Input matching analysis

The technique of filter design is employed for wideband input impedance matching. The two kinds of the most common used filter design technique are image parameter method and insertion loss method. The first one, image parameter method, consists of a cascade of simpler two-port filter sections to provide the desired cutoff frequencies and attenuation characteristics. Thus, although the procedure is relatively simple, the design of filters by image parameter method must often be iterated many times to achieve the desired results and that will result in large chip area. The other one, insertion loss method, uses network synthesis techniques to design filters with a completely specified frequency response. The design is simplified by beginning with low-pass filter prototypes that are normalized in terms of impedance and frequency. Transformations are applied to convert the prototype designs to the desired frequency range and impedance level [9]. The insertion loss method is used to design the broadband input matching for diminishing the implement costs. The Butterworth (Maximally flat) and Chebyshev (Equal ripple) filter design are two familiarly practical filter responses by used insertion loss method. The Butterworth design offers a smooth response curve with maximal flatness at zero frequency. The Chebyshev design offers a steeper response curve at the 3 dB cutoff frequency and requires fewer components. In this work, to have precipitous response curve at 3 dB cutoff frequency, the Chebyshev filter design is chosen. The filter designs can be scaled in terms of impedance and frequency, and converted to bandpass characteristics. This design process is illustrated in Fig. 2.3.1.

Fig. 2.3.1 The process of filter design by the insertion loss method The filter response is defined by its insertion loss, or power loss ratio, PLR:

)2 even function of ω; therefore it can be expressed as a polynomial in ω². Thus

)

where M and N are real polynomials in ω². Substituting this form to (2-1) gives the following:

)

Thus, for a filter to be physically realizable its power loss realizable its power loss ratio must be of the form in (2-3). Notice that specifying the power loss ratio simultaneously constrains the reflection coefficient, Γ(ω).

In this design, the Chebyshev polynomial is used to specify the insertion loss of an N-order low-pass filter as

The passband response will have ripples of amplitude 1+k², as shown in Fig. 2.3.2, since TN(x) oscillates between ±1 for |x|≦1. Thus, k² determines the passband ripple level.

Fig. 2.3.2 Chebyshev (equal-ripple) low-pass filter response (N=3)

From the power loss ratio equation of Chebyshev filter, the normalized element values of L and C of low-pass filter prototypes is shown in Fig. 2.3.3, and the normalize values are listed in Table 2.3.1.

(a)

(b)

Fig. 2.3.3 Ladder circuits for low-pass filter prototypes and their element definitions. (a) Prototype beginning with a shunt element. (b) Prototype beginning with a series element.

Table 2.3.1 Element values for equal-ripple low-pass filter prototypes (g0=1, ωc=1, N=1 to 3, 0.5dB ripple) [10]

N g1 g2 g3 g4

1 0.6986 1.0000

2 1.4029 0.7071 1.9841

3 1.5963 1.0967 1.5963 1.0000

Low-pass prototype filter designs can be transformed to have the bandpass response. If ω1

and ω2 denote the edges of passband, then a bandpass response can be obtained using the following frequency substitution:

Δ is the fractional bandwidth of passband. The center frequency, ω0, could

be chosen as geometric mean of ω1 and ω2, i.e.ω₀ = ω₁ω₂ . The low-pass prototype transfers to the band-pass filter type. The elements based on Table 2.1 are converted to series or parallel resonant circuits. The series inductor, Lk, is transformed to a series LC circuit with element value:

The shunt capacitor, Ck, is transformed to a shunt LC circuit with element value:

k

Fig. 2.3.4 shows the complete transformation circuit of low-pass filter converted to band-pass filter.

Fig. 2.3.4 Components convert from low pass filter to band-pass filter

Fig. 2.3.5 Small signal equivalent circuit of the inductive source degeneration structure In Fig. 2.3.5, since the input impedance of the MOS transistor with inductive source degeneration can be seen as a series RLC circuit

s of our third-order Chebyshev L-C filter structure can then absorb this MOS input impedance into its network. The size of M1 determines not only third-order L-C tank of band-pass filter but also the noise performance. According to these basic formulas, the models of authentic inductor and capacitor, and trading off noise performance, we can then omit the capacitor C2’ that shunts with the inductor L2’, and the capacitor C3’ is wholly replaced by the capacitance Cgs of M1 without connecting additional capacitor, as shown in Fig. 2.3.6 [11]. The inductor L3 is replaced by the inductors Lg and Ls. Besides, because the frequency of input signal is up to 10GHz, the electromagnetic effect of transmission lines changes the characteristic of the input matching network. The effect of transmission lines between components is considered and simulated by the software, Sonnet. The whole input matching network is shown in Fig.

2.3.7. The block of S2P vin means the equivalent S-parameter model of the transmission line

between input node and the inductor L1. The block of S2P net01 means the equivalent S-parameter between the inductor L1 and the capacitor C1, and so on. The capacitor Cpad is the parasitic capacitance from the RF signal pad to ground. Therefore, the input network has lower complexity and good reflected coefficient from 3.1GHz to 10.6GHz. The Smith chart of the simulated return loss (S11) from 3.1 to 10.6 GHz is shown in Fig. 2.3.8.

Fig. 2.3.6 Basic schematic of the LNA input network

Fig. 2.3.7 The whole schematic of the LNA input network

Fig. 2.3.8 The Smith chart of the simulated return loss (S11) from 3.1 to 10.6 GHz