Calculating The Parameters
A circuit model was created with the help of Microwave Office. The equivalent-circuit parameters can be calculated from the S-parameters based on an electromagnetic (EM) simulation. Once the S21 and S11 parameters have been computed at the resonant frequency by the AWR simulator, the required circuit parameters can be defined by using the relationship between the S-ABCD patameters and the Y-parameters as described by the following equations:
A = [(1 + S11)(1 - S22) + S12S21]/2S21 = 1 +Yp/Ys (1)
B = [(1 + S11)(1 + S22) - S12S21]/2S21 = 1/Ys (2)
C = (1/Z0)[(1 - S11)(1 - S22) - S12S21]/2S21 = 2Yp + (Y2p/Ys) (3)
D = (1/Z0)[(1 - S11)(1 + S22) + S12S21]/2S21 = 1 + (Yp/Ys) (4)
Ys = the series admittance of the π equivalent circuit;
Yp = the parallel admittance of the π equivalent circuit; and
Z0 = the characteristic impedance of the transmission line;
Ys = 1/Zs = 1/B (5)
Ys = 1/Zs = B = RsZLC/( Rs + ZLC), Rs = → ∞ (6)
Zs = ZLC = [(jωLs)(1/ jωCs)]/[jωLs + (1/jωCs)] (7)
Zs = ZLC = jωLs/[1 - ω2LsCs] (8)
YLC = [1 - ω2L3C3]/jωLs = j(ωCs – 1/ωLs) = jBLC (9)
BLC|ω = ωc = ω0Cs[(ωc/ω0) - ω0/ωc] (10)
Cs = BLC/ω0[(ωc/ω0) - ω0/ωc] and Ls = 1/(ω02Cs) (11)
Yp = (A - 1)/B = 1/Rp + jBRC ≈ jBRC = jωCp 12)
ω = ωc → Cp = BRC/ωc (13)
Figure 3 shows some of the electrical properties (resonance and cut-off frequencies) for changes in the length and width of the DGS head. The square head and its area (a x b) represents inductance, while the slots and distances between them (c, d) form capacitances. Values for the cutoff frequency, fc, and resonant frequency, f0, can be found from the transmission characteristics of the quasi-Yagi slot.6,7 The dimensions of the DGS (a = 8 mm, b = 3 mm, e = 1 mm, c = 2.5 mm, and s = 2.5 mm) were computed and optimized with the aid of the Microwave Office and Tex-line software packages.
To verify the dependence of the equivalent-circuit elements (capacitance and inductance) on the PCB surface as part of the distribution EM field, EM simulations were performed with results shown in Figs. 4(a) and (b). For this analysis, the microstrip structure was divided into two regions. In region I, the electric field is highly concentrated in the gap, hence any change in the gap’s dimensions affects the effective capacitance of the structure. In region II, the electric field nearly vanishes. On the other hand, the current is distributed throughout the whole structure. Therefore, any change in the length of the square area strongly affects the magnetic field distribution and, hence, the surface current. This in turn leads to a change in the effective inductance of the structure. Therefore, region I corresponds to capacitance and region II corresponds to inductance. The full structure corresponds to an LC resonator.
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