Cavity model analysis for impedance of DMPA

Figure 3.8 shows a DMPA example, as well as SMPA. Usually, the feed position of SMPA is along the middle line of the patch (x1 = a/2). Naturally, the first DMPA example is also considered as feed position along the middle line x=a/2. Here, we call it middle-fed DMPA.

31

3. DIFFERENTIAL MICROSTRIP PATCH ANTENNA

They1 andy2 are symmetrical along they =b/2. Therefore, the two feed positions of DMPA are (a/2, y1) and (a/2, y2), respectively.

Since the antenna has two feeding points, each feeding point is injected with RF signals which have a 180-degree phase difference. The natural choice from single-ended microstrip patch antenna (SMPA) is to put two feeding points symmetrically according to patch. Feeding distance is defined as the distance from the edge of the patch to the feeding position. The dual feed points are symmetrical around the liney=b/2.

(a) SMPA configuration. (b) DMPA configuration.

Figure 3.8: Single-ended microstrip patch antenna (SMPA) (a) and differential feed microstrip patch antenna (DMPA) (b).

The rest of this section will present the cavity model analysis of DMPA. Two different feed mechanisms are discussed, and one practical DMPA in mmW applications is shown in detail.

From Section3.3, the application of DMPAs are extended to arrays, and both E-plane array and H-plane array are shown. In Section 3.4, a wide bandwidth transition, which is implemented in far-field measurement of DMPA, is shown. A short summary with different DMPA/array in applications is given at the end.

A. Middle-fed DMPA

The cavity model in the previous section is also suitable for DMPA since the assumptions are still valid. The antenna impedance of DMPA can be derived from multi-feed analysis [91]

[98].

The antenna impedance of SMPA (ZSM P A) has a cosine-squared relationship with different feeding positions (y_i). This can be derived simply from the cavity model.

First, let us recapitulate Equation (5) in [91] for antenna impedance with single-ended feed.

Zs = jωµ0t

∞

X

m,n=0

φ²_mn(x₁, y₁)j₀²(mπd_e 2ae

)

k_mn² −k²_e (3.16)

whereω is the angular frequency,µ₀is the permeability of vacuum, andd_eis the “effective width” of a uniform strip ofz directed source current of 1 A. Since the patch exhibits fringing effects, the physical dimensions of the patch (a,b) are replaced with the effective dimensions 32

(ae,be).

φ_mn(x, y) = p

ε_0mε_0n/a_eb_ecos(mπx/a_e) cos(nπy/b_e) (3.17a)

k_mn² = (mπ/a_e)²+ (nπ/b_e)² (3.17b)

k²_e = ε_r(1−jδ_e)k₀² (3.17c)

k₀ = ω

c = 2πf

c (3.17d)

j0(x) = sin(x)

x (3.17e)

wherec is the velocity of light in free space,f is frequency,δ_e is the effective loss tangent of dielectric,εr is the relative permittivity, andε0m= 1 form= 0 and 2 form6= 0.

The subscripts of Zs denotes single-ended feed. The subscript (m,n) indicates the mode indices in the xand y axis. Typically, x1 is selected as ae/2, andy1 represents feed distance measured from the edge of the patch to the feed point.

Z_sis a sum of series and can be rewritten as Z_s =

∞

X

m,n=0

Z_s,mn (3.18)

where

Zs,mn = jωµ0t

φ²_mn(x1, y1)j²₀(mπde

2ae

)

k_mn² −k_e² (3.19)

Zs,mn represents the antenna impedance under TMmn mode and is called mode impedance here. Comparing (3.17a)-(3.17e), for fixed mode number (m,n), Re[j/(k_mn² −k_e²)] reaches maximum value when k_mn² =εrk²₀. Therefore, the peak value of Re[Zs] occurs at its resonant frequency (fmn) and drops quickly away from the resonant frequency. In other words, the value of Re[Zs] around the fundamental resonant frequency (f01) is dominated by the value of Re[Zs,01].

The resonant resistanceRsis defined as the value of Re[Zs] atf01, and it can be calculated as

Rs(y=y1) = Re[Zs]f₀₁ ≈ Re[Zs,01]f₀₁ ∝ cos²(πy1/be) (3.20) Comparing (3.17a) with (3.20), we obtain the following relationship forRs(y=y1) over the feed distance (y1):

Rs(y=y1) = Rs(y=y0) cos²(πy1/be) (3.21) where Rs(y = y0) represents the antenna impedance with feed point at the edge of patch [109]. It proves that the impedance of patch antenna with single-ended feed exhibits cosine-squared behavior over the feed distance. Does the antenna impedance of DMPA have a similar behavior? Next, we address antenna impedance of DMPA. It may be calculated using the Z-parameters:

33

3. DIFFERENTIAL MICROSTRIP PATCH ANTENNA

Zd = V_d

I = 2(Z11−Z12) = 2(Z22−Z21) (3.22) where

Z11 = jωµ0t

∞

X

m,n=0

φ²_mn(x₁, y₁)j₀²(mπd_e 2ae

)

k²_mn−k_e² (3.23a)

Z₁₂ = jωµ₀t

∞

X

m,n=0

φmn(x1, y1)φmn(x2, y2)j₀²(mπde

2a_e )

k_mn² −k²_e (3.23b)

The subscript dofZd denotes differential feed. Z11 and Z12 are called self-impedance and mutual impedance, respectively. Comparing (3.23a) and (3.23b) with (3.16), we can find that Z11 is the same asZs, whileZ12 is different. Now we compareZ12 withZ11.

Similar toZ_s, we introduce mode impedanceZ_d,mn forZ_d:

Z_d =

∞

X

m,n=0

Z_d,mn = 2

∞

X

m,n=0

(Z_11,mn−Z_12,mn) (3.24)

where

Z11,mn = jωµ0t

φ²_mn(x1, y1)j₀²(mπd_e 2ae

)

k²_mn−k_e² (3.25a)

Z12,mn = jωµ0t

φ_mn(x₁, y₁)φ_mn(x₂, y₂)j₀²(mπd_e 2ae

)

k_mn² −k²_e (3.25b)

Obviously, the self-mode impedanceZ11,mnis equal toZs,mnin (3.19). It has a relationship with the mutual mode impedanceZ_12,mnas follows:

Z11,mn

Z12,mn

= φmn(x1, y1)

φmn(x2, y2) = cos(mπx1/ae) cos(nπy1/be)

cos(mπx2/ae) cos(nπy2/be) (3.26) For the typical feed positions,x1=x2=ae/2, andy1=be−y2, we can obtain

Z11,mn

Z12,mn

= cos(nπy1/be)

cos(nπ(be−y1)/be) =

( 1, n is even number

−1, n is odd number (3.27) Insert (3.27) into (3.24):

Z_d =

∞

X

m,n=0

Z_d,mn= 4

∞

X

m=i,n=2i+1,0

Z_11,mn= 4

∞

X

m=i,n=2i+1,0

Z_s,mn (3.28)

Up to this point, we have built the relationship betweenZdandZsusing the mode impedance Z_11,mn(Z_s,mn). Equation (3.28) implies thatZ_d,mnis zero if mode index n is an even number, while Zd,mn is four times of Zs,mn if n is an odd number. The fundamental mode of the antenna is TM₀₁, and the next two higher-order modes are TM₂₀ and TM₂₁. Here, we assume 34

that 1.5be> ae> be. Therefore,Zd,20 andZd,21 are zero, whileZd,01 is four times ofZs,01. Similar to R_s, we define the resonant resistance R_d as Re[Z_d] at f₀₁. The assumption of (3.20) is still valid. Therefore, we calculate theRd as follows:

Rd(y=y1) ≈Rd,01 = 4Rs,01 ∝ cos²(πy1/be) (3.29) Equation (3.29) implies some interesting results:

First, the resonant resistance of the antenna with differential feed exhibits cosine-squared be-havior over the feed distance, which is same as for antenna with single-ended feed:

Rd(y=y1) = Rd(y=y0) cos²(πy1/be) (3.30) whereRd(y= 0) represents the antenna impedance with differential feed points at the edges of patch.

Secondly, the impedance match position for the antenna with differential feed can be calcu-lated fromR_s:

R_d(y=y₁) ≈4R_s(y=y₁) = 4R_s(y= 0) cos²(πy₁/b_e) (3.31) Equation (3.31) shows that the resonant impedance of the differential feed antenna is four times that of the single-ended feed antenna for the same feed distance (y₁).

Usually, the reference impedance of the differential feed antenna is selected as 100 Ohm, which is double that of the single-ended feed antenna. Thus, the impedance match feed distance of the differential feed antenna (yd,100Ω) and that of the single-ended feed antenna (ys,50Ω) are related by comparing (3.21) and (3.31):

cos(πy_d,100Ω/b_e) = cos(πy_s,50Ω/b_e)/√

2 (3.32)

If we normalize the feed distancey by ˜y =y/be, Equation (3.32) may be rewritten as

cos(π˜yd,100Ω) = cos(π˜ys,50Ω)/√

2 (3.33)

Obviously, the normalized feed distance of the DMPA ˜y_d,100Ω is larger than that of the SMPA ˜ys,50Ω. That means the feed point of the former antenna is close to the patch center compared with that of the latter one.

Full-wave simulation of the DMPA example – DMPA with middle feeding points

Here is an example of DMPA antenna at 79 GHz.

The dimension of the patch can be calculated according to [89]. The width of a single patch is given by Equation 3.34

W = c

2fr

r 2

εr+ 1 (3.34)

35

3. DIFFERENTIAL MICROSTRIP PATCH ANTENNA

withcas the free-space velocity of light,εr( = 3) as the relative permittivity, andfr as the resonance frequency. The lengthLof a single patch is calculated to Equation3.35

L = 1

2λ_r−2∆l (3.35)

∆l

h = 0.412(εef f + 0.3)(^W_h + 0.264)

(εef f −0.258)(^W_h + 0.8) (3.36) with εef f as the effective permittivity and λr as the relative wavelength. They can be calculated according to [89]. In Equation3.36,h( = 0.127 mm) is the height of the substrate.

The patch parameters are calculated as follows: L1 = 0.98 mm andW1 = 1.3 mm.

A model of DMPA, as in Figure 3.8(b), was built in CST MWS. Two ports are set as 50 Ohm lumped port. The differential antenna impedance are calculated from mixed-mode methods [110].

The feed distance (y₁) varies from 0 mm to 0.4 mm, with 0.1 mm step. The patch impedances are simulated by CST MWS. Figure3.9shows the normalized input resistance of DMPA versus feed distance. Taking the fringing effects (∆) into account, theb in Figure 3.8 is replaced by be. It showscos² behavior of the resistance, which is same as SMPA.

Figure 3.9: Normalized input resistance of DMPA – middle feed.

Another comparison is with SMPA. For a fair comparison, the same size of patch (0.98 mm, 1.3 mm) is fed with single-ended signal. The ratio of the resonant resistances of DMPA and SMPA is shown in Figure3.10. It fulfils the prediction in Equation3.31.

Figure 3.11 shows the antenna impedance of DMPA with middle feed, 3.11(a) real part and 3.11(b) imaginary part. The dashed lines are the self impedance Z11,s (blue lines) and the mutual impedance Z_21,s (green lines). The solid lines are the differential mode impedance (black lines). It shows clearly the mode cancellation for differential feed antenna.

The impedance matching feed distance for DMPA and SMPA is 0.2 mm and 0.3 mm, respectively.

Till now, we have derived three important conclusions between DMPA and SMPA:

1) The resonant resistance of the fundamental mode of DMPA is four times that of SMPA;

36

Figure 3.10: Resistance ratio between DMPA and SMPA – middle feed.

(a) Real(Z)

(b) Imag(Z)

Figure 3.11: Antenna impedance of DMPA with middle feed: (a) real part and (b) imaginary part.

2) The resonant resistance of DMPA exhibits cosine-squared behavior over the feed distance similar to SMPA;

3) The impedance matching feeding point (100 Ohm) of DMPA is roughly 1.4 times the impedance matching feeding point (50 Ohm) of SMPA.

B. Edge-fed DMPA

The middle feed mechanism of DMPA is suitable for coaxial cable feed, where the cable can be easily connected from the back of the patch [105]. In microstrip line feed structure, it re-quires long feed line and bending before the patch [96] [107]. In mmW, this bending introduces 37

3. DIFFERENTIAL MICROSTRIP PATCH ANTENNA

additional loss and radiation. Therefore, another feeding mechanism, which shifts the feeding position from the middle of the patch (x=a/2) to the edge of the patch (x= 0), provides a more compact solution for the RF front-end in mmW systems.

Figure3.12shows the comparison of both feeding methods. In both configurations, the dual feed points are symmetrical around the line y = b/2. In Figure 3.12(a), the feed points are arranged along the vertical center line of the patch (a/2,yi). In Figure3.12(b), they are placed at the edge of the patch (0,y_i).

(a) middle feed (b) edge feed

Figure 3.12: DMPA with (a) middle feed and (b) edge feed.

The discussion in the previous section can be extended for the edge feed DMPA.

Here, we recapitulate Equation 3.18for ended feed antenna impedance. The single-ended feed antenna impedance (Zs) can be calculated by using the mode impedanceZs,mn:

Zs=

∞

X

m,n=0

Zs,mn (3.37)

The subscript (m,n) indicates the mode indices in the x and y axis. It was shown in the previous section that: (i) the differential antenna impedance Zd,mn can also be calculated as the sum of impedances over all modes; (ii) for symmetrical differential feed mechanisms, only the modes (m, n= 2i+ 1) exist [111].

Z_d =

∞

X

m,n=2i+1

Z_d,mn (3.38)

Different feed points introduce different mode impedances. For instance, since TM11mode requires zero E-field along x=a/2, it cancels out in a middle feed configuration but exists for the edge-feed antenna. Thus, the differential antenna impedance with middle feed (Zd,center) can be written as

38

Z_d,center=

∞

X

m=2i,n=2i+1

Z_d,mn (3.39)

and the differential antenna impedance with edge feed (Zd,edge) as

Z_d,edge=

∞

X

m=0,n=2i+1

Z_d,mn (3.40)

Comparing Equations3.39and3.40, we observe that

Zd,edge =

∞

X

m=0,n=2i+1

Zd,mn

=

∞

X

m=2i+1,n=2i+1

Z_d,mn+

∞

X

m=2i,n=2i+1

Z_d,mn

=

∞

X

m=2i+1,n=2i+1

Z_d,mn+Z_d,center (3.41)

It is shown in former section that Z_d,mn is zero if the mode index n is even, while Z_d,mn is four timesZ11,mn ifnis odd. Z11,mn denotes the mode impedance of the Z-parameterZ11. It is also identical to the mode impedance of the single-ended feed antenna impedance Z_s,mn. Therefore,

Im Zd,edge

=

∞

X

m=2i+1,n=2i+1

Im Zd,edge

+ Im

Zd,center

(3.42)

= 4

∞

X

m=2i+1,n=2i+1

Im Z_s,mn

+ Im

Z_d,center

(3.43)

Since Im Zs,mn

is always positive (m, n > 1) around frequency f01, f01 is the resonant frequency for the TM₀₁mode. Thus, Im

Z_d,edge

is bigger than Im

Z_d,center

aroundf₀₁. This implies that for the same feed distanceyi,

(i) the fundamental resonant frequency of the edge-feed differential antenna is higher than that of the middle-fed, and

(ii) the edge-feed antenna requires a larger value of the electrical separation condition ξ.

Full-wave simulation of the DMPA example – DMPA with edge feeding points

Same as before, full-wave simulation results are provided for DMPA with the edge feeding method. The same patch (L = 0.98 mm, W = 1.3 mm) shown in the previous section is reused for comparison.

Figure 3.13shows the real part of antenna impedance of DMPA as well as single-ended multi-feed antenna. The simulation results show that a couple of modes from single-ended multi-feed are cancelled in DMPA. For instance, the mode TM10, TM20, and TM30are cancelled with differ-ential feed mechanism.

39

3. DIFFERENTIAL MICROSTRIP PATCH ANTENNA

Figure3.14shows the real part of the antenna impedance of DMPA with edge feed and middle feed. It shows that (i) the edge feed MPA has more modes than middle feed, for instance, TM11, TM₃₁, etc., and that (ii) around the fundamental mode TM₀₁, both DMPAs have a similar real part of antenna impedance Real(Z).

Figure3.15shows the imaginary part of the antenna impedance of DMPA with edge feed and middle feed. It shows that around the fundamental mode TM01, the Imag(Z) of edge feed is higher than that of middle feed DMPA. The reason is that the higher-order mode TM11 raises the Imag(Z).

Table3.1shows the simulated results of the patch. For the same patch, the edge feed requires lessy_xfor the impedance matching. That is mainly because of the influence of the higher-order mode TM11.

The theory of edge-feed analysis has been published by the author in [112].

Table 3.1: Simulated DMPA with middle feed and edge feed

Port type Feed method W L y_x

lumped middle feed 1.30 mm 0.98 mm 0.3 mm lumped edge feed 1.30 mm 0.98 mm 0.2 mm

Figure 3.13: Real(Z) of DMPA with edge feed.

Figure 3.14: Real(Z_{DM P A}) of DMPA middle and edge feed.

Im Dokument Differential feed antenna in millimeter wave Radar applications / submitted by Ziqiang Tong (Seite 45-54)