Prof. David R. Jackson ECE Dept. Spring 2014 Notes 12 ECE

Tube of Surface Current Top view: An infinite hollow tube of surface current, flowing in the z direction: 1 2 x y a From superposition, we know we can solve the problem using (TM z ) y x z a 1 2 2

Tube of Current (cont.) Represent J sz (  ) as a (complex exponential) Fourier series: The coefficients c n are assumed to be known. 3

  <  a Apply B.C.’s (at  = a ) :   >  a Tube of Current (cont.) 4

The first B.C. gives The second B.C. gives Hence we have, from the first BC, Tube of Current (cont.) 5

From the second BC we have Substituting for a n from the first equation, we have Next, look at the term Tube of Current (cont.) 6

This may be written as (using x = ka ) Hence, (This follows from the Wronskian identity below.) Tube of Current (cont.) 7

or Using we have Tube of Current (cont.) 8

  <  a   >  a Tube of Current (cont.) 9 Generalization: a phased tube of current

Inductive Post Equivalence principle: remove coax aperture and probe metal 2a2a h I0I0  The magnetic-current frill is ignored in calculating the fields. Coax feed 2b2b The probe current is assumed to be uniform (no z or  variation). 10 Parallel-plate waveguide I0I0

Inductive Post (cont.) Tube problem: I0I0 For  a: The probe problem is thus equivalent to an infinite tube of current in free space. Tangential electric field is zero where (There is no z or  variation of the fields.) 11

Hence and Inductive Post (cont.) 12

so Inductive Post (cont.) We then have 13

Complex source power radiated by tube of current (height h ): or Inductive Post (cont.) 14

or where Inductive Post (cont.) 15

so Inductive Post (cont.) Equivalent circuit seen by coax: 16 + I0I0 Z in Coax V0V0 -

For ka << 1 Therefore We thus have the “probe reactance”: Inductive Post (cont.) 17

or X in Inductive Post (cont.) 18 Probe inductance:

Summary Inductive Post (cont.) 19 Probe inductance: Probe reactance:

Inductive Post (cont.) Comments:  The probe inductance increases as the length of the probe increases.  The probe inductance increases as the radius of the probe decreases.  The probe inductance is usually positive since there is a net stored magnetic energy in the vicinity of the probe.  This problem provides a good approximation for the probe reactance of a microstrip antenna (as long as kh << 1 ). 20

21Example Practical patch antenna case:

22 R LC Z0Z0 LpLp Microstrip Antenna Model f 0 = resonance frequency of RLC circuit The frequency where R is maximum is different from the frequency where the reactance is zero, due to the probe reactance.

23 Lossy Post If the post has loss (finite conductivity), then we need to account for the skin effect, which accounts for the magnetic energy stored inside the post.

24 Example (Revisited) Practical patch antenna case: Assume:

25 Accuracy of Probe Formula H. Xu, D. R. Jackson, and J. T. Williams, “Comparison of Models for the Probe Inductance for a Parallel Plate Waveguide and a Microstrip Patch,” IEEE Trans. Antennas and Propagation, vol. 53, pp , Oct HFSS