reciprocity and emc measurements - IEEE Electromagnetic ...

5“If E a , H a are the field vectors due to a periodicdisturbance from a source A 1 located at O 1 and E b , H bare the corresponding field vectors due to adisturbance originating in A 2 from a source located atO 2 , then∫1+2( Ea× Hb) ⋅ds =1+2( E× H) ⋅ds(11)the surface integrals as indicated by the subscripts 1+2being taken over closed surfaces 1 and 2 surroundingthe sources A 1 and A 2 respectively.”Not directly given in [24] is the additional equality alsofollowing from the reciprocity considerations∫D( Ea⋅ Jb(12)where D is the volume containing the sources A 1 and A 2represented by the source vectors J a and J b , respectively.In Eq.(11) the surface of D is indicated by thesubscripts 1+2. Equation (12) will be used in thederivation of the hybrid reciprocity theorem in Section5. As will be noted below, Eqs.(11) and (12) areconnected to a special case of the general reciprocityrelation Eq.(19).So the theorem combines two field states (a and b),comparable to the discussed reciprocity theorem for anN-port Kirchhoff network. The derivation of Eqs. (11)and (12) starts from sets of two Maxwell’s equationsexpressed in the frequency domain for the states a andb:aacurlE= − jω µ Haa a(13a,b)curlH= + jωεE + JandbbcurlE= − jω µ Hbb b (14a,b)curlH= + jωεE + JNow the trick is to remember the existence of the vectorrelation div(X×Y)=Y⋅curlX−X⋅curlY and to writediv(div(EEab× H× Hba) = H) = H(15)(16)An expression for H b ⋅curlE a in Eq.(15) can be found bymultiplying the left and right hand members of Eq.(13a)by H b (be careful with the order of the variables)H(17)In a similar way, expressions can be found for the otherterms on the right hand side of Eqs.(15) and (16) bymultiplying Eqs.(13b), (14a) and (14b) by the propervectors. After substitution of all resulting expressions inEqs.(15) and (16) and subtracting the two finalequations, we find thatdiv(18)This equation, in which the ω-terms no longer appear, issometimes called the local form of the reciprocitytheorem. To find the general form, Eq.(18) has to be∫∫b a)dv = ( E ⋅ J ) dvbaD⋅curlE⋅curlEba⋅ curlE= − jω µabHb− E− Ebab⋅ Ha⋅curlH⋅curlHa b b a b a a b( E × H − E × H ) = E ⋅ J − E ⋅ Jabaintegrated over the volume D containing all sourcesrepresented by J a and J b , yieldinga b b a∫ div( E × H − E × H ) ⋅Da b b a∫ ( E × H − E × H ) ⋅dsSb a a b∫ ( E ⋅ J − E ⋅ J ) ⋅dvDd v =(19)We had to expect an integration as the Maxwell equationsconsider field derivatives whereas an expressionfor the fields is needed. The most left hand member ofEq.(19) has been converted from an integral over avolume D into an integral over the surface S of D byapplying Gauss’s theorem. This action also ‘removes’the div operation. Equation (19) is the general form ofthe Lorentz reciprocity theorem, a form that can even beextended [14], although this extension is not needed inthe context of this paper.A special case is the situation in which the relationbetween the E and the H vector of the field is fixed,such as in the far-field of an antenna. Then the outcomeof the integrals in Eq.(19) is equal to zero, a value givenby Lorentz as he considered the propagation propertiesof light. In the far-field, the field propagates as a planeor quasi-plane wave so that the E and the H vector ofthe field are perpendicular to each other and have aconstant ratio, η=√(µ/ε), the wave impedance (377 Ω inair). In vector notation the latter means that ν×E= ηH,where ν is the unit vector perpendicular to the planeformed by E and H. After application of this relation toE a ×H b and to E b ×H a and after application of the vectorrelation X×(Y×Z)= Y⋅(X⋅Z)−Z(X⋅Y) we find thata b b aE × H = E × H = η(E(20)a b b aso that E × H − E × H = 0 and, consequently,the integrals in Eq.(19) are equal to zero and Eqs.(11)and (12) given at the start of this section automaticallyfollow.5. The hybrid reciprocity theoremThe experimentalist can only humbly lift his hat at themathematical fireworks presented in Section 4 and thenpass to the order of the day. An application is what isneeded to catch his or her interest. As mentioned in theintroduction, a very useful application was alreadygiven in 1929 by Ballantine [10], referring to the workof Wilmotte [11].This application, resulting in thehybrid reciprocity theorem, can also be found in Section4.5 of a more recent textbook [16]. The theorem givesan expression for the voltage U i induced by an incidentfield E i in an antenna or structure acting as an antenna,e.g. an EUT (Equipment Under Test) and its connectedcables. Since the transmission and reception ofelectromagnetic waves is of interest in EMC fieldmeasurements, the two states a and b will now bedenoted by t (of transmission) and r (of reception).In the following, two wire antennas A 1 and A 2 areconsidered (see Fig.6). Each wire antenna consists ofb⋅ E=a)