Microstrip Antennas- Rectangular Patch

Transcription

1 April 4, 7 rect_patc_tl.doc Page of 6 Microstrip Antennas- Rectangular Patc (Capter 4 in Antenna Teory, Analysis and Design (nd Edition) by Balanis) Sown in Figures Easy to analyze using transmission line or cavity models Most common type of patc or microstrip antenna Transmission line model Te rectangular microstrip antenna is represented as two slots or apertures (of widt and eigt ) separated by a low impedance transmission line of lengt L (see Figure 4.). Fringing of te fields, particularly te electric field, at te edges of te patc is an issue of concern because of te finite dimensions involved. Figures 4., 4.3, and 4.5 illustrate fringing. Figure 4.5 Microstrip line and its electric field lines, and effective dielectric constant geometry. (From Balanis, Antenna Teory, Analysis and Design (Second Edition))

2 April 4, 7 rect_patc_tl.doc Page of 6 Fringing makes te patc seem bigger (electrically) tan te pysical dimensions of te patc. Tis impacts te resonant frequency of te patc. It is dependent on te dielectric constant r of te substrate as well as te pysical dimensions L, and. To account for te fringing of te electric field above te microstrip (in te air above te substrate), an effective dielectric constant < < r is defined. As most of te field lines are confined between te patc and te ground plane (like a capacitor), tends to be closer to r. Te effective dielectric constant allows te microstrip to be modeled as if it were in a omogeneous dielectric medium of. Figure 4.6 sows te effective dielectric constant as a function of frequency for several substrates. Note tat r as te frequency increases, i.e., te electric field concentrates in te substrate. Figure 4.6 Effective dielectric constant versus frequency for typical substrates. (From Balanis, Antenna Teory, Analysis and Design (Second Edition))

3 April 4, 7 rect_patc_tl.doc Page 3 of 6 Te initial (low frequency) value of is were / >. r r To account for fringing, te pysical lengt of te rectangular patc is extended at bot ends by a lengt L to give an effective lengt of L eff = L + L (see Figure 4.7). Figure 4.7 Pysical and effective lengts of rectangular microstrip patc. (From Balanis, Antenna Teory, Analysis and Design (Second Edition))

4 April 4, 7 rect_patc_tl.doc Page 4 of 6 For te dominant TM mode (patc acts as a cavity resonator), te resonant frequency is f (uncorrected) r c L r f c r,c L eff c = q L r (corrected) were q is te fringing factor Solving for L eff, we get L eff q f r,c f r c f r,c were is te guided wavelengt (Note: want (f r,c ) = f r ). Now, we can return to te transmission line model were we will represent te two radiating slots wit parallel equivalent admittances Slot # Y G j B Slot # Y G j B separated by a microstrip transmission line of lengt L wit

5 April 4, 7 rect_patc_tl.doc Page 5 of 6 caracteristic admittance Y c = /Z c (see Figure 4.8). Moreover, for rectangular patces, te slots are identical Y Y G G and B B Figure 4.8 Rectangular microstrip patc and its equivalent circuit transmission model. (From Balanis, Antenna Teory, Analysis and Design (Second Edition)) How can we calculate te slot admittances? A simple derivation (based on infinitely long slots) tat assumes tat te electric field is uniform across te slot(s) yields were k G B k for 4.636ln k for is te free space wave number. Tis result c is accurate only wen >> and <<.

6 April 4, 7 rect_patc_tl.doc Page 6 of 6 A more accurate formula for te conductance (based on te cavity model) is were G k cos Prad sin cos V P rad sin k sin cos 3 V cos sin 3 d d were is te impedance of free space. Te integral can be evaluated numerically (best option), or solved to get sin k G cosk k Si k k were S i ( ) is te sine integral (see Appendix III of Balanis). For extremely wide or narrow slots G 9. A plot of G versus / is sown in Figure 4.9.

7 April 4, 7 rect_patc_tl.doc Page 7 of 6 Figure 4.9 Slot conductance as a function of slot widt. (From Balanis, Antenna Teory, Analysis and Design (Second Edition)) To get te input admittance Y in, te admittance of slot # (Y ) must be translated across te lengt of te rectangular patc to te location of slot # and added to Y (tey are in parallel). If tis lengt (somewere between L eff and L < /) is properly selected, te translated slot # admittance is * Y G jb G jb Y. Ten, te input admittance and impedance become and Z Y Y Y G in in R Y. in in G

8 April 4, 7 rect_patc_tl.doc Page 8 of 6 Taking into account mutual coupling between te slots requires an adjustment yielding Z in R in G G were G is te mutual conductance between te slots and te plus (+) sign is used for modes wit asymmetric voltage distributions (e.g., te dominant TM mode) and te minus (-) sign is used for modes wit symmetric voltage distributions. Te mutual conductance can be calculated using k sin cos 3 G J klsin sin d cos were J ( ) is a Bessel function of te first kind of order (zero). Usually, G << G. Typically, R in is in te range of 5 to 3. To matc a feeding microstrip transmission line to tis impedance would require it to be very narrow; moreover, 5 lines are a de facto standard in te RF circuit world.

9 April 4, 7 rect_patc_tl.doc Page 9 of 6 Terefore, te question arises, How can we adjust R in? R in can be decreased by increasing. Tis action is limited to /L < because te aperture efficiency of te slots drops for /L >. An alternative is to use an inset or recessed microstrip feed (see Figure 4.). Tis works because te voltage is a maximum and te current a minimum at te edges of te patc, leading to large impedance values. As we go into te patc, te voltage drops and te current increases, leading to smaller impedances until we reac te midpoint of te patc. Te input resistance of te inset feed is given by G B B Rin ( y) cos y sin y sin y G G L Y L Y c,feed c,feed L were Y c,feed = /Z c,feed and Z c,feed is te caracteristic impedance of te feeding microstrip transmission line (widt ). Note: te inset distance y must be in te range < y < L/. Te caracteristic impedance Z c for any microstrip transmission line of widt (i.e., te feed or patc) can be calculated using Z c 6 8 ln ln.444

10 April 4, 7 rect_patc_tl.doc Page of 6 If G /Y c,feed << and B /Y c,feed << (te typical case since te widt of te feeding microstrip is muc less tan tat of te rectangular patc), te input resistance of te inset feed becomes R ( y ) cos y R ()cos y in in G G L L wic is plotted in Figure 4. (normalized by R in ()). Figure 4. Recessed microstrip-line feed and variation of normalized input

11 April 4, 7 rect_patc_tl.doc Page of 6 resistance. (From Balanis, Antenna Teory, Analysis and Design (Second Edition)) If we set R in (y ) equal to te caracteristic impedance of te feeding transmission line Z c,feed and note tat R in () R in in tis case, we can solve for te inset lengt y in( ) Z c,feed L R y L cos cos R () R in in Te notc widt n on eiter side of te inset feed introduces some capacitance. Tis can impact te resonant frequency sligtly (%) and can cange te input impedance. Also, te feeding microstrip transmission line will perturb te radiation from slot # (i.e., Y canges) wic argues for minimizing n. As a starting point, select te notc widt n to fall in te range. < n <.5. To get truly accurate results, a full-wave numerical model of te antenna sould be run after using te design based on te transmission line model to get accurate lengts and widts. A similar process applies if a coaxial probe feed is used, i.e., te input resistance can be decreased by moving te coaxial probe in from te edge of te patc.

12 April 4, 7 rect_patc_tl.doc Page of 6 Design Procedure ) Specify ε r and of substrate, te desired resonant frequency f r, and te impedance Z c,feed of te feeding transmission line. ) Calculate widt of patc using c f r (selected to give good radiation efficiency). Strictly, te patc widt sould be less tan te lengt (i.e., < L) to ensure operation only in te TM mode. Practically, > L can be used if te patc is excited/driven so as not to excite oter modes (e.g., TM mode is dominant wen > L). can be canged so long as /L <, avoid aperture efficiency decrease. 3) Calculate 4) Calculate r r L.4 r ) Calculate te effective lengt and guided wavelengt. c Leff f r L eff

13 April 4, 7 rect_patc_tl.doc Page 3 of 6 6) Calculate te patc lengt L. L Leff L L, L/ and calculate & ceck tat /L <. 7) Calculate G G and B B. G B,est k for 4.636ln k for,est G were k k sin cos cos B G B,est G,est sin 3 d is te free space wave number. c 8) Calculate te caracteristic impedance Z c,ant and admittance Y c,ant for te rectangular microstrip antenna. Z 6 8 ln ln.444 c,ant

14 April 4, 7 rect_patc_tl.doc Page 4 of 6 and Y c,ant Z c,ant 9) (Optional) Use Smit cart or direct calculation to verify * Y G jb G jb Y. ) Calculate te mutual conductance between te slots k sin cos 3 G J klsin sin d cos ) Calculate R in (used plus (+) sign in original equation). Z in R in G G ) If an inset microstrip feed is required (i.e., R in Z c,feed ), calculate lengt y of te inset needed to matc te rectangular patc to te feeding transmission line. en G /Y c,feed << and B /Y c,feed <<, R y L Z,feed cos c. R in Tis answer can be cecked using y G B y B y in ( y) cos sin sin G G L Y L Y c,feed c,feed L If R in (y ) is not equal to Z c,feed, te lengt y sould be iteratively adjusted until R in (y ) = Z c,feed.

15 April 4, 7 rect_patc_tl.doc Page 5 of 6 3) Determine te widt of te feeding microstrip transmission line. Metod : If available, use information/tools from te manufacturer of te substrate/pcb. For example, te Rogers Corporation as an on-line JAVA calculator for determining te widt of a microstrip transmission line based on desired Z and te particular substrate at ttp:// Metod : Iteratively (i.e., guess a starting value of < ) determine te widt of te feeding transmission line using and Z c,feed r r ln.444 ln. Note tat te effective permittivity must be recalculated for te feeding transmission line because it as different dimensions tan te patc antenna (i.e., it is muc narrower).

16 April 4, 7 rect_patc_tl.doc Page 6 of 6 Metod 3: Use te design equation A 8e A e.6 B B B r r were and r ln ln( ).39 Zc,feed r r. A.3 6 r r B. Z c,feed r 4) Select te notc widt n in te range. < n <.5. 5) Draw resulting design. Top View. n y n L