ab 2 Radio-frequency Coils and Construction Background: In order for an MR transmitter/receiver coil to work efficiently to excite and detect the precession of magnetization, the coil must be tuned to the resonance (precession) frequency of the magnetization precession and be impedance matched to the pre-amplifier (50 Ohms). Recall that a linear B1 magnetic field can be written as B B cos(ω t) xˆ 1 1 The objective is to produce the highest B1 possible using an RF coil that matches the subject configuration. Practical MR Coil: The simplest MR coil that will produce a magnetic field is a simple loop with a series capacitor (a series-resonance circuit) When the inductor () is resonant with the capacitor (C), the impedance of this series-resonance circuit is Z R + ix + ix R R + i ω C 1 ωc 1 when ω ωc Where R is the resistance of the inductor loop, therefore, the resonance frequency of this MR coil circuit is given by f 1 2π C The inductor value,, is fixed by the loop structure. The capacitor C can be variable to allow the circuit frequency to vary. Tuned and matched resonance circuit: 1
The practical implementation of a tuned and matched MR coil can be described by the following circuit diagram where the B1 field is produced by the inductor. This is an appropriate description for an MR coil that is not close to a conductive object (e.g. biological object) When the inductor,, is resonant with the capacitor, parallel section of the circuit becomes Z p RC p C p, the impedance of this The use of the parallel capacitor allows the circuit impedance to be tuned to a desired value. However, the complete circuit response is complicated. The variable series capacitor, Cs, and the variable parallel capacitor, Cp, allow exact tuning to the desired resonance and impedance matching to 50 Ohms. Quality factor: ratio of energy stored to energy dissipated (power loss) ω Q( ω ω ) 0 0 Δω ( 3dB) or ( ) X s ω 1 Q ω ω 0 0 R R R C s Basic RF measurements 2
An electromagnetic (or any other) wave experiences partial transmittance and partial reflectance when the medium through which it travels suddenly changes. 1. Reflection coefficient The reflection coefficient is the ratio of the amplitude of the reflected wave to the amplitude of the incident wave. In particular, at a discontinuity in a transmission line, it is the complex ratio of the electric field strength of the reflected wave ( V ) to that of the incident wave ( V ). This is typically reflected incident represented with a Γ (capital gamma) and can be written as: Γ V reflected V incident The reflection coefficient may also be established using other field or circuit quantities. The reflection coefficient can be given by the equations below, where Z S is the impedance toward the source, Z is the impedance toward the load: Γ Z Z s Z + Z s The absolute magnitude of the reflection coefficient (designated by vertical bars) can be calculated from the standing wave ratio, SWR: 3
Γ SWR 1 SWR + 1 Γ is a complex number that describes both the magnitude and the phase shift of the reflection. The simplest cases, when the imaginary part of Γ is zero, are: Γ 1: maximum negative reflection, when the line is short-circuited, Γ 0: no reflection, when the line is perfectly matched, Γ + 1: maximum positive reflection, when the line is open-circuited. 2. Voltage standing wave ratio The voltage component of a standing wave in a uniform transmission line consists of the forward wave (with amplitude V f ) superimposed on the reflected wave (with amplitude V r ). Reflections occur as a result of discontinuities, such as an imperfection in an otherwise uniform transmission line, or when a transmission line is terminated with other than its characteristic impedance. The reflection coefficient Γ is defined thus: Γ V r V f For the calculation of VSWR, only the magnitude of Γ, denoted by ρ, is of interest. At some points along the line the two waves interfere constructively, and the resulting amplitude V max is the sum of their amplitudes: V max V + V V + ρ V V ( 1+ ρ) f r f f f At other points, the waves interfere destructively, and the resulting amplitude V min is the difference between their amplitudes: V min V V V ρv V ( 1 ρ) f r f f f The voltage standing wave ratio is then equal to: SWR V max Vmin 1+ ρ 1 ρ As ρ, the magnitude of Γ, always falls in the range [0,1], the VSWR is always +1. The SWR can also be defined as the ratio of the maximum amplitude of the electric field strength to its minimum amplitude, i.e. E max / Emin. 4
3. Transmission coefficient The transmission coefficient is defined in terms of the incident and transmitted wave according to: V T transmitted V incident where T incident is the voltage magnitude in the wave incident upon the load/barrier and T transmitted is the voltage magnitude in the wave moving away from the barrier on the other side. Measurement of MR coil response: Using a network analyzer, the unloaded (free space) MR coil responds in the following manner. 1. S11 reflection measurement 5
2. S21 transmission (no loading) measurement 3. S21 transmission (with loading) measurement 6
The loaded MR coil responds in the following manner, where loading in this context means that the coil is interacting with nearby conducting objects (human body). Note the resonance frequency shift and lower Q. When the MR coil is near a conductive object (e.g. a biological subject), the object becomes part of the electronic circuit as illustrated in the revised circuit diagram below. RF Coil Body Therefore, the MR coil must be retuned to the desired resonance frequency and rematched to the impedance of 50 Ohms. Coil Circuit elements and radio frequency response C: capacitor values are measured in Farad (F) units. : Inductor values are measured in Henry (H) units. 7
ω : Resonance frequency is Hertz (Hz). Note that a unit prefix μ is 6 10 and p is 10 12 Coil Construction I: Solenoid Coil Using the wire provided, wind the following three coils (inductors) and tape each to the plastic sheet without compressing the spring. Then solder a fixed capacitor of the same value across the ends of each coil to create a series resonant circuit. Wire: A length of 16-gauge enamel coated copper wire with approximately 5mm of enamel scrapped off each end. Capacitor: Use a fixed capacitor of the same value (e.g. 130pF or 270pF ) for each coil. ook for the printed number on the capacitor and record this in your note. 1. Two-turn coil: appropriate length of wire wound twice around the tube 2. Four-turn coil: appropriate length of wire wound four times around the tube Coil Construction II: Surface Coil Using a quarter-inch copper tape to make a square (4-5 inches of length in one side). Solder on the overlapped corner, cut a notch on one side and solder a capacitor. Measurements: Using the network analyzer and the pickup look, measure and record the resonance frequency for each coil. Then calculate the apparent inductance of each coil, assuming the coils can be treated as a simple series-resonance circuit. 8
ab assignment: 1. Tape the coils to the plastic sheet so that the windings are pressed gently on top of each other (for multiple turns) 2. With the pickup loop and network analyzer, measure and record the resonance frequency (using S11) for the solenoid coils of 2 turns and 4 turns. 3. With the pickup loop and network analyzer, measure and record the resonance frequency (using S11) and the quality factor (using S21) for each coil the surface coil. 4. Rest your arm on the surface coil, measure and record the resonance frequency (S11) and the Q factor. 5. Calculate the inductance of each coil in μh units 6. Discuss the results and answer the question: a. Why are the frequencies different for the two solenoid coils? b. Why are the frequencies and the quality factor different for the surface coil when loaded and unloaded? Instructor demonstration of coil mutual-inductive coupling and decoupling (if time allows) 9