the Schmitt chart, TL impedance and reflection coefficient

- TL applications in MRI: Impedance matching, balanced to unbalanced connection, power splitting and monitoring, phase shifting, preamplifier decoupling, and coil design.

- TL practical considerations and TL lumped element equivalents

- Non-conventional power transfer in MRI

During both the excitation and acquisition phases of the MRI experiment, RF signals need to travel efficiently between the scanner system and the MR coils. In the conventional MRI setup, the coil and the scanner are interconnected through coaxial cables, which is a type of shielded transmission line that allows power to be transferred from one point to the other, ideally without radiation losses. As an example, Figure 1 shows a local transmit (TX) receive (RX) setup in the MRI scanner. Power is transferred from the power amplifier (RFPA) located in the equipment room to the coil plug located in the scanner bore through a long coaxial connection. There are multiple RF interconnections between the system and the coil, built from transmission line segments (with lengths of a fraction of the signal wavelength) or their lumped element equivalent implementation (LC network). The impedance of interconnecting devices and coil (coupled to a nominal load) are matched to the characteristic impedance of the coaxial cable (MRI standard 50 Ω) for maximum power transfer. An RF switch (Transmit/Receive switch) is used to isolate the low-power receiver’s electronics, mainly the receive amplifier (or preamplifier), from the high-power RF transmit signal. An RF coupler can be inserted in the transmit line to have information about the actual power delivered to the coil (for example under a change in load impedance). A balun (balanced to unbalanced) is a device used in coaxial connections to suppress RF currents on the cable’s shield, which can reduce system performance and compromise safety.

A differential segment of a lossless transmission line can be modeled as a series inductance connected to a shunt capacitance as shown in Figure 2. L and C are the distributed (in units per length) inductance and capacitance respectively. The TL (wave) equations (shown in the figure) can be formulated from this differential length model applying Kirchhoff’s current and voltage laws (KCL and KVL).

A lossless TL can transport electromagnetic energy from one point to the other in the form of a pure transverse electromagnetic (TEM) wave. This means the electrical and magnetic field components are both zero in the direction along the line (z). The termination at both ends of the line (source and load impedance) will determine how energy is absorbed and reflected along the lines (Figure 3).

From the wave solution, the impedance at the input of the line can be calculated as:

$$ Z_{i n}=Z_{0}\frac{Z_{L}+jZ_{0}\tan\frac{2\pi}{\lambda}l}{Z_{0}+jZ_{L}\tan\frac{2\pi}{\lambda}l}, Z_{0}=\sqrt{\frac{L}{C}}, \beta=\frac{2\pi}{\lambda}=\frac{2\pi f}{v_{p}}, $$

where Z₀ is the characteristic impedance of the line, Z_L the load impedance, β the propagation constant and λ the signal wavelength. When the source and the load are matched to the characteristic impedance of the line (R_S =R_L=Z0 ), all transmitted power is absorbed by the load. In contrast, when one or both impedances are different than Z₀, power is reflected back and forth along the line (standing wave pattern) and less power is delivered to the load. Normally, in MRI we use coaxial connections with Z₀ = 50 Ω, therefore to ensure maximum power transfer to the loaded coil we perform 50 Ω matching almost everywhere along the transmit and receive chains. In practice, coaxial cables are lossy TLs (they have an attenuation factor) and this cannot be ignored as frequency and/or length of the connection increase. This is the case for the long high power coaxial connection between the RFPA and the coil plug. For example in a 7T system, even with impedance matching, roughly 50 % of the transmit power reaches the coil.

For a quarter-wavelength transmission line,

$$Z_{i n}(n\frac{\lambda}{4})=Z_{0}\frac{Z_{L}+jZ_{0}\tan n\frac{\pi}{2}}{Z_{0}+jZ_{L}\tan n\frac{\pi}{2}}=\frac{Z_0^2}{Z_{L}}$$

The transmission line behaves as an open circuit when Z_L=0 and as a short circuit when Z_L=∞. This impedance transformation is used in many of the interconnecting devices and an example is presented in the next section. On the other hand for a half-wavelength transmission line there is not impedance transformation:

$$Z_{i n}(n\frac{\lambda}{2})=Z_{0}\frac{Z_{L}+jZ_{0}\tan n \pi}{Z_{0}+jZ_{L}\tan n \pi}=Z_{L}$$

We will discuss these and other interesting cases of impedance transformation for MRI applications.

As an introduction to this presentation, and because of their importance to ensure RF safety and system performance, I will discuss briefly two of the aforementioned interconnecting subsystems: the balun and the directional coupler.

A balun is a device that transforms a balanced signal to an unbalanced one. The quarter wavelength balun (or bazooka balun) is a nice example to show the practicality of the impedance transformation property of a TL. It can be placed, for example, on the coaxial cable that connects the coil (balanced port) to the T/R switch (unbalanced port) to suppress common mode currents in the outside of the coaxial shield. This is extremely important to ensure safety of coaxial connections placed near the patient as well as to avoid cable cross talking when other coaxial connections are nearby. An outer sleeve (normally implemented by a connection to the shield of a coaxial segment) is connected to the shield of the main coaxial connection at a distance of λ/4 forming a short circuited TL. Because of the impedance transformation (Z_in(λ/4)), a current circulating in the coaxial shield will see a very high impedance at the balanced connection point as it is shown in Figure 4.

To ensure RF safety and system performance, power delivered to the coil can be monitored through a directional coupler located in the transmit chain. A MR compatible coupler can be built from a pair of coupled microstrip lines as shown in Figure 5a. Using one more time the impedance equation (Zin) and performing a differential and common-mode excitation analysis we can derive the coupling factor and the equivalent characteristic impedance of the microstrip coupler modeled as two TLs as shown in Figure 5b.

$$C(\frac{\lambda}{4})=\frac{Z_{0cm}-Z_{0d}}{Z_{0cm}+Z_{0d}}, Z_{0}=\sqrt{Z_{0cm}Z_{0d}}, Z_{0cm}=f(w,h,s,\epsilon_{r}), Z_{0d}=g(w,h,s,\epsilon_{r})$$

C is the coupling factor between the input and the coupled port (normally above 10 dB), Z_0d and Z_0cm are the differential and common-mode impedances respectively which are (non-linear) functions of the geometry of the coupled microstrip lines (Figure 5a). A figure of merit of the directional coupler is its directivity which is a measure of how much power is coupled when power is injected in the output port (reversed connection). Microstrip couplers are tuned couplers and have poor directivity in general. As alternative, transformer based couplers present higher directivity (> 15 dB) and insertion losses are minimal (<2 dB) across the MR spectrum. However, these are magnetic and should be left out the MRI room. During my presentation I will discuss further these and other devices for power measurement.This presentation will focus on TL concepts and how this concepts are used in the design of the different interconnecting devices located in the transmit and the receive chains. Conventionally in MRI the RF signal travels along these chains through coaxial connections. However, the complexity of these connections (cabling layout, coupling and balun construction) increases with the number of transmit and/or receive channels. Therefore there are ongoing research efforts to replace coaxial connections by optical links or wireless communications. Miniaturization of the electronics is another important factor at the time of construction of high density arrays. To this end, transmission line segments are replaced by their lumped element equivalent. However a discrete component implementation becomes difficult when the MR frequency falls in the ultra-high frequency range (300 MHz-3 GHz). Finally, new RF power amplification approaches have been presented that eliminate the need of some of the conventional interconnections inserted between the amplifier and the coil to ensure efficient power transfer. I will discuss these non-conventional approaches further during the last part of my presentation.