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Arbitrary Dual-Band RLC Circuits Mohammad Khalaj-Amirhosseini, Mahmoud Moghavvemi and Aliyar Attaran Abstract— A new approach to design a dual-band RLC circuits is proposed. In this approach, a single-band circuit is designed and then converted to a dual-band circuit by substituting its inductors and capacitors for a proper parallel or series LC resonator. The validity of the proposed approach is verified by designing, implementing and measuring a dual-band RLC power divider at frequencies 10 and 20 MHz as well as a dual-band tank circuit. The proposed approach can be used for designing arbitrary dual-band circuits such as narrowband bandpass filters, bandstop filters, power dividers and combiners, phase shifters etc. Index Terms— Dual-Band Circuits; Frequency Scaling; Conversion of Elements; Power Divider; Tank Circuit

——————————  ——————————

1 INTRODUCTION

D

UAL-BAND circuits are advantageous in many modern electronic and communication systems having two frequency bands. The main approaches to achieve dual-band microstrip circuits are replacing each transmission line with a composite right/left-handed transmission line (CRLH) [1, 2] or cascading two single-band circuits [3-6]. The only approach to achieve dual-band RLC circuits is considering a suitable circuit with regard to its application and finding the values of the components. In this article, a new and fundamental approach is introduced to achieve an arbitrary dual-band RLC circuit. The process starts with design of an RLC single-band circuit which is then converted to a dual-band circuit by substituting its inductors and capacitors for a proper parallel or series LC resonator. Brief explanation of this approach is provided in the following sections. A dual-band power divider at frequencies 10 and 20 MHz is designed and constructed and an excellent agreement is obtained between the theoretical and practical results.

2 SINGLE-BAND TO DUAL-BAND CONVERSION If the reactance of each passive element (non-resistive) of an RLC circuit at frequency 2 is equal and opposite of that at frequency 1, a frequency scaling equal to 1 will exist between the responses of the circuit at these two frequencies. In this case the following relationship will be held. (1) H ( j2 )  H ( j1 )  H  ( j1 ) The relation (1) means that the amplitude of frequency response at two frequencies 1 and 2 are equal and consequently the circuit functions as a dual-frequency or as a narrowband dual-band circuit at two desired frequencies 1 and 2. The simplest passive elements which have 1 frequency scaling at two distinct frequencies are an inductor and a

capacitor in parallel or in series with each other, i.e. LC resonators. Fig. 1 depicts how the reactance of a parallel or series resonator at two frequencies 1 and 2 equals to +1 or 1 times of reactance of an inductor at frequency 1, respectively. Also, Fig. 2 depicts how the reactance of a series or parallel resonator at two frequencies 1 and 2 equals to +1 or 1 times of reactance of a capacitor at frequency 1, respectively. Tables 1 and 2 show the required values of resonator elements with +1/1 or 1/+1 frequency scaling at frequencies 1 and 2, respectively, where  = 2 1. It is seen that the resonance frequency of all resonators is equal to 12 which is the geometric average of two desired frequencies. According to either one of these tables, each inductor or capacitor in a single-band circuit is substituted for a proper resonator to achieve a dual-band circuit. So, two possible dual-band circuits are achievable from a single-band circuit. It is notable that the fundamental elements of the resulted dual-band circuit are LC resonators instead of inductors and capacitors. Therefore, the number of passive elements of the dual-band circuit is twice that of the single-band circuit.

3 EXAMPLES AND RESULTS To validate the proposed approach, an RLC power divider is designed, fabricated and measured at two frequencies f1 = 10 MHz and f2 = 20 MHz, supposing Z0 = 50 . Fig. 3 shows an RLC power divider working at a single frequency 1 [7] provided that L01  (C 01 ) 1  Z 0 . For

f1 = 10 MHz, the component values will be L0 = 795.8 nH and C0 = 318.3 pF. Fig. 4 shows two RLC power divider working at both frequencies f1 = 10 MHz and f2 = 20 MHz converted from Fig. 3 with regard to Tables 1 and 2. The S parameters of the circuits shown in Figs. 3 and 4 can be ———————————————— obtained by even and odd mode analysis [8]. Figs. 5-7  Mohammad Khalaj-Amirhosseini is with Iran University of Science and illustrate the amplitude of the scattering parameters of Technology, Tehran, Iran.  Mahmoud Moghavvemi and Aliyar Attaran are with Department of Elec- three designed power dividers. The deep nulls at desired trical Engineering, University of Malaya (UM), Kuala Lumpur, Malaysia. frequencies are observable for the input and output return losses as well as the output isolation. It is seen that © 2010 JOT http://sites.google.com/site/journaloftelecommunications/

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two dual-band circuits operate at two desired frequencies although their bandwidth are less than that in the primary single-band circuit. The designed power divider shown in Fig. 4(a) was fabricated using inductors with quality factor of 25 at 10 MHz and 5% tolerance. Fig. 8 shows the measured amplitude of the scattering parameters of the fabricated dual-band power divider compared with theoretical results. As shown, there is a good agreement between theoretical and measurement results. The input and output return losses as well as the isolation between the outputs are at least 25 dB at both desired frequencies.

Table 1. The required values of resonator elements with +1/1frequency scaling An alone A substitute Equations for two The values of element in resonator in frequencies resonator singledualelements frequency frequency circuit circuit

C0

L

L1 

C

1 1  C1 C01

C  C0

1 1 L 2   C 2 C01

L0

L

C1  C2 

C

L

1 1  L1 L01 1 L 2



C

L01

2

1 C0 1



L  L0

1



2

1 L0 1

Table 2. The required values of resonator elements with 1/+1 frequency scaling An alone A substitute Equations for two The values of element in resonator in frequencies resonator singledualelements frequency frequency circuit circuit

L

C0

C1  C2 

Figure 1. The reactance of a parallel and series resonator along with reactance of an inductor

C L0

L

C

1  C01 L1 1 L 2

C  C0

 C01

L

1   L01 C1

L  L0

L 2 

1   L01 C 2

C

795.8 nH

#2 397.9 nH

#1

50 318.3 pF

397.9 nH

318.3 pF 159.1 pF

#3

795.8 nH

(a) 397.9 nH

Figure 2. The reactance of a parallel and series resonator along with reactance of a capacitor

#2 318.3 pF

50

795.8 nH

C0 #2 L0

Z0

159.1 pF

#1 159.1 pF 397.9 nH

795.8 nH

#3

#1 C0

L0

318.3 pF

#3 Figure 3. A single-band RLC Wilkinson power divider

C012 2

L1 

159.1 pF

(b) Figure 4. Two dual-band RLC power dividers (a) +1/1 frequency scaling (b) 1/+1 frequency scaling

1

 

1

 

L012 2

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Figure 5. The amplitude of the scattering parameters of single-band power divider shown in Fig. 3

Figure 8. The measured scattering parameters (dots) of the fabricated dualband power divider shown in Fig. 4(a) compared with theoretical results

As a second example, consider Fig. 9 as a dual-band Tank circuit composed of two inductors L1 and L2, two capacitors C1 and C2 and one resistor R. In fact, it is a conventional Tank circuit (L1 and C1) connected in parallel to an anti-resonant circuit (L2 and C2), obtained from converting a conventional Tank circuit to a dual-band one. The input admittance of this circuit is as follows

  2 L1C1  1 C2  (2)  j  2  L2C2  1   L1 To make resonance at both frequencies 1 and 2 with the Yin 

1 1  Bin   R R

same relative bandwidth, two following relations are achieved after some mathematical manipulations. 1 (3) L1C1  L2 C 2  Figure 6. The amplitude of the scattering parameters of dual-band power divider shown in Fig. 4(a)

1 2 C1 L2 1 2 k    C2 L1 ( 2  1 ) 2 ( k  1) 2

 1 2  B  B1  B2   in k R     1 

(4)

1

(5)

(k  1) 1 1  2 k  1 R (k  1) C1  kC2 (k  1) RC1 where B1 and B2 are the frequency bandwidth at frequencies 1 and 2, respectively and also 2

2

k



2 1



(6)

has been defined as the resonance frequency ratio. It is seen from (3) that the resonance frequency of two series and parallel resonators is equal to 12 which is the

Figure 7. The amplitude of the scattering parameters of dual-band power divider shown in Fig. 4(b)

geometric average of two desired frequencies. This frequency is the anti-resonance frequency of the proposed Tank circuit too. From (3)-(5), one can determine the value of all elements of the proposed Tank circuit, knowing one of them such as R. To demonstrate the validity of the proposed dual-band Tank circuit, two dual-band Tank circuit are designed at frequencies (f1 = 10 MHz and f2 = 15 MHz) and (f1 = 10 MHz and f2 = 12 MHz) with 10% relative bandwidth for each one. Table 3 shows the values of elements for four cases determined from (3)-(5), assuming R = 1.0 k. Fig.

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10 illustrates the normalized magnitude of impedance of two designed dual-band Tank circuits. It is seen that there is two resonances at desired frequencies with approximately the same desired bandwidth.

C2 R

C1 L1 L2

Figure 9. The proposed dual-band Tank circuit

[2]

[3]

[4] [5]

[6] [7] [8]

Figure 10. The normalized magnitude of impedance of two designed dualband Tank circuits Table 3. The values of elements of two designed dual-band Tank circuits f1 , f2 [MHz] C1 [pF] C2 [pF] L1 [H] L2 [H] 10 , 15 63.66 10.61 2.65 1.59 10 , 12 72.34 2.41 2.92 87.5

4 CONCLUSION A new approach was introduced to design dual-band RLC circuits. In this approach a single-band circuit is designed and then converted to a dual-band circuit by substituting its inductors and capacitors for a proper parallel or series LC resonator. Two possible dual-band circuits are achievable from a single-band circuit. The proposed approach was verified by designing, fabricating and measuring a dual-band RLC power divider at frequencies 10 and 20 MHz as well as a dual-band tank circuit. The results obtained clearly indicate that the proposed approach can be used for designing any arbitrary RLC dual-band circuits.

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