1. Prof. David R. Jackson Dept. of ECE Fall 2015 Notes 29 ECE 6340 Intermediate EM Waves 1

2. + - Mutual Impedance The mutual impedance Z 21 between two antennas is calculated. + - + - 2

3. The open-circuit voltage V 2 is put in the form of a reaction. Mutual Impedance (cont.) + - + - + - E 1 = field radiated by J 1 i in the presence of open-circuited antenna 2. 3

4. The equivalence principle is used to replace antenna 1 with its surface current. The antenna current J 1 ant is that excited on antenna 1 when it is in the presence of open-circuited antenna 2. Mutual Impedance (cont.) 4

5. Reciprocity is invoked, and then the equivalence principle. Mutual Impedance (cont.) 5 The antenna current J 2 ant is that excited by I 2 on antenna 2 when it is by itself (antenna 1 is absent).

6. Reciprocity is invoked one more time. Mutual Impedance (cont.) 6 The mutual impedance is then

7. Summary 7 Mutual Impedance (cont.) J s1 ant = current on antenna 1 when it is excited by I 1 in the presence of open-circuited antenna 2. J s2 ant = current on antenna 2 when it is excited by I 2 in the absence of antenna 1.

8. If the two currents come together to one surface, we have the self- impedance formula for a single antenna: 8 Self Impedance J ant = current on the antenna when it is excited by I. Note: This is a variational expression for the input impedance of an antenna.

9. The concept of a “transfer function” is very useful in biomedical problems, for calculating the electric field (and hence the heating) at a point inside a body. 9 Transfer Function Body Lead Device The goal is to calculate the electric field inside the body at point r 0, when an external source is radiating. Source

10. We use reciprocity, selecting the b source to be a testing dipole at the observation point, in the p direction. 10 Transfer Function (cont.) a source b source Body Lead Device

11. From reciprocity, we have 11 Transfer Function (cont.) a source b source Body Lead Device

12. Next, apply the equivalence principle to remove the device and the lead, and keep the currents that are on them. 12 Transfer Function (cont.) a source b source Body The set of currents called b consists of the dipole b plus the currents on the device and lead that are set up by the radiating dipole b (in the absence of source a ).

13. Next, apply reciprocity one more time. 13 Transfer Function (cont.) a source b source V L is the volume containing the lead and the device. Body

14. For the volume integral we have: 14 Transfer Function (cont.) a source b source C L is the contour of the lead Body I L is the current on the lead (in the direction of the contour) due to the unit-amplitude dipole at r 0. S D is the surface of the device J sD is the current on the device due to the unit-amplitude dipole at r 0.

15. We then have: 15 Transfer Function (cont.) Source Note: The field in the a situation means the field of the a sources radiating in the presence of the body, but without the lead or the device present. Body The transfer function is the current on the lead (in the direction of the contour) due to the unit-amplitude dipole at r 0.

16. Summary 16 Transfer Function (cont.) Source Body Lead Device E inc = the field produced by the external source radiating in the presence of the body, but without the lead or the device present.