The input network can be represented by chain-scattering parameter matrices as follows:
where m denotes the number of transmission lines and
T
ijk(i,j = 1,2) denotes the matrix element of the k-th component of the input network. The output network is given as matrix element of the l-th component of the output network.
The chain-scattering matrix of the overall amplifier system can be obtained by sequential multiplications of the chain-scattering matrices of the input network, microwave transistor, and output network, which is given as
11 12 11 12 11 12 11 12
The scattering parameters of the overall amplifier system can be obtained by the following conversion:
⎥⎥
The parameter Ss21 is considered as the system function of the amplifier. To design an amplifier embedded with a band-pass filter, Ss21 is manifested to fit the
“magnified” band-pass filter prototype in the sense of the least squares error. We set the target amplitude response
S
21idea[ ] i
of the “magnified” band-pass filter as a fifth-order Chebyshev band-pass filter with a constant power gain in the pass-band, as shown in Fig. 5. The range of |S21|2 of GaAs FET transistor FSX017LG in data sheet is 8<|S21|2 <10. In order to set an achievable gain over the wide pass-band, we select the gain of the amplifier as 9dB for the frequencies from 3.1GHz to 10.6GHz.The least squares error method is employed to minimize the deviation of Ss21 fromS
21idea[ ] i
as follows:∑
=Fig. 5 Target gain response
We select four short stubs, two open stubs, four two-section open-circuited stubs, as well as series unit lines as the basic matching networks. The configuration of the amplifier design is shown in Fig. 6. Notice that the circuit configuration in Fig.6 is selected to meet the required circuit response S21ideal shown in Fig.5. To simplified the optimization algorithm requirements for both S11 and S21 are neglected for the present consideration. A close look of Fig.6 indicates that there are six nulls in the frequency range of interest. Four equal-length two-section open stubs are employed to implement four zeros located at 0.125π,0.19π,0.71π and 0.79π. Four one-section stubs are used to the null at 0π, while two one-section open stubs are used to realized the null at π. Both of Rf and Cf are used to improve the stability of the microwave transistor so that the transistor maintains the unconditionally stable condition over the entire frequency range. As stated in section 2.2. We have Rf=350Ω and Cf=33pF.
Fig. 6 Configuration of ultra-wideband (UWB) amplifier embedded with band-pass filter
Upon substituting (8)–(11) and (17) into (14), The scattering parameters could be displayed as below:
Out
The general scattering parameters of configuration of Fig.6 can be obtained in the parallel form as follows:
∑
The direct form in (19) could be obtained by making reduction of fractions to a
common denominator of (18). The upper limits of sigmas are obtained by matlab numerical simulation in which the cascade connection of each micro-strip elements and the z-polynomial representation of FET. It is noteworthy that aij
(n) and b
ij(m) in
(18) and (19) are functions of characteristic impedances of the input and output transmission lines. As the result, the characteristic impedances of the transmission lines are obtained in the optimization process.∑
∑
=
−
=
−
=
10951
1 1108
1
1
) (
) ( )
(
n
n ij
m
m ij
S ij
Z n a
Z m b z
T
(19) The high orders in both the numerator and the denominator of the polynomial in (19) can cause numerical noise [11]. Fig. 7 shows the simulation scattering parameters of the amplifier, which are obtained from the direct-form expression in (19). The numerical noise is generated by quantization process in the modeling. The quantization noise can arise in the digital implementation of the arithmetic operations involving the binary data due to the finite word length limitations of the registers storing the numbers and the results of the arithmetic operations.
Now consider how the roots of the denominator and numerator polynomials are affected by the errors in the coefficients. Clearly, each polynomial root is affected by all of the errors in the coefficients of the polynomial. Thus, each pole and zero will be affected by all of the quantization errors in the denominator and numerator polynomials, respectively. More specifically, Kaiser(1966) showed that if the Poles and zeros of high-order z-polynomials are located tightly, small errors in the coefficients of the denominator or nominator can cause large changes in poles and zeros in this direct-form expression. Therefore, poles and zeros of the direct-form structure are quite sensitive to the variation of the coefficients of high-order z-polynomials. Fig. 8 shows that the parallel-form function in (18) is generally much less sensitive to coefficient variation than the equivalent direct-form function. To reduce the numerical noise during the system simulation, the parallel-form polynomials of chain-scattering parameters were adopted in this study.
Fig. 8 Simulation results of S parameter frequency for parallel-form z-polynomials For the input matching circuit, the obtained characteristic impedances of open stubs, from left to right, are [145, 145 Ω]. The characteristic impedances of short stubs, from left to right are [145, 145, 145, 55.8 Ω]. The characteristic impedances of series unit lines, from left to right, are [31, 56, 29, 34, 48, 50, 27.3 Ω]. For the output
matching circuit, the characteristic impedances of two-section open-circuited stubs, from the left to the right, are (Z2(1),Z1(1)) = (90, 623 Ω), (Z2(2),Z1(2)) = (145, 591 Ω), (Z2(3),Z1(3)) = (145, 14 Ω), and (Z2(4),Z1(4)) = (145, 6 Ω). The characteristic impedances of series lines at the output, from the left to the right, are [68, 43, 35, 25, 31, 37.3 Ω].
The unattainable characteristic impedances of transmission-line sections are modified by the frequency-bandwidth-impedance relation of a notch filter. The relations between characteristic impedance, bandwidth Δ, and center frequency ΩN of a notch filter implemented with a two-section open stub are [12]
1 0
2
2 tan ( ) 2
Z Z
=
−Δ
(20) and
1 1 2
1 2
1 /
cos ( ),
1 /
N
Z Z Z Z
−
−
= +
Ω
(21)
where Z0 is the reference characteristic impedance, which is 50 Ω. When the normalizing frequency is doubled and both Δ and ΩN are reduced to one half, the new attainable characteristic impedances are (Z′2(1),Z′1(1)) = (130, 67 Ω) and (Z′2(2),Z′1(2)) = (138, 55 Ω); when the normalizing frequency is reduced to one half and both Δ and ΩN are doubled, the new characteristic impedances are (Z′2(3),Z′1(3))
= (74.4, 37 Ω) and (Z′2(4),Z′1(4)) = (86, 20 Ω).
Notice that the characteristic impedance of the serial reference characteristic impedance is Zo for all basic elements, as shown in table1. The shunt elements are placed at the junctions between two finites lines. As a result, the formulas in both (20) and (21) relating the characteristic impedance, bandwidth Δ and center frequency of a notch filter are applicable. It is pertinent to point out the simulation results of using converted characteristic-impedances transmission lines are the same as those of using unattainable characteristic-impedance transmission lines for the circuit shown in Fig.6.
In other word, the outer impedances seen from the stubs are Zo.