Spring 2015 Notes 1 ECE 6345 Prof. David R. Jackson ECE Dept. 1

Overview In this set of notes we discuss the CAD model of the microstrip antenna.  Discuss complex resonance frequency  Derive formula for Q  Derive formula for input impedance  Derive formula for impedance bandwidth 2

CAD Model of Microstrip Antennas LpLp (probe inductance) Tank (RLC) circuit RLC 3 The circuit model is justified from the eigenfunction method in the cavity model, discussed later. Probe-fed patch antenna

Tank Circuit: complex resonance frequency Transverse Resonance Equation (TRE): RLC + - V The complex resonance frequency is denoted as  0. 4

Resonance Frequency (cont.) so choose + sign 5 TRE:

Resonance Frequency (cont.) Denote: 6

Assume We then have: Resonance Frequency (cont.) 7 (a good resonator)

(Take V = 1) In the time domain: so Natural Response (no source) RLC + - V 8 The complex resonance frequency is  0.

Natural Response (cont.) 9

Stored Energy RLC + - v(t)v(t) i(t)i(t) For the inductor: Therefore, (Take V = 1) 10 For the capacitor:

Stored Energy (cont.) Note: RLC + - v(t)v(t) i(t)i(t) Also, note that 11

Stored Energy (cont.) Hence 12

Q of Cavity = energy dissipated per cycle (period T ) = average power dissipated or 13 (this includes radiation loss)

Q of Cavity (cont.) 14

Q of Cavity (cont.) We then have Recall that Hence 15

Q of Cavity (cont.) Hence 16 so

Q of Cavity (Cont.) We can thus write 17

Input Impedance R LC The probe inductance is neglected here. 18

Input Impedance (cont.) or Then we have: Define where 19 (real resonance frequency)

Input Impedance (cont.) Hence, we have Define: 20

Input Impedance (cont.) Define: We then have 21

Input Impedance (cont.)

Reflection Coefficient R L C Z0Z0 23

Bandwidth Bandwidth definition is based on SWR < S 0 24 x0x0 x0x0  S0S0 S x 1.0 f2f2 f1f1 S0S0 S f f0f0 The value S 0 is often chosen as 2.0.)

Bandwidth (cont.) so We can solve for f r in terms of x: Recall that 25 Fractional bandwidth:

Hence To determine correct sign, enforce that Therefore (So choose the plus sign.) Bandwidth (cont.) 26

Now we need to solve for x 0 : Hence, so Also, Bandwidth (cont.) 27

Therefore or Bandwidth (cont.) 28 so Thus we have

Bandwidth (cont.) The solution is: 29

Hence, We then have For S 0 = 2 we have: Bandwidth (cont.) 30

Note: -9.5 f f0f0 Bandwidth (cont.) 31

Complete Model R LC Z0Z0 LpLp (This will be derived in a HW problem.) 32

Complete Model (cont.) Define In terms of the normalized variable x, the resonance frequency f res where the input impedance is purely real, corresponds to If then (This follows from a binomial expansion of the square-root term in the numerator.) (This will be derived in a HW problem.) 33

Complete Model (cont.) At the resonance frequency, the input resistance is then 34

Complete Model (cont.) Given a specified value of the input resistance at resonance (e.g., R in res = 50  ), we wish to solve for the corresponding value of R. Note that the probe reactance changes the input resistance at resonance. Note that the CAD formula for resonant input resistance (in the short-course notes) gives us the value of R in terms of the feed location. 35

Complete Model (cont.) To solve for R, use and solve iteratively: Zero iteration: First iteration: Second iteration: 36