Design of Impedance Matching Network for Low-Power, Ultra-Wideband Applications

: This paper addresses the design of ultra-wideband (UWB) impedance matching networks operating in the unlicensed 3.1–10.6 GHz frequency band for low-power applications. It improves the simplified real frequency technique (SRFT) by adding a realizability check and employing an iterative approach with different initial guesses in optimization to achieve realizable solutions under the requirements of UWB, low-power consumption, and a minimum number of circuit components. The comparison of solutions obtained using the SRFT with published solutions based on the Chebyshev filter theory is presented. It is shown that the optimal SRFT solution requires fewer components in the impedance matching network, maximizes the RF power delivery over the UWB spectrum with a reflection coefficient below − 10 dB, and allows for circuit optimization to reduce power consumption. Using the improved SRFT, it demonstrates a systematic approach to find the strategies and limitations of designing the input matching networks for low-power UWB applications using GlobalFoundries 90 nm BiCMOS technology.


Introduction
Ultra-wideband (UWB) technology is characterized by exceptional features, such as high data rate, low power spectral density, enhanced target recognition, precise localization, and ranging capabilities [1].These qualities make UWB a desirable choice for a wide variety of applications, including wireless communication, radar, and imaging.In these applications, the primary focus is ensuring the optimal transmission and reception of signals within the operational bandwidth.Hence, the design of broadband impedance matching networks (IMN) has emerged as a significant area of interest for engineers [2].
To match a source impedance Z S to an arbitrary load impedance Z L , as illustrated in Figure 1, various theories and techniques have been developed.To meet this objective, Fano [3] and Youla [4] invoked the gain-bandwidth limitation theory and utilized the Darlington representation, where the load Z L is represented by a lossless two-port terminated in a unit resistor.Analytical models for determining the IMN are also used, but they require an analytical model of the load impedance [5] and often result in suboptimal solutions in terms of circuit performance metrics and network complexity [6].
Another approach to the design of passive IMNs relies on filter theory [7][8][9][10].The approach employs either a cascaded L-section topology consisting of high-pass and lowpass filters [7,8] or a bandpass filter [9,10].These fixed topologies, however, may not necessarily provide a minimum number of elements in the IMN for a specific load [6].A related approach employs series or tank resonators, and it is based on manipulations of the resonant frequencies [11,12].When these resonant frequencies can be strategically spaced apart while maintaining an acceptable gain variation, it is possible to attain a broadband IMN.However, it may not prove effective for high fractional bandwidth (FBW) requirements, where many resonators are required.Design approaches for IMNs with no prior assumption about the network topology, no need for analytical load modelling, and leading to a minimum element count have always been desired.To fulfill this need, Carlin [5] developed the so-called real frequency technique (RFT) utilizing the real-frequency (e.g., experimental) load-impedance data along with numerical optimization.The initial version of the RFT technique was to solve a single matching problem, i.e., a complex load impedance ZL with a source impedance Zs = 50 Ω [5].The subsequently developed RFTs can be classified into four distinct categories, namely the line segment technique (RFT-LST), the direct computational technique (DCT), the parametric approach, and the simplified real frequency technique (SRFT) [2].The line segment technique, direct computational technique, and parametric approach formulate the objective function utilizing the unknown input immittance (driving point admittance or impedance) across the entire angular real-frequency ω axis.The procedure starts with the real part of the input immittance, as the imaginary part would be found with the Hilbert transformation or the Gewertz procedure.A lossless two-port IMN can be synthesized by finding an optimized input immittance.The line segment technique formulates the real part of the input immittance as a linear combination of straight lines in 1 ( ) . On the other hand, the direct computational technique defines the real part of a positive real minimum immittance as a rational even function . The parametric approach represents this rational function in a parametric form in 1 ( )


and takes the even function of Zin as the real part of Zin [14].Finally, the simplified real frequency technique simplifies the task by defining the two-port scattering parameters of an IMN as rational functions [15], which helps in handling the double-matching problems [13,16] and streamlining the computation process [17].
In this paper, we present the simplified real frequency technique in detail in Section 2 and apply this technique to the design of the IMN for low-power applications in Section 3, followed by the conclusions.

Simplified Real Frequency Technique (SRFT)
The SRFT starts with defining the scattering parameters, SIMN, of an IMN containing lossless reciprocal passive elements based on three polynomials: h, g, and f.Because the IMN should satisfy the paraunitary requirement † IMN IMN ⋅ = S S I [17][18][19], where I is the identity matrix, SIMN is expressed as Design approaches for IMNs with no prior assumption about the network topology, no need for analytical load modelling, and leading to a minimum element count have always been desired.To fulfill this need, Carlin [5] developed the so-called real frequency technique (RFT) utilizing the real-frequency (e.g., experimental) load-impedance data along with numerical optimization.The initial version of the RFT technique was to solve a single matching problem, i.e., a complex load impedance Z L with a source impedance Z s = 50 Ω [5].The subsequently developed RFTs can be classified into four distinct categories, namely the line segment technique (RFT-LST), the direct computational technique (DCT), the parametric approach, and the simplified real frequency technique (SRFT) [2].The line segment technique, direct computational technique, and parametric approach formulate the objective function utilizing the unknown input immittance (driving point admittance or impedance) across the entire angular real-frequency ω axis.The procedure starts with the real part of the input immittance, as the imaginary part would be found with the Hilbert transformation or the Gewertz procedure.A lossless two-port IMN can be synthesized by finding an optimized input immittance.The line segment technique formulates the real part of the input immittance as a linear combination of straight lines R in = ∑ N k=1 a k (ω)R k [5].On the other hand, the direct computational technique defines the real part of a positive real minimum immittance as a rational even function D(ω 2 ) [13].The parametric approach represents this rational function in a parametric form and takes the even function of Z in as the real part of Z in [14].Finally, the simplified real frequency technique simplifies the task by defining the two-port scattering parameters of an IMN as rational functions [15], which helps in handling the double-matching problems [13,16] and streamlining the computation process [17].
In this paper, we present the simplified real frequency technique in detail in Section 2 and apply this technique to the design of the IMN for low-power applications in Section 3, followed by the conclusions.

Simplified Real Frequency Technique (SRFT)
The SRFT starts with defining the scattering parameters, S IMN , of an IMN containing lossless reciprocal passive elements based on three polynomials: h, g, and f.Because the IMN should satisfy the paraunitary requirement where I is the identity matrix, S IMN is expressed as where h(s) and g(s) are polynomials of order n, n is the number of elements in an IMN, and k = 0, 1, 2, . .., n.The polynomial g(s) is strictly Hurwitz, which has all poles on the left-hand side of the complex plane to avoid impractical or unrealizable solutions.The polynomial f (s) provides the information about the transmission zeros of the IMN.Transmission zeroes are defined as the frequencies at which the output signal is completely cut off.In consideration of a transmittance function S 21 that may possess finite zeros at ±jω i , zeros at infinity, or zeros at the origin, its general form is expressed as Such form implies that f (s) can be either an odd function or an even function.The presence of finite zeros, such as in the modified Chebyshev design [20], often increases the number of elements and complicated synthesis [21].Therefore, it is preferable for f (s) to be in the s k form.The parameter k may assume a value of zero in low-pass scenarios and a value greater than one in bandpass and high-pass cases [22].The lossless condition introduces a relationship among the polynomials f, g, and h.Knowing f (s) allows for expressing g(s) as a function of h(s) as The objective in designing the IMN for a known load can be achieved by either maximizing the transductor power gain G T , ideally approaching unity, or minimizing the reflection coefficient Γ in , as follows from and The SRFT utilizes optimization techniques to determine the optimal polynomial h(s) and, consequently, the corresponding polynomial g(s).Optimization aims to minimize the error function based on the objective function (4) or (5) for load impedance data.

Comparing SRFT and Chebyshev Filter-Based Solutions
Bode and Fano demonstrated a physical limitation on the maximum achievable bandwidth for loads comprising resistors and reactive elements [3,23].For example, in the case of a load with a resistor R and capacitor C connected in series, this limitation can be found by [24] ∞ 0 1 It is evident from ( 6) that efficient broadband impedance-matching structures inherently exhibit filter-like characteristics [23,24].For optimal matching performance, all reflection coefficients |Γ in (ω)| within the frequency band of interest are preferred to be small while |Γ in (ω)| outside this frequency range are one (i.e., perfect reflection).So, the sharper the cut-off rate of the filter is, the better the performance of the IMN will be [25].It is well known that the Chebyshev filter offers a sharper cut-off; therefore, it is chosen for comparison with the SRFT method.In [9], a UWB Chebyshev filter theory was used to design the IMN for a load modelled by a 50 Ω resistor in series with a 650 fF capacitor.The IMN comprises five components, with their values (from [9]) depicted in Figure 2. The reflection coefficient resulting from this design is shown in Figure 3, obtained by simulating the circuit in Figure 2 using the Cadence Virtuoso Spectre circuit simulator with ideal elements from the analog library.
whole UWB spectrum has been found, as shown by the green symbols in Figure 3.This solution is better than the solution obtained using the Chebyshev filter theory, which features a maximum reflection coefficient of −12.3 dB [9].Moreover, the solution obtained using the SRFT requires only four elements, as demonstrated in Figure 4, as opposed to the five elements required by the Chebyshev design.This is beneficial, particularly in this example, saving space since an inductor is eliminated.Another way to verify the performance of an IMN solution for a particular load is to check how close it is to the Bode-Fano limit, πRC in (6).As |Γin(ω)| in ( 6) cannot be determined using an analytical expression in SRFT, it is assumed that |Γin(ω)| remains constant and equal to an average value Γavg across the specified frequency band.This approximation can simplify (6) as [24] 2 avg 0 where  whole UWB spectrum has been found, as shown by the green symbols in Figure 3.This solution is better than the solution obtained using the Chebyshev filter theory, which features a maximum reflection coefficient of −12.3 dB [9].Moreover, the solution obtained using the SRFT requires only four elements, as demonstrated in Figure 4, as opposed to the five elements required by the Chebyshev design.This is beneficial, particularly in this example, saving space since an inductor is eliminated.Another way to verify the performance of an IMN solution for a particular load is to check how close it is to the Bode-Fano limit, πRC in (6).As |Γin(ω)| in ( 6) cannot be determined using an analytical expression in SRFT, it is assumed that |Γin(ω)| remains constant and equal to an average value Γavg across the specified frequency band.This approximation can simplify (6) as [24] 2 avg 0 where On the other hand, when designing a UWB IMN using SRFT, the choice of the optimization method plays a significant role in obtaining a solution with better performance.Most reported RFT and SRFT techniques use Levenberg-Marquart optimization [15,17,26], a local optimization algorithm that aims to find the minimum of a function in the vicinity of an initial guess.Thus, it is best suited for problems where a solution is expected to be found in the proximity of the initial guess [27].Selecting a good initial guess in such a highly nonlinear optimization process is critical, and it substantially impacts the ability to reach the optimal IMN [15,22,28].
To see if we can obtain a better solution than the Chebyshev filter approach, we applied the SRFT to the same load as in [9].It is observed that the SRFT may yield solutions that cannot be realized by LC elements, even if they demonstrate superior matching performance.We characterize them as unrealizable solutions.One such solution is shown by the red symbols in Figure 3.The realizability of a solution can be verified by examining the input reflection coefficients observed at the input port of the IMN when its output port is terminated with either a short or an open circuit.For example, when the IMN comprises solely reactive elements, and when the output port is open-circuited, its input reflection coefficient Γ in,OC or input impedance Z in,OC can be expressed in terms of f (s), h(s), and g(s) as and Z in,OC (s) = 1+Γ in,OC (s) where N(s) and D(s) are the numerator and denominator of the input reflection coefficient.
In (8), the input impedance Z in,OC is expressed in a partial-fraction form, where k i is the coefficient of s i , k pi and k ′ pi are residues, and ω pi are poles.Since the input impedance Z in,OC only comprises reactive components, it should exhibit the following three characteristics.First, it must be an odd function (k 0 = 0) with the numerator and denominator differing by one degree.Second, it should contain only simple zeros and poles, all interlaced on the jω axis.Third, the residues of Z in,OC for all poles must be real-positive [29].If a solution violates any of these features, it cannot be implemented only by reactive elements.For instance, the red curve in Figure 3 is not realizable because of its k 0 < 0.
To find a realizable solution, an exhaustive search with various initial guesses using the Levenberg-Marquardt optimization method is required.In the SRFT, it assigns initialguess values to the n + 1 coefficients of h(s).Each coefficient value is randomly selected from a continuous uniform distribution within the bounds defined by l b and u b for quick optimization convergence [30].If the solution fails to satisfy the feasibility conditions outlined above, another initial guess is used until a realizable solution is found.Based on this iterative approach, a solution with a maximum reflection coefficient of −13.7 dB in the whole UWB spectrum has been found, as shown by the green symbols in Figure 3.This solution is better than the solution obtained using the Chebyshev filter theory, which features a maximum reflection coefficient of −12.3 dB [9].Moreover, the solution obtained using the SRFT requires only four elements, as demonstrated in Figure 4, as opposed to the five elements required by the Chebyshev design.This is beneficial, particularly in this example, saving space since an inductor is eliminated.reflection coefficient Γavg is 0.115.This results in 78 ps, closer to the Bode-Fano limit of 102 ps compared with the 72 ps value obtained from [9].
As an alternative, one may consider utilizing global optimization techniques.These methods can offer the lowest possible reflection coefficient at the expense of significantly longer computations.RFT-based techniques predominantly employ local minimum optimization methods, yielding a solution quickly.As exemplified above, SRFT may result in an unrealizable solution, in which case another search must be initiated.With a global optimization method, e.g., the genetic algorithm (GA), the algorithm may bypass a realizable solution and converge to an unrealizable solution due to its inclination towards achieving the lowest minimax objective.Indeed, [31] reports an approach using the GA; however, impractical responses are sometimes obtained.It is clear that synthesizing a realizable solution for a UWB IMN (topology and component values) is problematic.While the optimization may indicate the existence of a solution, it may be difficult or even impossible to synthesize a practical network, particularly in scenarios involving intricate configurations and components like transformers.As shown in [32], some preassumption is required to find the right synthesis.

IMN for Low-Power Applications
Assuming the antenna provides a source impedance ZS = 50 Ω in the frequency band of interest, the improved SRFT can find an IMN to match a load ZL to the 50 Ω source impedance, as shown in Figure 1.The inductor-degenerated topology is a commonly used technique [33] for amplifiers using bipolar junction transistors (BJT) [34] and field-effect transistors (FET) [35].Figure 5 shows the frequently used common emitter amplifier with a degeneration inductor Le to obtain the required input resistance for narrowband [34] and wideband matching [10].Here, C1 serves as a DC blocking capacitor, and L1 is an RF choke to isolate the biasing circuit from the RF port.We demonstrate the designs of the input matching networks based on the improved SRFT for different bias conditions and circuit topologies using GlobalFoundries 90 nm BiCMOS technology.
For low-power applications, the circuit's input reactance sets the lower power consumption limit.When we reduce the power consumption by decreasing the base-to-emitter voltage applied to the BJT, the base-emitter junction capacitor (Cbe) becomes smaller, resulting in a smaller Cin (or a bigger absolute value of input reactance) and a lower Bode- Another way to verify the performance of an IMN solution for a particular load is to check how close it is to the Bode-Fano limit, πRC in (6).As |Γ in (ω)| in (6) cannot be determined using an analytical expression in SRFT, it is assumed that |Γ in (ω)| remains constant and equal to an average value Γ avg across the specified frequency band.This approximation can simplify (6) as [24] BW where BW = ω max − ω min , ω 0 = √ ω max ω min , and ω min and ω max are the lower and upper limits of the frequency band.For the realizable solution shown in Figure 3, the averaged reflection coefficient Γ avg is 0.115.This results in 78 ps, closer to the Bode-Fano limit of 102 ps compared with the 72 ps value obtained from [9].
As an alternative, one may consider utilizing global optimization techniques.These methods can offer the lowest possible reflection coefficient at the expense of significantly longer computations.RFT-based techniques predominantly employ local minimum optimization methods, yielding a solution quickly.As exemplified above, SRFT may result in an unrealizable solution, in which case another search must be initiated.With a global optimization method, e.g., the genetic algorithm (GA), the algorithm may bypass a realizable solution and converge to an unrealizable solution due to its inclination towards achieving the lowest minimax objective.Indeed, [31] reports an approach using the GA; however, impractical responses are sometimes obtained.It is clear that synthesizing a realizable solution for a UWB IMN (topology and component values) is problematic.While the optimization may indicate the existence of a solution, it may be difficult or even impossible to synthesize a practical network, particularly in scenarios involving intricate configurations and components like transformers.As shown in [32], some preassumption is required to find the right synthesis.

IMN for Low-Power Applications
Assuming the antenna provides a source impedance Z S = 50 Ω in the frequency band of interest, the improved SRFT can find an IMN to match a load Z L to the 50 Ω source impedance, as shown in Figure 1.The inductor-degenerated topology is a commonly used technique [33] for amplifiers using bipolar junction transistors (BJT) [34] and field-effect transistors (FET) [35].Figure 5 shows the frequently used common emitter amplifier with a degeneration inductor L e to obtain the required input resistance for narrowband [34] and wideband matching [10].Here, C 1 serves as a DC blocking capacitor, and L 1 is an RF choke to isolate the biasing circuit from the RF port.We demonstrate the designs of the input matching networks based on the improved SRFT for different bias conditions and circuit topologies using GlobalFoundries 90 nm BiCMOS technology.
equivalent input capacitance Cin at 3 GHz due to the reduction in Cbe.The lowest collector current for a realizable IMN solution is at IC = 27.4 mA.Further reducing Cin is prohibited by the Bode-Fano limit in (9).The Rin of this circuit exhibits frequency-dependent variations due to the inherent characteristics of the BJT transistor in this technology.The intrinsic base resistance varies from approximately 50 Ω to 17 Ω across the frequency band of interest.The adjustment of this variation around 50 Ω with the assistance of Le poses an additional challenge in the quest for an IMN compared to a circuit that provides a constant 50 Ω over the bandwidth.In addition, comparing the circuit's cut-off frequency (fT) and the transistor's fT at these bias points indicates that the emitter degeneration inductor Le enhances the fT of the circuit slightly and follows the fT of the transistor.For low-power applications, the circuit's input reactance sets the lower power consumption limit.When we reduce the power consumption by decreasing the base-to-emitter voltage applied to the BJT, the base-emitter junction capacitor (C be ) becomes smaller, resulting in a smaller C in (or a bigger absolute value of input reactance) and a lower Bode-Fano limit, which makes it more challenging to find an IMN solution with Γ in < −10 dB.To examine the lowest power consumption for the inductive degeneration topology shown in Figure 5 and ensure the circuit has sufficiently high cut-off frequency f T to cover the 3.1-10.6GHz band, Figure 6 shows the equivalent C in at 3 GHz, the f T of the transistor, and the f T of the amplifier (with L e in the range of 30-40 pH) at different collector currents I C .The transistor used in the simulation has a length and width of 90 nm and 10 µm, respectively, and its collector is biased at 1.0 V. To be able to compare with the result in [9] and ensure that we can obtain a realizable IMN solution with Γ in < −10 dB over 3.1-10.6GHz, we start with the collector current I C = 32.8mA, at which C in is about 650 fF, which is the same as the value reported in [9].As expected, the reduction in the collector current decreases the equivalent input capacitance C in at 3 GHz due to the reduction in C be .The lowest collector current for a realizable IMN solution is at I C = 27.4 mA.Further reducing C in is prohibited by the Bode-Fano limit in (9).The R in of this circuit exhibits frequency-dependent variations due to the inherent characteristics of the BJT transistor in this technology.The intrinsic base resistance varies from approximately 50 Ω to 17 Ω across the frequency band of interest.The adjustment of this variation around 50 Ω with the assistance of L e poses an additional challenge in the quest for an IMN compared to a circuit that provides a constant 50 Ω over the bandwidth.In addition, comparing the circuit's cut-off frequency (f T ) and the transistor's f T at these bias points indicates that the emitter degeneration inductor L e enhances the f T of the circuit slightly and follows the f T of the transistor.
sic base resistance varies from approximately 50 Ω to 17 Ω across the frequency band of interest.The adjustment of this variation around 50 Ω with the assistance of Le poses an additional challenge in the quest for an IMN compared to a circuit that provides a constant 50 Ω over the bandwidth.In addition, comparing the circuit's cut-off frequency (fT) and the transistor's fT at these bias points indicates that the emitter degeneration inductor Le enhances the fT of the circuit slightly and follows the fT of the transistor.To further reduce the power consumption, C in is to be established by adding a capacitor C p in parallel at the amplifier input, as shown in Figure 7. Incorporating C p mitigates variations in R in with respect to frequency as it is placed in parallel with the intrinsic impedance of the transistor.This arrangement brings R in closer to 50 Ω.When reducing the collector current I C , we adhere to two essential criteria: (1) ensuring that the IMN remains below −10 dB across the bandwidth and (2) maintaining the circuit's f T above 100 GHz to ensure a circuit bandwidth up to 10.6 GHz.C p and L e are adjusted to ensure that Z in and f T meet these requirements.Figure 8 shows the equivalent C in at 3 GHz, the cut-off frequencies f T of the transistor and the amplifier (with L e in the range of 60-140 pH) at different collector currents I C .It is observed that, although adding C p decreases the f T of the amplifier, we can further reduce the collector current I C to 6.1 mA.When comparing with the initial I C = 32.8mA, we observe an 81.3% reduction in collector current while maintaining the f T at around 100 GHz to ensure the amplifier operates at 10.6 GHz.A further decrease in the collector current I C would yield a circuit with a reduced f T or Γ in surpassing −10 dB.Table 1 shows the component values of the IMN, C p , and L e for the bias conditions of the base-to-emitter voltage V BE and the collector current I C in Figure 8.
ensure a circuit bandwidth up to 10.6 GHz.Cp and Le are adjusted to ensure that Zin and fT meet these requirements.Figure 8 shows the equivalent Cin at 3 GHz, the cut-off frequencies fT of the transistor and the amplifier (with Le in the range of 60-140 pH) at different collector currents IC.It is observed that, although adding Cp decreases the fT of the amplifier, we can further reduce the collector current IC to 6.1 mA.When comparing with the initial IC = 32.8mA, we observe an 81.3% reduction in collector current while maintaining the fT at around 100 GHz to ensure the amplifier operates at 10.6 GHz.A further decrease in the collector current IC would yield a circuit with a reduced fT or Γin surpassing −10 dB.Table 1 shows the component values of the IMN, Cp, and Le for the bias conditions of the base-to-emitter voltage VBE and the collector current IC in Figure 8.Based on the amplifier shown in Figure 7, Figure 9 shows the complete amplifier design with its input IMN.Table 1 shows the component values of the circuit for the bias conditions in Figure 8.The ICmax and ICmin specify the tolerable range of the collector current to which the IMN solution can still maintain Γin < −10 dB over 3.1-10.6GHz.Maintaining the tolerance for ICmin at low bias currents is challenging because Cin and the resulting Bode-Fano limit reduce, and, therefore, it is hard to find a solution with Γin < −10 dB.Based on the amplifier shown in Figure 7, Figure 9 shows the complete amplifier design with its input IMN.Table 1 shows the component values of the circuit for the bias conditions in Figure 8.The ICmax and ICmin specify the tolerable range of the collector current to which the IMN solution can still maintain Γin < −10 dB over 3.1-10.6GHz.Maintaining the tolerance for ICmin at low bias currents is challenging because Cin and the resulting Bode-Fano limit reduce, and, therefore, it is hard to find a solution with Γin < −10 dB.Based on the amplifier shown in Figure 7, Figure 9 shows the complete amplifier design with its input IMN.Table 1 shows the component values of the circuit for the bias conditions in Figure 8.The I Cmax and I Cmin specify the tolerable range of the collector current to which the IMN solution can still maintain Γ in < −10 dB over 3.1-10.6GHz.Maintaining the tolerance for I Cmin at low bias currents is challenging because C in and the resulting Bode-Fano limit reduce, and, therefore, it is hard to find a solution with Γ in < −10 dB.

Conclusions
We demonstrated that the SRFT is a highly efficient design method for UWB IMNs, which provides solutions with the minimum component count in the IMN.On the other hand, we also show that not all optimal reflection-coefficient responses obtained with the SRFT can be assured to be realizable or synthesizable.The choice of the optimization method and the initial guess are essential for uncovering solutions in RFT-based methodologies.With local optimization, which is usually employed by the SRFT methods, an improper initial point can result in a solution that is optimal in a mathematical sense but is physically unrealizable with components such as capacitors and inductors.On the other hand, local optimization methods converge fast, thus allowing for exhaustive searches with various initial points.Such searches are not feasible with global optimization methods, which are slower and more difficult to steer through an initial guess and converge to a realizable solution.Nonetheless, there exists an opportunity for further investigation to rigorously define the underlying SRFT model features leading to unrealizable solutions, thereby providing guidelines for the selection of the initial guess as well as the formulation of constraints to steer the optimization away from such solutions.Finally, we have demonstrated that the improved SRFT ensures realizable IMN solutions.The systematic approach in this paper enables us to find a UWB IMN solution with an 81.3% power reduction and an amplifier f T at around 100 GHz to ensure the amplifier operates at 10.6 GHz.

Figure 1 .
Figure 1.Matching an arbitrary load impedance ZL to a source impedance ZS.

Figure 1 .
Figure 1.Matching an arbitrary load impedance Z L to a source impedance Z S .
ω and max ω are the lower and upper limits of the frequency band.For the realizable solution shown in Figure3, the averaged

Figure 3 .
Figure 3. Reflection coefficient of the design using the Chebyshev filter theory (•)[9], the SRFT-based realizable solution (▲), and the SRFT-based unrealizable solution (■).The maximum Γ in of each response is indicated by the blue color symbols.

Figure 6 .Figure 6 .
Figure 6.The equivalent input capacitance Cin () at 3 GHz and the fT () of the inductive degeneration common-emitter amplifier in Figure 5 with Le in the range of 30-40 pH and the fT () of the transistor at different collector currents IC.The transistor used in the simulation has a length and width of 90 nm and 10 µm, respectively, and its collector is biased at VCC = 1.0 V.

Figure 7 .
Figure 7.An inductively degenerated common-emitter amplifier with a capacitor Cp connected in parallel to the input of the amplifier.

Figure 7 .
Figure 7.An inductively degenerated common-emitter amplifier with a capacitor C p connected in parallel to the input of the amplifier.

Figure 8 .
Figure 8.The equivalent input capacitance Cin () at 3 GHz and the fT () of the inductive degeneration common-emitter amplifier with CP and Le in Figure7and the fT () of the transistor at different collector currents IC.The transistor used in the simulation has a length and width of 90 nm and 10 µm, respectively, and its collector is biased at VCC = 1.0 V.

Figure 8 .
Figure 8.The equivalent input capacitance C in (▲) at 3 GHz and the f T (•) of the inductive degeneration common-emitter amplifier with C P and L e in Figure 7 and the f T (■) of the transistor at different collector currents I C .The transistor used in the simulation has a length and width of 90 nm and 10 µm, respectively, and its collector is biased at V CC = 1.0 V.

Figure 8 .
Figure8.The equivalent input capacitance Cin () at 3 GHz and the fT () of the inductive degeneration common-emitter amplifier with CP and Le in Figure7and the fT () of the transistor at different collector currents IC.The transistor used in the simulation has a length and width of 90 nm and 10 µm, respectively, and its collector is biased at VCC = 1.0 V.

Table 1 .
Component values for the amplifier in Figure9with V CC = 1.0 V.