Reciprocity Principle

 

Ergin Atalar, Ph.D.

 

In calculating the amount of received signal in magnetic resonance (MR), the reciprocity principle is commonly used. In this text, the definition of this principle and its proof will be given.

Figure 1 Measured voltage across an antenna is related with the magnetic field generated by the antenna when a unit current is applied to its terminal.

Assume an antenna and an MR signal source placed in a heterogeneous medium. The open circuit voltage, , induced by the MR signal source can be expressed in terms of the right hand polarized component of magnetic field, , generated by the antenna at the position of the MR signal source, if a unit current were applied to the terminal of the antenna (see Figure ).

                                                                                                              

where is the transverse component of the magnetization, is the complex number , is the larmor frequency, is the permeability of the free space.

 

The proof of the reciprocity principle is a modified version of the proof given by Vesselle [1] .

 

Since in the magnetic resonance, spins are rotating clockwise around the main magnetic field, the magnetization field distribution a point MR signal source (rotating spins) can be written as:

                                                                                                         

 where is the position of the point source. Assume that this point source is giving rise an electromagnetic field distribution ( , ). This field should satisfy the Maxwell’s equation:

                                                                                                                      

                                                                                                        

where is the complex permittivity.

            On the other hand, the electromagnetic field, ( , ), generated by a unit current applied across the terminal of the antenna should also satisfy the Maxwell’s equation.

                                                                                                     

                                                                                                                   

where is the position of the current source and is the unit vector in the orientation of the terminals of the current source.

                

After simplification, we find that

                              

Integrating the left hand side of the equation over the sphere, S, is equivalent to integral on the surface of it:

                                   

and the radius of the sphere goes to infinity, the surface integral vanishes because both sets of electromagnetic field satisfy the radiation condition. Since the left hand of Eqnis zero when integrated over the whole space, the right hand of the equation will result in the same:

                                                   

Evaluating these integrals using the Eqn result in the following identity:

                                                                      

Proving the validity of the reciprocity principle:

                                                                                                  

More generally, if the MR signal received is not due to a point source, rather due to a distribution, the voltage across the antenna can be expressed in terms of a volume integral over the whole space:

 

                                                                                            

Reference:

1.         Vesselle, H. and R.E. Collin, The signal-to-noise ratio of nuclear magnetic resonance surface coils and application to a lossy dielectric cylinder-- part i: Theory. IEEE Trans. on Biomedical Engineering, 1995. 42: p. 497-506.