Microwave Phase Shifters and Filters Based on a Combination of LH and RH Transmission Lines

Filippo Capolino/Applications of Metamaterials _C Finals Page  -- # 13 Microwave Phase Shifters and Filters Based on a Combination of Left-Handed and Right-Handed Transmission Lines I. B. Vendik St. Petersburg Electrotechnical University D. V. Kholodnyak St. Petersburg Electrotechnical University P. V. Kapitanova St. Petersburg Electrotechnical University . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . Definitions and General Equations for RH and LH TLs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . Microwave Phase Shifters Based on Switchable Left-Handed and Right-Handed Transmission Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . Microwave Filters with Resonators Based on Composition of LH and RH Transmission Line Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - 13.1 Introduction Electromagnetic (EM) metamaterials are usually defined as artificial effectively homogeneous structures with specific properties, which cannot be observed in natural materials. A classical exam- ple of such a metamaterial is a structure exhibiting simultaneously negative values of the dielectric permittivity ε and the magnetic permeability μ. Very often the concept of left-handedness is used for structures with backward EM waves in contrast to conventional materials with forward EM waves where the electric field, the magnetic field, and the propagation vector form the right-handed (RH) triad. Different approaches are used for a description of the fundamental EM properties of metamateri- als and the practical realization of these materials as well. Among them, the transmission line (TL) approach gives an efficient design tool for microwave applications providing a correct description of physical properties of metamaterials [–]. A conventional TL with a positive phase velocity behaves as a right-handed transmission line (RH TL). An artificial RH TL can be formed as a ladder network of capacitors connected in shunt- and series-connected inductors.The unit cell of the RHTL is shown in Figure .a.The dual TL can be designed as a ladder network of inductors connected in shunt- and series-connected capacitors as in the unit cell in Figure .b. This line has a negative phase velocity and is referred to as the left-handed transmission line (LH TL). A backward wave propagates along the LH TL, which can be considered as a one-dimensional metamaterial. A more general model of a 13-1 © 2009 by Taylor and Francis Group, LLC Filippo Capolino/Applications of Metamaterials _C Finals Page  -- # 13-2 Applications of Metamaterials LH TL is the composite right-/left-handed structure, which includes RH effects []. In many practi- cal applications, the influence of RH effects is negligibly small and many interesting features can be observed when a combination of RH and LH TLs is used. The most important feature of the LH and RHTLs is that their dispersion characteristics are described by different equations, which can be used for many beneficial applications. Further we comprehensively consider properties of LH and RHTLs and different combinations of these lines, which can be used for a design of miniature microwave devices with improved performance and enriched functionality. Among them are microwave phase shifters and microwave filters exhibiting enlarged functionality. 13.2 Definitions and General Equations for RH and LH TLs The homogeneous RH TL presented as a cascaded connection of unit cells (Figure .a) is described by the telegraph equations ∂V ∂z = −L ∂I ∂t ∂I ∂z = −C ∂V ∂t (.) where L and C are the inductance and capacitance per unit length, correspondingly V is the voltage I is the current Both V and I are periodical with respect to time t and coordinate z, along which the I–V wave propagates. In the case of sinusoidal waves, ∂/∂t = iω and the wave equations look as ∂V ∂z = −ωLCV (.) ∂I ∂z = −ωLCI (.) V2V1 V2V1 L1 Δz C1 –1 Δz L1 –1 Δz C1 Δz Δz (a) (b) Δz I1 I1 I2I2 FIGURE . Unit cells of (a) RH TL and (b) LH TL. © 2009 by Taylor and Francis Group, LLC Filippo Capolino/Applications of Metamaterials _C Finals Page  -- # Microwave Phase Shifters and Filters 13-3 The solutions to Equations . and . are V = V exp [i (ωt − kz)] (.) I = I exp [i (ωt − kz)] (.) where the wave number, k = kR, is defined as kR = ω √ LC > . (.) Both phase and group velocities are positive Vph = ω kR =  √ LC >  (.) Vg = ( ∂kR ∂ω ) − = Vph =  √ LC >  (.) Therefore, the forward wave propagates in the RH TL. In line with Equation ., the dispersion law is linear. The characteristic impedance is defined as Z = V I = √ L C (.) For a homogeneous perfect LH TL formed as a cascaded connection of the unit cells (Figure .b), the telegraph equations in the case of sinusoidal waves look as follows: ∂V ∂z = −  iω ⋅ (  C )  I (.) ∂I ∂z = −  iω ⋅ (  L )  V (.) where (/C) and (/L) are the inverse capacitance and the inverse inductance per unit length. The wave equations are written as ∂V ∂z = −  ω ⋅ (  L )  ⋅ (  C )  V (.) ∂I ∂z = −  ω ⋅ (  L )  ⋅ (  C )  I (.) The solutions to Equations . and . are the same as in Equations . and ., with the wave number defined as kL = −  ω ⋅ √ (  L )  ⋅ (  C )  < . (.) In this case, the phase velocity is negative Vph = −ω  √ ( L ) ⋅ (  C ) <  (.) © 2009 by Taylor and Francis Group, LLC Filippo Capolino/Applications of Metamaterials _C Finals Page  -- # 13-4 Applications of Metamaterials whereas the group velocity is positive Vg = ( ∂kL ∂ω ) − = ω  √ ( L ) ⋅ (  C ) >  (.) hence the backward wave propagates in the LH TL. In line with Equation ., the wave number (propagation constant) is inversely proportional to the frequency and the dispersion law is nonlinear. The characteristic impedance of the LH TL is defined as Z = √ ( C ) ( L ) (.) Equations . through . describe the homogeneous, infinitely long TLs with distributed param- eters. A section of such a line of length l can be described by the electrical length θ defined as θR,L = kR,L ⋅ l . (.) In accordance with Equation ., the frequency dependence of the electrical length of a section of the RH TL is θR (ω) = θR ω ω (.) where θR = kR l >  is the electrical length at the frequency ω. In the case of a section of LH TL, the electrical length can be found from Equation . and the frequency dependence is written as θL (ω) = θL ω ω (.) where θL = kL l <  is the electrical length at the frequency ω. In practice, the artificial RH and LH TLs can be composed as a periodical structure containing one inductive and one capacitive component in the unit cell of the length l defined by a real length of the lumped components (Figure .). In this case, one has to use the translation symmetry for a L0 (a) (b) C0 C0 C0 C0C0 L0 L0 Vn C0 Vn Vn+1 Vn+1 Vn+2 C0 Vn+2C0 C0 L0 L0 L0L0 L0 L0 In+1 In+2In l l l l In In+1 In+2 FIGURE . Artificial lumped element TLs: (a) RH TL and (b) LH TL. © 2009 by Taylor and Francis Group, LLC Filippo Capolino/Applications of Metamaterials _C Finals Page  -- # Microwave Phase Shifters and Filters 13-5 description of the I − V wave propagating along the one-dimensional structure [,]. The telegraph equations for the RH TL are written as Vn+ − Vn = −iωLIn+ (.) In+ − In = −iωCVn (.) The voltage and current are Vn = Ve−inθ , In = Ie−inθ (.) with θ = kl , n = , , . . . Substituting Equation . into Equations . and ., one obtains after some transformations the dispersion equation sin ( θ  ) =   ωLC = ω ωc (.) with the cutoff frequency ωc =  √ LC (.) For ω >ωc, θ is an imaginary quantity and the wave attenuates: the higher the frequency, the more is the attenuation. In the low-frequency limit (ω≪ ωc), ω ωc = ± sin( θ  ) ≈ ± θ  (.) and k = ω lωc = ω √ LC (.) with L = L/l andC = C/l .The artificial lumped element RHTL behaves at ω≪ ωc as an infinitely long, perfect TL with the linear dispersion law. In a wide frequency range, the artificial periodic RH TL (Figure .a) is considered as a low-pass lumped element TL. The same consideration of the artificial LH TL (Figure .b) leads to the dispersion equation sin ( θ  ) =    ωLC = ωc ω (.) with the cutoff frequency ωc =   √ LC (.) For ω < ωc, θ is an imaginary quantity and the wave attenuates: the lower the frequency, the more is the attenuation. In the high-frequency limit (ω≫ ωc), ωc ω = ± sin( θ  ) ≈ ± θ  (.) and k = − ω ⋅ √ (  L )  (  C )  (.) with ( L ) = (  L ) /l and ( C ) = (  C ) /l .The artificial lumped element LH TL (Figure .b) behaves as an infinitely long, perfect TL with k inversely proportional to ω. The artificial periodic LH TL is considered as a high-pass lumped element TL. The dispersion characteristics of the RH and LH TLs are shown in Figure .. In further consideration, the frequency range is limited by the inequalities ω≪ ωc for the RH TL and ω≫ ωc for the LH TL. © 2009 by Taylor and Francis Group, LLC Filippo Capolino/Applications of Metamaterials _C Finals Page  -- # 13-6 Applications of Metamaterials (a) –θ –θ θθ–π π0 0 (b) ω ω ω0 ω0 FIGURE . Dispersion characteristics of (a) RH lumped element TL and (b) LH lumped element TL. 13.3 Microwave Phase Shifters Based on Switchable Left-Handed and Right-Handed Transmission Lines The transmission-type phase shifter is a lossless two-port providing a change in a phase response of the EMwave under the control signal (current or voltage).The digital phase shifters using switchable channels are well known []: the EM waves propagate in turn along two channels formed by TL sections of different electrical lengths. These lines are characterized by a different phase response, which is used for obtaining the differential phase shift. The channels are switched by two single pole double throw (SPDT) switches. As a rule, the p-i-n diode switches, MEMS switches or field-effect transistors are used. The devices are controlled by current or voltage; the optical control also can be used. Switching between a low-pass network and a high-pass network is commonly used to design broadband phase shifters []. The phase shift is flat over a wide frequency range due to parallel runs of the phase characteristics for two channels. However, a simultaneous control of the required phase shift and a suitable input matching within the same bandwidth is complicated. Inappropriate mis- match causes significant loss level of the device. On the other hand, the phase shifters based on two switchable TL sections of different lengths exhibit a broadbandmatching, while the operational band- width is limited by a significant variation of the phase shift because of nonparallel phase responses in the two states. Using specific dispersion properties of RH and LHTLsmakes it possible to combine the benefits of switchable channel phase shifters of both kinds. It was shown [–] that using cascaded connection of RH and LH TL sections, it is possible to obtain a similar slope of the phase response of two TLs of different electrical lengths. This means that in principle, it is possible to design a controllable phase shifter based on switchable metamaterial TLs, providing a flat differential phase response. Let us consider a digital phase shifter based on switching between RH TL and LH TL sections.The operational principle of a digital phase shifter using switchable RH and LH TL sections is illustrated by Figure .. In one state, the signal goes through the RHTL section with a negative phase response φ, whereas in another state, it propagates through the LH TL section with a positive phase response φ. The differential phase response (phase shift), Δφ = φ − φ, is obtained by switching the signal path using two SPDT switches. Switching between the RH and LH TL sections with the electrical lengths, which are the same by absolute value at the central frequency and differ in sign, results in providing almost a constant phase shift over a fairly large bandwidth. It was theoretically estimated that in the case of the ideal RH and LH TLs switched by the perfect SPDT switches, the phase shift error is ±% in one octave bandwidth and about ±.% over two octaves for any value of the phase © 2009 by Taylor and Francis Group, LLC Filippo Capolino/Applications of Metamaterials _C Finals Page  -- # Microwave Phase Shifters and Filters 13-7 SP DTIn OutSP DT RH TL LH TL FIGURE . Structure of a digital phase shifter based on switchable TL sections of different electrical lengths. RH TL LH TL LR,T LR,T CL,T CL,π CL,T CR,T LL,T LR,π LL,π LL,πCR,π CR,π FIGURE . Lumped-element equivalent circuits of the RH and LH TL sections. shift [,]. Moreover, if the characteristic impedance of the both TLs is equal to the port impedance Z, the perfect matching is provided in any frequency range for both the states. It is reasonable to form the phase shifter containing switchable RH and LH TL unit cells as T- or Π-networks (Figure .). For different bits of a digitalN-bit phase shifter giving the phase shift ΔΦm , the equivalent electrical length, θ, of the both RHTL and LHTL sections should be chosen as follows ∣θ∣m = ΔΦm/ (.) where ΔΦm = ΔφLH − ΔφRH (.) and m = , , . . . ,N is the bit number. In general, the TL section can be described by the ABCDmatrix [] [ A B C D] TL = [ cos θ iZ sin θ i sin θ/Z cos θ ] (.) Taking into account the symmetry of the section of a homogeneous TL, it is possible to replace it by the symmetric lumped element T- or Π-circuits. The unit cells presented in Figure . can be used as equivalent circuits of the sections of RH and LH TLs. It is supposed that the geometrical length of the lumped element unit cell is equal to zero. Comparing the matrix (Equation .) with theABCD © 2009 by Taylor and Francis Group, LLC Filippo Capolino/Applications of Metamaterials _C Finals Page  -- # 13-8 Applications of Metamaterials matrices of T- and Π-circuits, one can find the LC components of the equivalent lumped element unit cells [] LR,T = Z tan (θR/) ω , CR,T = sin θR ωZ (.) LR,Π = Z sin θR ω , CR,Π = tan (θR/) ωZ (.) for the RH TL and LL,T = Z ω sin ∣θL∣ , CL,T =  ωZ tan (∣θL∣/) (.) LL,Π = Z ω tan (∣θL∣/) , CL,Π =  ωZ sin ∣θL ∣ (.) for the LH TL. In the frequency range close to the chosen frequency ω, the characteristics of the T- or Π-circuits with LC parameters defined by Equations . through . will be the same as for the correspond- ing RH/LHTL sections with known Z and θ.This equivalent presentation of the RH/LHTL sections is used in designing microwave devices. The phase shift is determined by the LC parameters of T- or Π-circuits used in the channels of the phase shifter. Using equivalence of the ABCDmatrix of a TL section with the matrices of T- and Π-circuits, the phase characteristics of the RH and LH TLs can be found as φRH (ω) = − arccos (A)T,Π = − arccos ( − ω LRCR) (.) φLH (ω) = arccos (A)T,Π = arccos [ −  ωLLCL ] (.) The products LRCR and LLCL are the same for the T- and Π-sections for both the RH and LH TLs and are calculated using Equations . through .. Equations . and . can be considered as dispersion equations. The main characteristic of interest is the slope parameter of the phase characteristics, which can be determined by the differentiation of Equations . and . with respect to ω: dφRH dω = − √ LRCR √  − (ωLRCR/) (.) dφLH dω = − √ /LLCL ω ⋅  √  − (/ωLLCL) (.) The phase response of a TL section relates to the electrical length as φ (ω) = −θ (ω). For the maxi- mum available digital phase shift, ΔΦ = φLH − φRH = ○ (m = ), the electrical lengths of both RH and LH channels are ∣θ∣ =  ○, and consequently ωLRCR = ωLLCL = . At the central frequency of the operational bandwidth ω dφRH dω ∣ω=ω = dφLH dω ∣ω=ω = − √ LR,LCR,L (.) This equality is valid at the central frequency only.The frequency dependence of the slope is different for LH and RH TLs. © 2009 by Taylor and Francis Group, LLC Filippo Capolino/Applications of Metamaterials _C Finals Page  -- # Microwave Phase Shifters and Filters 13-9 0.5 1 1.5 f/f0 2–60 –50 –40 –30 –20 –10 0 ∂Δ j ∂ω FIGURE . Frequency dependence of derivative ∂(Δφ)∂ω in [deg] for the natural TL section (solid line), T-section of lumped element RH TL (dash-dotted line), and T-section of lumped element LH TL (dashed line); the electrical length ∣θ∣ =  ○ at f / f = . For m > , the terms in Equations . and ., ωLRCR/ ≪  and /ωLLCL ≪ , both rapidly decrease when m arises. Thus, one can simplify Equations . and . for m >  dφRH dω = − √ LRCR (.) dφLH dω = − √ /LLCL ω (.) The frequency dependencies for the slope parameter of the distributed RH TL section for the natural TL section, lumped element section of RH TL, and lumped element section of LH TL with electrical length ∣θ∣ =  ○ (m = ) at f / f =  are shown in Figure .. The frequency-dependent slope of the RH TL section presented by the T (Π)-single cell is very close to the frequency-independent slope of the natural RH TL section of the same electrical length θ. The difference in the slope parameters of the LH and RH TL sections is remarkably pronounced at a lower frequency range (ω << ω) and is less at higher frequencies, though rises at ω >> ω. The simulations revealed that the smaller the θ, the smaller is the difference between the slope parameters of lumped element LH and RH TL sections. Thus, one can conclude that for a design of the broadband phase shifter on switchable RH and LH TL sections, it is reasonable to use single cells with a small value of the equivalent electrical length. This follows the conclusion that for a lower m, the RH and LH branches should be designed as a cascaded connection of RH and LH TL single cells (T or Π) having a small electrical length. As an example, Figure . presents the theoretical phase response of the RH and LH TL sections for different numbers of LC sections (single cells in Figure .) providing a phase shift of ○ while switching the RH and LH channels. In the normalized bandwidth of one octave (.–.), the deviation of the phase shift for ideal RH and TL sections is ±.○. The higher the number of single cells, the closer is the characteristic of the artificial lumped-component TL to the characteristic of the © 2009 by Taylor and Francis Group, LLC Filippo Capolino/Applications of Metamaterials _C Finals Page  -- # 13-10 Applications of Metamaterials 0.5 –180 –135 –90 –45 0 45 90 135 180 Ph as e r es po ns e ( de g) LH TL RH TL Normalized frequency (ω/ω0) LH TL RH TL 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 FIGURE . Phase characteristics versus normalized frequency for different numbers of LC sections: single LC section, dashed lines; two LC sections, dash-dotted lines; and ideal TL section, solid lines. ideal T-section.There is a remarkab