ISSN 0735-2727, Radioelectronics and Communications Systems, 2014, Vol. 57, No. 12, pp. 531–541. © Allerton Press, Inc., 2014. Original Russian Text © I.N. Prudyus, V.I. Oborzhytskyy, 2014, published in Izv. Vyssh. Uchebn. Zaved., Radioelektron., 2014, Vol. 57, No. 12, pp. 16–29.

Dual-Band Devices Based on Coupled-Stripline Section with Cross-Symmetrical Loads I. N. Prudyus* and V. I. Oborzhytskyy** Lviv Polytechnic National University, Lviv, Ukraine *e-mail: [email protected] **e-mail: [email protected] Received in final form May 19, 2014

Abstract—An equivalent representation of a circuit formed by coupled lines section with cross-symmetrical loads and additional reactive elements is suggested in the form of partial one-port networks with in-phase or antiphase excitation. Analytical relations for the electrical parameters calculation of elements of the circuit, which enables one to implement devices with dual-band operation mode, have been obtained based on the input impedances of these one-port networks. The examples of application of relations obtained for the simulation of dual-band phase shifters with fixed phase difference, impedance transformers and bandpass filters are provided. The simulation results are confirmed by experimental investigations of these devices. DOI: 10.3103/S0735272714120024

Implementation of telecommunication systems, that operate in multiple frequency bands according to the accepted standards, require the development of microwave devices with two and more operation frequency bands. Utilization of dual-band devices allows one to reduce the dimensions and the cost of designs developed. In this case significant attention is paid to the designing of passive electronic components, which include such microwave integrated form devices as power dividers/combiners, directional couplers, bridge circuits, impedance transformers, crossovers (intersections of transmission lines), filters. They are also used in the designing of dual-band active devices (amplifiers, generators, mixers). Sections of single striplines are most often utilized to develop the majority of abovementioned dual-band designs. Researches connected with the utilization of coupled transmission lines sections, using which one can decrease the dimensions of the device, are encountered more rarely, for example, in the designing of dual-band power dividers [1–4], impedance transformers [1–5], directional couplers [6–9]. Sections consisting of two transmission lines with electromagnetic coupling, cross-opposite ends of which are open- or short-circuited, are encountered only in designs with single operation frequency band. Most often these are bandpass filters with parallel coupled half-wavelength resonators [10], sections of hybrid ring circuit, which provide 3p/4 phase shift [11], as well as transformers of active load’s impedance into specified value of input impedance [12–14]. Since conductive coupling between input and output is absent in such sections of coupled transmission lines, then it allows one to use them simultaneously as direct current blocker [14]. One of the reasons prohibiting the designing of devices with dual-band operation mode is the existence of a variety of restrictions in known variants of equivalent representation of coupled transmission lines section. Equivalent circuit should possess the possibility of taking into account of reactive elements and stubs, that are introduced into the structure of the section in the form of additional and cross-symmetrical loads for the implementation of dual-band devices for different purposes. Creation of novel designs, as well as the development of analytical techniques for the calculation of electrical parameters of their elements based on the suggested equivalent circuit determine the purpose of this article. EQUIVALENT REPRESENTATION OF COUPLED TRANSMISSION LINES SECTION WITH CROSS-SYMMETRICAL LOADS Section consisting of two similar closely spaced striplines, between which electromagnetic coupling exists, is widely used in the mode of two-port network, in the case when two of its four terminals are conventionally open-, short-circuited or connected with one another. 531

532

PRUDYUS, OBORZHYTSKYY Ze/2, qe

l 1 Zw

1

2 Ze, Zo, qe, qo

Zw

-1:1

2

Zo

q

q 2

1 Zt, q

Zw

Zo/2, qo

(a)

Zo

(b) Fig. 1.

Zw

(c)

Figure 1a presents the variant of two-port network, open terminals of which are located on the diagonally opposite sides, and feeding transmission lines with wave impedance Zw are connected to two other terminals. Several equivalent representations are known for such section of coupled lines with length l, which has wave impedance, phase constant and electrical length in even-mode operation (in-phase excitation) Ze, be, qe and in odd-mode operation (antiphase excitation) Zo, bo, qo, where qe,o = be,ol. Equivalent circuit of one of them, for which transmission matrix is presented in [15], is shown in Fig. 1b. Another widespread variant of equivalent circuit for the case of open-circuited sections is depicted in Fig. 1c [16], where Zt = (Ze – Zo)/2. Common disadvantage of such equivalent circuits consists in the fact that the variation of terminal loads, for example, from open- to short-circuit or connection of additional elements to the lines makes their implementation impossible. Besides, presented circuit (Fig. 1c) does not take into account the presence of different phase velocities of even and odd modes i.e. electrical length of the sections is taken to be equal to averaged value of q. Analysis of potential distribution in the structure with cross-symmetry (Fig. 1a) shows the following. In the case of in-phase excitation of terminals 1, 2 in central vertical cross section of lines the potentials are equal, and in the case of antiphase excitation of the terminals they have opposite signs in this cross section. Consequently, according to the even-odd mode analysis technique [17] initial symmetric two-port network can be decomposed on two partial one-port networks: one with in-phase (even) excitation with connected to the terminals sections of l/2 length and antiphase (odd) excitation with terminals of sections with l/2 length, which are connected via phase inverter. Figure 2a presents the circuit of coupled lines section, which is supplemented by reactive impedances (loads) jXldi, that are connected with cross-opposite ends of lines, and also with reactive impedances jXi, connected with the center of each of the lines. Subscript “i” in designations of circuit elements indicates their frequency dependence. Equivalent circuits of one-port networks with in-phase or antiphase excitation, which are obtained with the consideration of abovementioned potential distribution and which take into account the presence of additional reactive elements, are presented in Fig. 2b, c. In the former case terminals of the sections are connected by transmission line with zero electrical length q¢, and in the latter case connecting line has electrical length q¢ = p, that provides phase opposition of potentials at the terminals. Each of partial one-port networks (Fig. 2) can be considered as symmetric two-port network formed by the section of coupled transmission lines with pair of connected terminals, which are also connected with reactive impedances j2Xi. Output ports of these two-port networks are loaded by reactive impedances jXldi, which are transformed by the two-port network at the input into input reactive impedances jXei and jXoi on condition that losses in transmission lines are negligible. Therefore, in order to calculate electrical parameters of elements of two-port network with cross-symmetry (Fig. 2a) in general case one needs to: 1) define the values of input impedances Zei, Zoi of partial one-port networks with in-phase and antiphase excitation based on the specified values of operation parameters of the device; 2) find the connection between input impedances of partial one-port networks and parameters of elements of their equivalent circuits by solving the problem of impedance transformation by a symmetrical two-port network. Wave scattering parameters of a symmetrical two-port network are connected with the input impedances of its partial one-port networks with in-phase and antiphase excitation as follows [18]: S 11i = S 22i = ( z ei z oi - 1) / D i , S 21i = ( z ei - z oi ) / D i , RADIOELECTRONICS

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jX³

j2X³

jXe³

jXld³

Zw 1

533

j2X³

jXo³

q¢ = p

q¢ = 0 2 Zw

jXld³ jX³

jXld³

j2X³

(a)

j2X³

jXld³

(b)

(c)

Fig. 2.

where D i = 1 + z ei + z oi + z ei z oi , z ei ,oi = rei ,oi + jx ei ,oi = Z ei ,oi / Z w are the values of input impedances, which are normalized to wave impedance of feeder lines. Using (1) input impedances of partial circuits are computed based on the values of wave parameters, which are determined from the specified values of operational characteristics. In the case depicted in Fig. 2 these are reactances jXei, jXoi. In order to find solution of the problem of impedance transformation, i.e. the conversion of complex (in general case) load impedance Zldi into input impedance Zini by a symmetrical two-port network we will use the expression from [17] connecting input reflection coefficient with reflection coefficient from the load, substitution of expressions (1) in which results in the following dependence [19]: z ( z + z oi ) + 2 z ei z oi , z ini = ldi ei z ei + z oi + 2 z ldi

(2)

where z ini ,ldi = rini ,ldi + jx ini ,ldi = Z ini ,ldi / Z w are the values of input impedance and load impedance, which are normalized to wave impedance of feeder lines. In the case of reactive transforming two-port network, when rei = roi = 0, based on (2) one can write the following expression for normalized values of input reactances of partial one-port networks: x e2i A + 2x ei B - C = 0, x ( x - x ldi ) - 2( rini rldi - x ini x ldi ) , x oi = ei ini x ldi - x ini + 2x ei

(3)

where A = rini - rldi , B = x ini rldi + x ldi rini , Ñ = rldi | z ini | 2 - rini | z ldi | 2 . It follows from (3), that in the case of transformation of active impedance into active one, when x ini = x ldi = 0, input reactances are equal to x ei = rei roi , x oi = -x ei . In the case of transformation of reactive impedance, when rini = rldi = 0, the value of input reactance xei must be given, and xoi is calculated. In the case of utilization of formulas (3) for transforming two-port networks as parts of partial circuits (Fig. 2) one should take into account that in both cases reactance jXldi/Zw is the normalized load impedance xldi, reactance jXei/Zw is the input normalized impedance xini for circuit shown in Fig. 2b, reactance jXoi/Zw is the input normalized impedance xini for circuit depicted in Fig. 2c, and input reactances of partial one-port networks, on which every two-port network can be decomposed using the even-odd mode analysis technique, are input reactive impedances xei, xoi. Equivalent circuits of such one-port networks are presented in Fig. 3. The first pair of networks (Fig. 3a, b) is obtained as a result of decomposition of two-port network (Fig. 2b), what is indicated by the first subscript “e” in the designation of input impedances for even jXeei and odd jXeoi excitation, and the second pair of networks (Fig. 3c, d) is obtained as a result of decomposition of two-port network (Fig. 2c), what is

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PRUDYUS, OBORZHYTSKYY Ze, qe³/2

Zo, qo³/2 j2X³

jXee³

(a)

Ze, qe³/2

Zo, qo³/2 q¢ = p / 2

jXeo³

jXoe³

(b)

q¢ = p / 2 jXoo³

j2X³

j2X³

(c)

(d)

Fig. 3.

indicated by the first subscript “o” in the designation of input impedances for even jXoei and odd jXooi excitation. Based on equivalent circuits of one-port networks one can write the following expressions for normalized values of input impedances: x eei = z e (2x i + z e t ei ) / ( z e - 2x i t ei ), x eoi = z o t oi , x ooi = z o (2x i + z o t oi ) / ( z o - 2x i t oi ), x oei = z e t ei ,

(4)

where z e,o = Z e,o / Z w are the values of normalized to Zw wave impedances for even and odd modes of coupled transmission lines; x i = X i / Z w is the normalized to Zw reactance jXi; t ei ,oi = tan(q ei ,oi / 2). On the other hand, based on the second expression (3) one can write the following formulas for input impedances of partial one-port networks with odd excitation by applying the substitution of xin,i to xei and xoi: x ( x - x ldi ) + 2x ei x ldi , x eoi = eei ei x ldi - x ei + 2x eei x ( x - x ldi ) + 2x oi x ldi . x ooi = oei oi x ldi - x oi + 2x oei

(5)

Having performed the substitution of (4) into (5), one gets two equations, which connect the values of parameters of circuit elements (Fig. 2a) with the input impedances xei, xoi of its partial one-port networks (Fig. 2b, c). Having written the expressions for xi from each of these equations, we obtain: z z p t + z e p ei t ei - 2d ei + 2 z e z o t ei t oi , x i = e o ei oi 2 z o p ei t ei t ei - z e p ei - 2d ei t ei - 2 z e z o t oi z - z o p oi t oi - z e p oi t ei + 2d oi - 2 z e z o t ei t oi , xi = o 2 - z e p oi t ei t oi + z o p oi + 2d oi t oi + 2 z e z o t ei

(6)

where p ei = x ldi - x ei , p oi = x ldi - x oi , d ei = x ldi x ei , d oi = x ldi x oi . Having equated both expressions (6), one obtains quadratic equation for variable xldi: 2 x ld i ( a1c1 - a 2 c 2 ) + x ldi ( a1d 1 - a 2 d 2 + b1c1 - b 2 c 2 ) + b1d 1 - b 2 d 2 = 0,

(7)

where

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a1 = z e ( z o t oi + z e t ei - 2x ei ),

535

a 2 = z o ( z e t ei + z o t oi - 2x oi ),

b1 = z e (2 z e z o t ei t oi - z o x ei t oi - z e x ei t ei ), b 2 = z o (2 z e z o t ei t oi - z e x oi t ei - z o x oi t oi ), ñ1 = z o - z e t ei t oi + 2x oi t oi , d 1 = 2 z e z o t ei - z o x oi + z e x oi t ei t oi ,

ñ 2 = z e - z o t ei t oi + 2x ei t ei , d 2 = 2 z e z o t oi - z e x ei + z o x ei t ei t oi .

Consequently, using (6), (7) and basing on specified by design considerations values of parameters l, Ze, Zo and on corresponding to them values of bei, boi of coupled lines section, as well as on the values of input reactances xei, xoi of partial one-port networks of the structure (Fig. 2), which are determined by required values of operation parameters of the device, one can calculate the values of reactive loads jXi, jXldi, using which the required operation mode of the device is provided. Having equated in (6) denominators to zero for the variant of a structure (Fig. 2a) without additional reactive impedances, which are connected to the center of the section (jXi = ¥), we obtain two expressions for xldi, one of which has the following form: z x t t - z e x ei + 2 z e z o t oi . x ldi = o ei ei oi z o t ei t oi - z e - 2x ei t ei

(8)

Having equated these expressions, we obtain quadratic equation for one of wave impedances of coupled transmission lines: z o2 a + z o b + c = 0,

(9)

where a = t oi ( t ei v i + 2 z e g ei ),

c = z e t oi ( z e t ei v i - 2x ei x oi g ei ), 2

v i = x ei - x oi , g ei = 1 + t ei ,

2

g oi = 1 + t oi ,

b = 2t ei g oi ( x ei x oi - z e2 ) - z e v i (1 + t e2i t o2i ) - 4 z e ( x ei t e2i - x oi t o2i ). In a similar manner one can obtain the expressions for the determination of xi è zo for the variant of the structure (Fig. 2a) without load impedances (jXldi = ¥). In this case the values of zo are computed as roots of cubic equation. In both situations the values of xei, xoi and the specified values of Ze, qei are the initial data for the calculations. Since in this case wave impedance of the odd mode Zo is unknown, then it is assumed that qei = qoi = qi and tei = toi = ti is substituted into (8), (9). Implementation of Dual-Band Operation Mode In order to operate in dual-band mode the designed device must provide the specified values of operational parameters and corresponding values of input reactances of partial networks xei, xoi in two frequency bands with center values fi, i = 1, 2. Two situations are possible in this case: – the values of input impedances are different at frequencies f1 and f2, i.e. xe2 ¹ xe1, xo2 ¹ xo1; – the values of input impedances are equal at frequencies f1 and f2, i.e. xe2 = xe1, xo2 = xo1. In the former case one needs to utilize the full variant of the structure (Fig. 2a) with two pairs of additional reactive elements. In this instance electrical lengths qei, qoi are calculated at each operation frequency fi based on specified value of section’s length l and on values bei, boi, which correspond to given values of Ze,

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Zo. Using them two pairs of impedances X1, X2 and Xld1, Xld2, which are further implemented as sections of transmission lines, are computed by formulas (6), (7). In the latter case of equal values of input impedances the solution finding is performed in the same way. However, for some problems one can use the approach of dual-band devices designing, that is connected with the reversal of sign of input reactive impedances at the second operation frequency [20]. It follows from (1) that at the reversal of sign of reactive components of input impedances of partial one-port networks (i.e. in the case of transition to complex conjugate values) the values of magnitudes of wave parameters of two-port network remain the same, only the sign of their arguments changes. The result will be the same in the situation of reciprocal exchange of complex conjugate values of input impedances. For a variety of devices such alterations of wave parameters do not lead to the variation of its operational parameters over another frequency band. For the network depicted in Fig. 2a it is seen from equations (6) that at reversal of signs of reactances xei, xoi the equations will hold only at reversal of signs of tei, toi and at the invariance of value of reactive impedance xi, but with an opposite sign. Consequently, if signs of tei, toi, and xi change in case of transition to f2, then this leads to the reversal of signs of xei, xoi, and so to abovementioned variations of wave parameters of devices. From equality te,o2 = –te,o1 it follows that q e,o2 / 2 = np - q e,o1 / 2, where n = 1, 2, 3, …. On the other hand, q e,o2 = (b e,o2 / b e,o1 )q e,o1. Taking into account the closeness in the case of striplines with dispersion of phase constants ratio to frequency ratio kf = f2/f1, where f1 < f2, it can be accepted that q e,o2 = k f q e,o1. Then we obtain the following formula for the electrical length of section at the frequency f1: q e,o1 = 2np / ( k f + 1).

(10)

It follows from (10) that the approach, which consists in the reversal of signs of input reactances at the second operation frequency, can be utilized only in the case of approximation qei = qoi = qi, which means tei = toi = ti. In this instance as a result of calculations using abovementioned relations we get two pairs of impedances X1, X2 = –X1 and Xld1, Xld2 = –Xld1, which are needed to be implemented further. The implementation of dual-band reactive elements with the values of input reactances equal to X1, X2 most often is realized by using stubs in the form of one or several transmission lines sections, which are open- or short-circuited at the end. If we utilize one section, then its electrical length q1 at the frequency f1 is determined by means of iterations from the expression [20]: tan(q 1 ) / tan( k f q 1 ) = a,

(11)

where a = X2/X1 for the open-circuited section and a = X1/X2 for the short-circuited section. If X2 = –X1, then in order to compute the electrical length q1 one uses (10), in which results (11) if a = –1. Wave impedance of the section is calculated using known relations based on input impedance X1 and its electrical length q1. Next, we will consider examples of the designing of dual-band devices based on coupled transmission lines with cross-symmetrical loads. PHASE SHIFTER WITH FIXED PHASE DIFFERENCE A device with fixed phase delay must be matched at the inputs (S11 = S22 = 0), has minimal losses and provide specified value j21 of phase shift of the output signal. It follows from (1) that the matching of symmetric two-port network is reached when zeizoi = 1. Then from (1) we obtain the transmission coefficient of two-port network: z - 1 1 - z oi . = S 21i = ei z ei + 1 1 + z oi

(12)

Based on (12) we get the expressions for the calculation of active rei and reactive xei components of input impedance zei using the specified value of |S21i| and required phase shift j21i:

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537

j21, deg.

S11 , dB –10

100

–20

0 2

j21

–30

1 –100

–40 0.5

1

1.5

2

–200 f, GHz

Fig. 4.

rei = (1 - | S 21i | 2 ) / B ei , x ei = 2 | S 21i |sin j 21i / B ei ,

(13)

where B ei = 1 + | S 21i | 2 - 2 | S 21i |cosj 21i . If the symmetric two-port network is purely reactive, then the matching condition and obtained from (13) expression for the calculation of input reactive impedance xei takes the following form: x ei x oi = -1, x ei = sin j 21i / (1 - cos j 21i ).

(14)

The value of xoi is determined from the matching condition. Therefore, the calculation of fixed phase shifter is realized by the following approach: using (14) for the frequencies f1 and f2 and basing on the specified values of phase delay j21i we compute the values of input reactances, which are used for the determination of values of additional reactive elements by formulas (6)–(9), that are implemented by the sections of transmission lines with parameters calculated by (10), (11). By way of example, in Fig. 4 we present the results of simulation of frequency characteristics for a microstrip fixed phase shifter, which provides phase delay j21i = +90°/–90° at operation frequencies fi = 0.95/2.15 GHz, and that can be used as the element of dual-band hybrid ring. Calculated by formulas (6), (7) electrical parameters of the circuit in case of implementation on a substrate with e = 2.1 having thickness 1.45 mm with specified values of wave impedances Zw = 70.71 W, Ze = 124 W, Zo = 62.26 W (bei = 0.0265/0.0601 rad/mm, boi = 0.025/0.0567 rad/mm) are as follows: qe1 = 110° (l = 72.39 mm), Xi = –32/32.18 W (the implementation is performed using an open-circuited section with wave impedance Z = 46.05 W and with electrical length, which equals q1 = 55.2° according to (11)), Xldi = –46.73/60.24 W (the implementation is realized using an open-circuited section with wave impedance Z = 72.79 W, q1 = 57.3°). Within the operation frequency bands, whose width at the matching level –15 dB equals to 100/80 MHz, phase response (curve 1) remains linear and it has values j21i = 88.5°/–91.4° at specified frequencies. Deviation from linearity at the edges of bands is caused by the influence of stubs at the center of a section (reactances jXi). This is confirmed by the curve 2, which is obtained for a similar phase shifter without stubs. Calculated by formulas (8)–(10) parameters of this device at specified values Zw = 70.71 W, Ze = 90 W and n = 1 are as follows: Zo = 24.05 W, q1 = 110.32°, Xldi = –14.06/14.06 W (the implementation is realized using an open-circuited section with wave impedance Z = 20.2 W, q1 = 55.17°). The disadvantage of such variant of the circuit consists in the fact that the result of the calculations is a low value of wave impedance Zo, which is difficult to implement physically.

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PRUDYUS, OBORZHYTSKYY

S11 , dB 2 1 –10 3 –20

–30 2.2

2.7

3.2

3.7

f, GHz

Fig. 5.

Fig. 6.

IMPEDANCE TRANSFORMER For the transformation of load impedance Zldi into specified value of input impedance Zini at two operation frequencies f1, f2 by a symmetrical reactive two-port network based on the section of coupled lines it is needed that at each frequency input reactances xei, xoi of its partial one-port networks with even-odd excitation are equal to the values calculated using (3). Based on these values, electrical parameters of circuit elements are determined using formulas (6)–(9) given hereinbefore. Frequency responses of reflection coefficients for two variants of transformation are presented in Fig. 5. Curve 1 refers to the circuit, that transforms at frequencies 2.4/3.9 GHz a complex load impedance, which is formed by the section of 50 W line with 100 W resistor at the end and which equals 28.7 – j15.9/28.5 + j11.4 W at these frequencies, into the input impedance equal 50 W. In this case calculated by (3) values of input reactances for Zini = 50 W are Xe = 84.6/16.2 W, Xo = –9.95/–69.25 W. Calculated by formulas (6), (7) electrical parameters of the circuit implemented on a substrate with the same parameters as those in the case of phase shifter with the same values of wave impedances, for which phase constants at specified operation frequencies equal bei = 0.0671/0.1093 rad/mm, boi = 0.0632/0.10276 rad/mm, are as follows: qe1 = 176° (l = 45.76 mm), Xi = –123.76/–48.26 W (the implementation is performed using an open-circuited section with Z = 103.8 W, q1 = 40°), Xldi = –25/118.85 W (the implementation is realized using an open-circuited section with Z = 141.72 W, q1 = 80°). A low level of matching at higher frequency band in case of transformation of complex impedances, that is observed on the curve 1, can be put to the disadvantages indicated for the investigation of such devices. Curves 2, 3 refer to the circuit, which provides the matching of active load impedance 100 W with wave impedance of feeder line Zw = 50 W at the same operation frequencies. As it follows from (3), in case of such transformation of active impedance by reactive symmetrical two-port network the values of input reactances at frequencies f1, f2 must be equal xei = –xoi = 1.414. To compute the parameters of the network one may use the approach connected with the reversal of the sign of input reactive impedances at the second operation frequency, because in this case load impedance is transformed into complex conjugate input impedance, which is purely active in this situation. Calculated by (6), (7) electrical parameters of the network with the same substrate parameters for the values of wave impedances Ze = 123.3 W, Zo = 61.8 W and for the electrical length, that equals q1 = 137.14° according to (10), are as follows: Xi = –51.8/51.8 W (the implementation is performed using an open-circuited section with Z = 132 W, q1 = 68.57°), Xldi = –63/63 W (the implementation is realized using an open-circuited section with Z = 160.57 W, q1 = 68.57°). The results of circuit simulation of such structure are shown in Fig. 5 (curve 2). Curve 3 represents frequency response of the device, that is obtained as result of electromagnetic simulation with some correction of dimensions, which provides the compensation of the influence of circuit layout irregularities (bends, dividers, variations of strip width). Marked by triangles (Fig. 5) results of measurement of reflection coefficient’s magnitude at the input of a transformer, photograph of prototype of which is presented in Fig. 6, are close to the simulation results. The advantage of the network (Fig. 2a), which is used as the impedance transformer, is the possibility of simultaneous direct current blocking for two frequency bands, that allows one to avoid the utilization of discrete capacitors in designing of active microwave hybrid integrated circuits. RADIOELECTRONICS

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539

S ij , dB –10 S 21 –20 S11

–30 –40 2.0

2.5

3.0

3.5

4.0

4.5

5.0

f, GHz

Fig. 7.

Fig. 8.

BANDPASS FILTER Bandpass filters with parallel coupled half-wavelength resonators usually represent a series connection of coupled lines sections, cross-opposite ends of which are open- (Fig. 1a) or short-circuited. In order to implement the dual-band operation of the filter, the structure of the from presented in Fig. 2a has been used. Filter designing is based on the equivalent substitution technique, according to which one chooses the basic variant of single-band filter with parallel coupled half-wavelength resonators, which provides specified operational parameters (bandwidth, attenuation levels) in the first operation frequency band with center frequency f1. Electrical parameters of such filter’s elements are calculated using known classic techniques [10, 16]. Next, the values of input reactances xe, xo of partial one-port networks with even-odd excitation are determined for each of constituent resonator segments in the form of a section of coupled transmission lines with open- or short-circuited cross-opposite ends. The formulas for the calculation of these parameters, which are obtained from (5) and substituting (4) in it and taking into account that jXi = ¥ at jXldi = {0, ¥} have the following form: x e = ( z ¢o t e t o - z ¢e ) / (2t e ), x o = ( z ¢e t e t o - z ¢o ) / (2t o )

(15)

for jXldi = ¥ and x e = 2 z ¢e z ¢o t o / ( z ¢e - z ¢o t e t o ), x o = 2 z ¢e z ¢o t e / ( z ¢o - z ¢e t e t o ) ¢ / Z w denotes wave impedances for even and odd modes of base section of for jXldi = 0, where z ¢e,o = Z e,o coupled transmission lines, which are normalized to wave impedance Zw of the lines at filter inputs; t e,o = tan(q ¢e,o / 2), q ¢e,o designates the electrical length of base section of coupled transmission lines for even and odd modes at the frequency f1. Obtained by (15) values of reactances xe, xo of each section of the filter prototype and their wave impedances are the initial data for the calculation of elements’ parameters of equivalent resonator segments ¢ (Fig. 2a), which form the dual-band filter, using formulas (6)–(9). One can assume that Ze,o equals to Z e,o when performing calculations. Since only one value is obtained for each of input reactive impedances xe, xo of equivalent section using (15), then during the calculations one can utilize approach of sign reversal at f2 for these reactances in the case of approximation tei = toi = ti, when q1 is calculated using (10), and X2 = –X1, Xld2 = –Xld1. If we use the circuit of a section without reactive impedances jXi in the case of approximation tei = toi = ti and accept the electrical length of a section of coupled transmission lines at the frequency f1 equal to value

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q 1 = np / ( k f + 1), i.e. in two times shorter than the one calculated by (10), then the equality t2 = 1/t1 will hold at frequency f2. In this case it follows from the expressions for input reactances of partial one-port networks that reciprocal inversion of values xe, xo will occur at frequency f2 under the condition of sign reversal of reactances jXldi (Xld2 = –Xld1) and the input reactance of partial network with odd excitation will be equal to –xe, and the reactance of partial network with even excitation will be equal to –xo. As it was in the previous case, the value of transmission coefficient of the section keeps the same, only the sign of its phase reverses. Under such changes of transmission coefficients of filter sections its selective properties remain the same in the second operation frequency band. Frequency responses of dual-band filter, that was developed by the indicated technique, are presented in Fig. 7, and the photograph of its prototype is shown in Fig. 8. As a prototype filter for the calculations we used a single-band filter with Chebyshev response, which provides in 3% band with center frequency 2.4 GHz ripples of attenuation at the level of 0.1 dB and 20 dB attenuation at the frequency 2.3 GHz. A filter with such response consists of three half-wavelength resonators (four segments of quarter-wave open-circuited at the ends sections of coupled transmission lines). The values of wave impedances of the first and of the fourth segments are equal to Ze = 60.47 W, Zo = 37.45 W, which for te = to = 1 (q ¢e = q ¢o = p/2) results in xe = –0.23, xo = 0.23 according to (15), and the values of wave impedances of the second and of the third segments are equal to Ze = 48.57 W, Zo = 43.03 W, which results in xe = –0.055, xo = 0.055. The results of simulation of frequency responses of wave parameters for filter prototype if it is implemented on a substrate with the parameters, that are similar to the ones from the previous examples, are shown in Fig. 7 by dashed lines. As a result of calculations of equivalent segments, for which a circuit without reactive impedances jXi and with shortened length q1 of sections has been chosen, the following values of parameters have been obtained for operation frequencies 2.4/5.2 GHz: for the first and the fourth segments Ze = 60.47 W, Zo = 32.97 W, Xldi = –71.51/71.51 W (the implementation is performed using an open-circuited section with Z = 109.46 W); for the second and the third segments Ze = 48.57 W, Zo = 41.95 W, Xldi = –69.28/69.28 W (the implementation is performed using an open-circuited section with Z = 106.04 W). The electrical length of all sections and stubs equals to q1 = 56.84°. The filter is designed on a substrate with the same parameters as those in the previous devices. Points in Fig. 7 mark the results of measurements of wave parameters magnitudes of a prototype, which are close to the results of filter simulation. As we can see from presented curves, such variant of the structure has additional bandpass in the range of central frequency f0 = (f1 + f2)/2. Its occurrence is associated with the fact that at this frequency the electrical length of the sections and stubs becomes equal to p/2 and the network operates as usual filter with parallel coupled half-wavelength short-circuited resonators. Besides, the higher bandpass of this filter is narrower than the lower one. The suppression of parasitic band and width equalization of operation bands can be obtained by using segments with different structure (containing reactances jXi with stubs short-circuited at the ends, with full length of coupled lines section). CONCLUSIONS It has been shown that a section of coupled striplines with cross-symmetrical loads can be utilized in the designing of various dual-band microwave devices. In this case the possibility to obtain the specified functional properties together with the direct current isolation between input and output is provided. The suggested equivalent representation of such section with even-odd excitation has allowed us to obtain simple analytical expressions for the calculation of electrical parameters of a section and additional elements connected to it based on the input reactive impedances of partial one-port networks, for which the relation with wave parameters of the designed device has been determined. A dual-band phase shifter with fixed phase difference, impedance transformer and bandpass filter have been simulated based on the relations obtained. Reliability of the calculations has been confirmed by the results of simulation and experimental investigation. REFERENCES 1. Myun-Joo Park, “Dual-band Wilkinson divider with coupled output port extensions,” IEEE Trans. Microwave Theory Tech. 57, No. 9, 2232 (Sept. 2009), DOI: 10.1109/TMTT. 2009.2027169. 2. Z. Lin, Q.-X. Chu, “A novel approach to the design of dual-band power divider with variable power dividing ratio based on coupled-lines,” PIER 103, 271 (2010), DOI: 10.2528/PIER10012202.

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3. Yongle Wu, Yuanan Liu, Quan Xue, “An analytical approach for a novel coupled-line dual-band Wilkinson power divider,” IEEE Trans. Microwave Theory Tech. 59, No. 2, 286 (Feb. 2011), DOI: 10.1109/TMTT.2010.2084096. 4. B. Li, X. Wu, N. Yang, W. Wu, “Dual-band equal/unequal Wilkinson power dividers based on coupled-line section with short-circuited stub,” PIER 111, 163 (2011), DOI: 10.2528/PIER 10110108. 5. I. N. Prudyus, V. I. Oborzhytskyy, “A simulation technique for the dual-band equal-path power dividers based on coupled transmission lines,” in Proc. of 21st Int. Crimean Conf. on Microwave Engineering and Telecommunications Technologies, CriMiCo, 12–16 Sept. 2011, Sevastopol, Ukraine (Veber, Sevastopol, 2011), Vol. 2, pp. 612–613. 6. L. K. Yeung, “A compact dual-band 90° coupler with coupled-line sections,” IEEE Trans. Microwave Theory Tech. 59, No. 9, 2227 (Sept. 2011), DOI: 10.1109/TMTT.2011.2160 199. 7. M.-J. Park, “Comments on “A compact dual-band 90° coupler with coupled-line sections” IEEE Trans. Microwave Theory Tech. 60, No. 1, 201 (Jan. 2012), DOI: 10.1109/TMTT.2011. 2175243. 8. V. Oborzhytskyy, O. Samsonyuk, “Design of dual-frequency TEM-mode coupled-line directional couplers,” in Proc. of XIth Int. Conf. on Modern Problems of Radio Engineering, Telecommunications and Computer Science, TCSET’2012, 21–24 Feb. 2012, Lviv-Slavske, Ukraine (IEEE, 2012), p. 179, INSPEC: 12725539. 9. Xi Wang, Wen-Yan Yin, Ke-Li Wu, “A dual-band coupled-line coupler with an arbitrary coupling coefficient,” IEEE Trans. Microwave Theory Tech. 60, No. 4, 945 (Apr. 2012), DOI: 10.1109/TMTT.2012.2185949. 10. G. L. Matthaei, Leo Young, E. M. T. Jones, Microwave Filters, Impedance-Matching Networks, and Coupling Structures, Vol. 2 (McGraw-Hill, New York, 1964). 11. S. March, “A wideband stripline hybrid ring,” IEEE Trans. Microwave Theory Tech. 16, No. 6, 361 (June 1968), DOI: 10.1109/ TMTT.1968.1126693. 12. A. Podcameni, “Symmetrical and asymmetrical edge-coupled-line impedance transformers with a prescribed insertion loss design,” IEEE Trans. Microwave Theory Tech. 34, No. 1, 1 (Jan. 1986), DOI: 10.1109/TMTT.1986.1133273. 13. T. Jensen, V. Zhurbenko, V. Krozer, P. Meincke, “Coupled transmission lines as impedance transformer,” IEEE Trans. Microwave Theory Tech. 55, No. 12, 2957 (Dec. 2007), DOI: 10.1109/TMTT.2007.909617. 14. Hee-Ran Ahn, T. Itoh, “Impedance-transforming symmetric and asymmetric dc blocks,” IEEE Trans. Microwave Theory Tech. 58, No. 9, 2463 (Sept. 2010), DOI: 10.1109/TMTT. 2010.2058936. 15. G. I. Zysman, A. K. Johnson, “Coupled transmission line networks in an inhomogeneous dielectric medium,” IEEE Trans. Microwave Theory Tech. 17, No. 10, 753 (Oct. 1969), DOI: 10.1109/TMTT.1969.1127055. 16. G. L. Matthaei, Leo Young, E. M. T. Jones, Microwave Filters, Impedance-Matching Networks, and Coupling Structures, Vol. 1 (McGraw-Hill, New York, 1964). 17. A. L. Feldstein, L. R. Yavich, Synthesis of Microwave Two-Port and Four-Port Networks (Svyaz’, Moscow, 1971) [in Russian]. 18. V. I. Oborzhytskyy, “Utilization of symmetry singularities of linear high-frequency devices in their synthesis techniques,” Digest of Scientific Papers of IAME. Ser. Simulation and Information Technologies 29, 129 (2005). 19. V. I. Oborzhytskyy, “Approach of calculation of parameters of symmetrical high-frequency two-port network in the problems of impedance transformation,” Digest of Scientific Papers of IAME. Ser. Simulation and Information Technologies 34, 131 (2005). 20. I. N. Prudyus, V. I. Oborzhytskyy, “Design principles of analytical methods for calculation of dual-band strip directional couplers with full structure symmetry,” Izv. Vyssh. Uchebn. Zaved., Radioelektron. 57(4), 19 (2014), http://radio.kpi.ua/article/view/S0021347014040025 [Radioelectron. Commun. Syst. 57(4), 159 (2014)], http://radioelektronika.org/article/view/S0735272714040025, DOI: 10.3103/S0735272714040025.

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