Chapter 4 Basic Balun Introduction
5.2 Balun Design Procedure
5.2.2 Mathematical Model Establishment of the Novel Balun
A proposed novel broadband fifth-order balun is an essential portion of this thesis. It is the extension based on the conventional Marchand balun prototype. The original distributed model is shown in Figure 5.6.
Z
2Figure 5.6 Fifth-order distributed Marchand balun prototype.
The original two-port form can be obtained by combining port-2 and port-3 in series connection. As shown in Figure 5.7(a), the two characteristic impedance of transmission-line Z3 and load impedance R are replaced with the new element values, 2Z3 and 2R, respectively. There are six transmission-line sections denote Z1, Z2, 2Z3, ZC, ZL1 and ZL2, which form the complete balun network. To satisfy the condition of exact synthesis theory, the complete network is all composed of commensurate lines and has no redundant element. In other words, the first necessary step for the exact
synthesis theory is to establish an ideal optimum multi-pole network form. The original equivalent circuit of the fifth-order Marchand balun in S-domain can be derived from Richard’s transformation and it is shown in Figure 5.7(b).
Z
22Z
3Z
C2
Z
L 1Z
LZ
11 PORT
2 PORT
Z
O2R
(a)
C
L
1Z
2Z
1Z
OL
22Z
32R
(b)
Figure 5.7 Original two-port form of fifth-order Marchand balun. (a) Distributed prototype. (b) Richard’s domain lumped element prototype.
The original equivalent network shown in Figure 5.7(b) can be derived to the optimum multi-pole form by using Kuroda’s identities [9] and some other equivalent circuits. The complete design procedures and equivalent network diagrams are
presented as follows.
(1) In Figure 5.8(a), based on the Kuroda’s identities, the inductor L1 can be transposed to right side of the UE Z2 by adding a transformer and introducing a scaling factor, denotes by n.
2 1
1 Z
n= + L (5.14) (2) Because of the practical consideration of implementation, the transformer must be moved to the last of network. The inductor L2 is also adjusted by n. The related diagram is shown in Figure 5.8(b).
(3.) The final equivalent model shown in Figure 5.8(c) is obtained by shunting the middle two inductors and absorbing transformer into load impedance. The new element values are expressed as
' 2
C
Figure 5.8 Equivalent circuits in S-domain.
After accomplishing the equivalent circuit model, the overall ABCD-matrix and squared magnitude of transmission coefficient can be easily derived as follows.
'
As discussed in early Chapter, the transfer function is analytically derived and regulated to exhibit the fifth-order Chebyshev equal-ripple frequency responses. The squared S21-magnitude can be expressed as
2 eight, six, four, square and zero, respectively. They are functions of the ripple-level ε and cutoff frequency S in Richard’s domain, which determine the return loss C performance and bandwidth of the balun. The coefficients can be expressed as
0, , , , ,2 4 6 8 10 ( , C)
F F F F F F = f ε S (5.21) According to the Equation (5.22), some numerical solutions will be obtained by using the method of coefficient comparison.
' 0, 2, 4,..., 2 ,...
N N
F =F N = n (5.22) In fifth-order case, there are only six equations but eight unknown parameters they are ε , S , C Z , 1 Z , 2' Z , 3' L , C and ' R . In common balun design, the ' impedance transformation ratio has been determined and usually equal to two. It means that all ports in balun have the same port impedance. In addition, the ripple-level and cutoff frequency are also the important parameters of the balun specifications and they are determined in advance. Because there are two degrees of freedom in the proposed network, two of the three key parameters can arbitrarily determined which we are most concerned about.
5.3 Simulation and Measurement Comparison
The circuit models, full-wave EM simulations and measured results are illustrated in this section. The detailed comparison of the parameters will be listed in established table later. In this thesis, based on most commonly used 50 to 100Ω impedance transformation ratio, we give two models with different Richard’s cutoff frequencies which lead to different bandwidth of baluns in real frequency domain.
When impedance transformation ratio and cutoff frequency are fixed, the other six unknown parameters including ripple-level and five element values can be solved to obtain some reasonable solutions. Finally, the original fifth-order balun network prototype is perfectly established.
To achieve better electromagnetic (EM) simulation results and faster simulation speed, we prefer to use the EM simulation tool HFSS than the Sonnet. In addition, to deal with the balun network which includes defected-ground-structure (DGS) configuration, HFSS will contribute better accuracy and save more simulation time compared to the Sonnet. The better simulation results can be achieved from reasonable adjustment and well-tuning procedures.
For the practical consideration of implementation, all of the following networks are fabricated on the RT/Duroid 6010 substrate with high permittivity εr =10.2 and thickness h=25mil. To simplify the calculation of the simultaneous equations, the load impedance (R ) of two-port equivalent circuit is set to be unity. The scaling ' factor (n) can be solved from the Equation (5.14) and it is equal to 1.1892. The detailed circuit parameters of these two equivalent and original networks are tabulated in Table 5.1.
'
Table 5.1 Theoretical prototype and circuit parameters with different cutoff frequency, 0.6 and 0.4. (θ =90 @0 fO and R'= ) 1
Based on the exact design procedures, arbitrary center frequency of the presented balun network can be approached by checking the prototype values listed in Table 5.1.
There are several solutions calculated from simultaneous equations, but most of them are unreasonable or unrealizable. Based on the criteria that all of the reasonable parameters must be real and positive numbers, the solutions can be exactly determined. Figure 5.9 depicts the schematic layout of the proposed fifth-order Marchand balun, where Z , 1 Z and 3 Z are microstrip lines, C Z is slotline, and 2
1
Z and L ZL2 are coplanar striplines. All of the transmission-line sections are equal to quarter-wavelength at the center frequency in the proposed balun.
_ top layer gnd
Z
1Z
3Z
3Z
CZ
2 1Z
LZ
L2/ 6010
10.2 25
r
RT Duroid
h mil
ε
==
Figure 5.9 Schematic layout of the fifth-order Marchand balun.
As illustrated in Figure 5.9, the microstrip line is formed on one side of the substrate, while the slotline and coplanar stripline are etched on the opposite side that is perpendicular with the microstrip line. The practical network topologies and physical dimensions of the fifth-order balun with SC = j0.6 are shown in Figure 5.10 and Table 5.2. The fabricated network photos include both front side and back side are shown in Figure 5.11. The series open stub is constructed by microstrip line with an open-circuited end, while the shunt short stubs are all formed by coplanar striplines on the ground. Compared to the slotline, the coplanar stripline structure shows less dispersive characteristic and produces less radiation loss. Furthermore, to compensate microstrip-to-slotline junction effect, all the lengths of microstrip lines and slotline are slightly increased. In addition, we should also increase the widths of microstrip lines due to the defected-ground-structure (DGS) configuration.
1
Figure 5.10 Practical systematic layout of the fifth-order balun withSC = j0.6.
L1 L2 L3 L4 L5 L6
Table 5.2 The physical dimensions of the proposed fifth-order balun with SC = j0.6 corresponding to the layout Figure 5.10. (in mil)
Figure 5.11 Fabricated layout of the fifth-order balun withSC = j0.6.
Figure 5.12 illustrates the S-paraters with magnitude and phase responses of measured, circuit model, and EM simulated results. The simulations are obtained from the circuit model and full-wave EM model by the ADS and HFSS simulation tools, respectively. The magnitude/phase imbalances are shown in Figure 5.12(b) and (c).
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8
1.0 1.3 1.6 1.9 2.2 2.5 2.8 3.1
0.7 3.3
-0.4 -0.2 0.0 0.2 0.4
-0.6 0.6
freq, GHz dB(S(7,8))-dB(S(9,10))
dB(S(5,4))-dB(S(6,4))
(b)
1.0 1.3 1.6 1.9 2.2 2.5 2.8 3.1
0.7 3.3
176 178 180 182 184
174 186
freq, GHz mag(phase(S(7,8))-phase(S(10,9)))
mag(phase(S(5,4))-phase(S(6,4)))
(c)
Figure 5.12 Fifth-order Marchand balun responses withSC = j0.6. (a) Predicted and measured S-parameters. (b) Magnitude imbalance. (c) Phase imbalance.
In this case, f is chosen to be 2GHz. The fractional bandwidth is 0 approximately equal to 131% and it can be easily obtained by the Richard’s transformation. The theoretical passband begins from 0.688GHz to 3.312GHz in real frequency domain. Now, we focus on the magnitude / phase imbalances over the frequency range. They are less than 0.1dB and 1.2D for full-wave EM simulated results and less than 0.5dB and 2.5D for measured results. The detailed comparisons of the specific parameters of circuit model, EM model and practical measurement are tabulated in Table 5.3. There are many uncontrollable factors, such as the frequency dispersion of the non-TEM slotline, the frequency dependent effective permittivity, the misalignment of the microstip / slot line junction, and the dimension tolerance in etching process.
Table 5.3 Comparisons for fifth-order balun with SC = j0.6
The practical network topologies and physical dimensions of the fifth-order balun with SC = j0.4 are shown in Figure 5.13 and Table 5.4. The ZL2 section is omitted because the equivalent circuit value is much larger than the intrinsic impedance in free-space, which is equal to 377Ω. The fabricated network photos are
shown in Figure 5.14.
14 L
Figure 5.13 Practical systematic layout of the fifth-order balun with SC = j0.4.
L1 L2 L3 L4 L5 L6 118 50 134 132 150 100
L7 L8 L9 L10 L11 L12
270 180 120 180 250 150 L13 L14 W1 W2 W3 W4
610 850 23 16 80 30
W5 W6 G1
10 20 40
Table 5.4 The physical dimensions of the proposed fifth-order balun with SC = j0.4 corresponding to the layout Figure 5.12. (in mil)
Figure 5.14 Fabricated layout of the fifth-order balun withSC = j0.4.
Figure 5.15 depicts the S-parameters with magnitude and phase responeses of the fifth-order balun with SC = j0.4. The circuit and full-wave EM prototypes are simulated from the ADS and HFSS simulation tools, respectively.
0.5 1.0 1.5 2.0 2.5 3.0 3.5
1.0 1.3 1.6 1.9 2.2 2.5 2.8 3.1
0.7 3.3
-0.2 0.0 0.2 0.4 0.6 0.8
-0.4 1.0
freq, GHz
dB(S(7,8))-dB(S(9,10)) dB(S(5,4))-dB(S(6,4))
(b)
1.0 1.3 1.6 1.9 2.2 2.5 2.8 3.1
0.7 3.3
176 178 180 182 184
174 186
freq, GHz mag(phase(S(7,8))-phase(S(10,9)))
mag(phase(S(5,4))-phase(S(6,4)))
(c)
Figure 5.15 Fifth-order Marchand balun responses withSC = j0.4. (a) Predicted and measured S-parameters. (b) Magnitude imbalance. (c) Phase imbalance.
In this case, f is chosen to be 2GHz. The fractional bandwidth is 0 approximately equal to 152% and the theoretical passband in real frequency begins from 0.484GHz to 3.516GHz. The magnitude/phase imbalances are less than 0.7dB and 0.3D for full-wave EM simulated results and less than 1.0dB and 4.0D for measured results. The detailed comparisons of the specific parameters of circuit model, EM model and practical measurement are tabulated in Table 5.5.
Synthesis EM Simulation
Table 5.5 Comparisons for fifth-order balun with SC = j0.4
5.4 Conclusion and Future Work
An exact broadband synthesis method has been systematically presented in this paper. General transfer functions derived from the equivalent circuits were formulated to realize the specific responses in a Butterworth or Chebyshev sense and to explicitly determine the normalized values of equivalent lumped elements in Richard’s domain under the required specifications. The practical distributed networks in the form of quarter-wavelength lines are easily derived from the equivalent models. The fifth-order Marchand balun is firstly introduced by the synthesis approach and its responses are exactly described in the Chebyshev polynomials. Practical networks are
realized on hybrid microstrip line, slotline and coplanar stripline structures.
Furthermore, the higher order model of balun can be obtained by increasing the non-redundant elements such as shunt open-circuited stubs, series short-circuited stubs or cascade transmission-lines. For example, the equivalent circuit model of sixth-order balun is described in Figure 5.16 and its normalized element values can be also derived by using the same procedures as previous analyses. Figure 5.17 depicts the equivalent optimum multi-pole form of the sixth-order balun and its corresponding distributed network. The higher order model can be obtained by repeating the unit set which composed of a shunt inductor and a unit element, or directly cascading several unit elements as shown in Figure 5.17(a). The series open-circuited stub fabricated by the quarter-wavelength open-end slotline can also be introduced to increase the order of balun. However, the structure is seldom used because of the serious radiation effect in slotline and it is also difficult to be realized on limited area of ground plane. Many other applications such as bandpass filters, bandstop filters and balun filters can be also definitely described and designed by the proposed exact synthesis approach. In addition, utilizing the broadband characteristic of the proposed balun, a wideband magic-T might be obtained. Moreover, the magic-T may be derived the complete mathematical solutions and equivalent circuit models for the required specifications as same as the exact synthesis process of balun.
Furthermore, to achieve smaller network volume and better spurious responses, each transmission-line sections in the proposed balun network can be replaced with stepped-impedance resonator (SIR) structure.
Z
2(b) Original lumped prototype. (c) Approximate responses.
C
Figure 5.17 Sixth-order or higher order balun.
(a) Lumped prototype. (c) Distributed prototype.
References
[1] S. B. Cohn,“Design considerations for high-power microwave filters,”IRE Trans. on Microwave Theory and Techniques, vol. MTT-7, pp. 149-153, Jan. 1959.
[2] L. Young,“Peak interval fields in direct-coupled-cavity filters,”IEEE Trans.
Microwave Theory and Techniques, vol. 8, no. 6, pp. 612-616, Nov. 1960.
[3] E. N. Torgow and G. E. Collins,“Band-stop filters for high-power applications,”
IEEE Trans. Microwave Theory and Techniques, vol. MTT-13, pp. 508-513, Jan.
1965.
[4] A. R. Harish and R. J. Cameron,“Peak voltage analysis in high power microwave filters,”IEE Microwave Filters and Multiplexers Colloquium, pp. 10/1-10/5, Nov.
2000.
[5] C. Ernst, V. Postoyalko, and N. Khan,“Relationship between group delay and stored energy in microwave filters, ” IEEE Trans. Microwave Theory and Techniques, vol. 49, no. 1, pp. 192-196, Jan. 2001.
[6] A. Sivadas, M. Yu, and R. J. Cameron,“A simplified analysis for high power microwave bandpass filter structures,”in Proc. Int. Symp. IEEE Microwave Theory and Techniques, Boston, MA, vol. 3, pp. 1771-1774, Jun. 2000.
[7] M. Yu,“Power-handling capability for RF filters,”IEEE Microwave Magazine, pp. 88-97, Oct. 2007.
[8] G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters, Impedance Matching Networks, and Coupling Structures, New York, McGraw-Hill, 1964.
[9] D. Pozar, Microwave Engineering, 3’rd edition, New York, John Wiley & Sons, 1998.
[10] J. P. Kim and W. S. Park, “Novel configurations of planar multilayer Magic-T using microstrip-slotline transitions,”IEEE Trans. Microwave Theory and Techniques, vol. 50, no. 7, pp. 1683-1688, Jul. 2002.
[11] K. U-yen, E. J. Wollack, J. Papapolymerou, and J. Laskar,“A broadband planar Magic-T using microstrip-slotline transitions, ” IEEE Trans. Microwave Theory and Techniques, vol. 56, no. 1, Jan 2008.
[12] K. U-yen, E. J. Wollack, J. Papapolymerou, and J. Laskar,“A compact low-loss planar Magic-T using microstrip-slotline transitions,”in IEEE MTT-S Int.
Microw. Symp. Dig., Honolulu, HI, pp. 37-40, Jun. 2007.
[13] K. U-yen, E. J. Wollack, S. Horst, T. Doiron, J. Papapolymerou, and J. Laskar,
“ Slotline stepped ciucular rings for low-loss microstrip-to-slotline transitions,”IEEE Microw. Wireless Compon. Lett., vol. 17, no. 2, pp. 100-102, Feb. 2006.
[14] K. S. Ang and Y. C. Leong,“Converting balun into broadband impedance transforming 180o hybrid,”IEEE Microwave Theory and Techniques, vol. 50, no. 8, pp. 1990-1995, Aug. 2002.
[15] G. J. Laughlin,“A new impedance-matched wideband balun and Magic-T,”
IEEE Microwave Theory and Techniques, vol. MTT-24, no. 12, pp. 523-528, Dec. 1980.
[16] W. K. Roberts,“A new wide-band balun,”Proc. IRE, vol.45, pp. 1628-1631, Dec. 1957.
[17] G. Oltman,“The compensated balun,”IEEE Microwave Theory and Techniques, vol. MTT-14, pp. 112-119, Mar. 1966.
[18] N. Marchand,“Transmission line conversion transformers,”Electronics, vol. 17, pp. 142-145, Dec. 1944.
[19] K. S. Ang and D. Robertson,“Analysis and design of impedance transforming planar Marchand baluns,”IEEE Microwave Theory and Techniques, vol. 49, no.
2, pp. 402-406, Feb. 2001.
[20] J. H. Cloete,“Exact design of the Marchand balun,”Microwave Journal, vol.
23, no. 5, pp. 99-103, May 1980.
[21] S. B. Cohn,“Slot line on a dielectric substrate,”IEEE Microwave Theory and Techniques, vol. MTT-17, pp. 768-778, Oct. 1969.
[22] E. A. Mariani, “ Slotline characteristics, ” IEEE Microwave Theory and Techniques, vol. MTT-17, pp. 1091-1096, Dec. 1969.
[23] B. Schuppert,“Microstrip / Slotline transitions : modeling and experimental investigation,”IEEE Microwave Theory and Techniques, vol. 36, no. 8, pp.
1272-1282, Aug. 1988.
[24] R. Janaswamy and D. H. Schaubert,“Characteristic impedance of a wide slotline on low-permittivity substrates, ” IEEE Microwave Theory and Techniques, vol. MTT-34, no. 8, pp. 900-902, Aug. 1986.
[25] E. A. Mariani, C. P. Heinzman, J. P. Agrios, and S. B. Cohn,“Slot line characteristics,”IEEE Microwave Theory and Techniques, vol. MTT-17, on. 12, pp. 1091-1096, Dec. 1969.
[26] K. C. Gupta, R. Garg, I. Bahl, and P. Bhartia, Microstrip Lines and Slotlines, 2’nd edition, Boston, Artech House, 1979.
[27] P. I. Richards,“Resistors transmission-line circuits,”Proc. IRE, vol. 36, pp.
217-220, Feb. 1948.
[28] H. J. Carlin and W. Kohler,“Direct synthesis of bandpass transmission line structures, ” IEEE Microwave Theory and Techniques, vol. MTT-13, pp.
283-297, May 1965.
[29] R. J. Wenzel,“Exact design of TEM microwave networks using quarter-wave lines,”IEEE Microwave Theory and Techniques, vol. MTT-12, pp. 94-111, Jan 1964.
[30] M. C. Horton and R. J. Wenzel,“General theory and design of optimum quarter-wave TEM filters,”IEEE Microwave Theory and Techniques, vol.
MTT-13, pp. 316-327, May 1965.
[31] H. Ozaki and J. Ishii,“Synthesis of transmission-line networks and the design of UHF filters,”IRE Trans. on Circuit Theory, vol. CT-2, pp. 325-336, Dec. 1955.
[32] R. G. Garcia and J. I. Alonso,“Systematic method for the exact synthesis of ultra-wideband filtering responses using high-pass and low-pass sections,”
IEEE Microwave Theory and Techniques, vol. 54, no. 10, pp. 3751-3764, Oct.
2006.
[33] Z. H. Tu and Q. X. Chu,“Determining transmission zeros of general Chebyshev low-pass prototype filter, ” Microwave and Millimeter Wave Technology, ICMMT Int. Conf., Apr. 2007.
[34] R. J. Cameron,“General coupling matrix synthesis methods for Chebyshev filtering functions,”IEEE Microwave Theory and Techniques, vol. 47, no. 4, pp.
433-442, Apr. 1999.
[35] R. J. Wenzel,“Wideband high-selectivity diplexers utilizing digital elliiptic filters,”IEEE Microwave Theory and Techniques, vol. MTT-15, no. 12, pp.
669-680, Dec. 1967.
[36] R. J. Wenzel,“Application of exact synthesis methods to multi-channel filter design,”IEEE Microwave Theory and Techniques, vol. MTT-13, pp. 5-15, Jan.
1965.