Negative Group Delay Circuit with Improved Signal Attenuation and Multiple Pole Characteristics
 Author: Chaudhary Girdhari, Jeong Junhyung, Kim Phirun, Jeong Yongchae
 Publish: Journal of electromagnetic engineering and science Volume 15, Issue2, p76~81, Apr 2015

ABSTRACT
This paper presents a design of a transmission line negative group delay (NGD) circuit with multiple pole characteristics. By inserting an additional transmission line into a conventional NGD circuit, the proposed circuit provides further design parameters to obtain wideband group delay (GD) and to help reduce signal attenuation. As a result, the number of gain compensating amplifiers can be reduced, which can contribute to stable operation when integrated into RF systems. The multiple pole characteristics can provide wider NGD bandwidth and can be obtained by connecting resonators with slightly different center frequencies separated by quarterwavelength transmission lines. For experimental validation, an NGD circuit with two poles GD characteristic is designed, simulated, and measured.

KEYWORD
Distributed Transmission Line , Low Signal Attenuation , Multiple Pole , Negative Group Delay

I. INTRODUCTION
Electromagnetic wave propagation in any medium obeys the fundamental physical laws described by Maxwell’s equation [1]. Most media exhibit normal propagation called subluminal, where the speed of propagation of individual timeharmonic components is slower than the speed of light,
c , in a vacuum at all frequencies. However, in a specific and narrow frequency band of signal attenuation (SA) or in an anomalous dispersion frequency, the group velocity is observed to be greater than thec . This abnormal wave propagation is called superluminal group velocity or even negative group velocity [1,2].The wave propagation in any medium can be characterized by group velocity and group delay (GD), which are same. The GD in a circuit can be investigated by examining transmission phase variation with respect to frequency and can be defined as a negative derivative of the signal transmission phase according to frequency, as shown in (1).
As seen from (1), when quantity
τ_{g} is positive, the peak of the output pulse suffers a positive delay with respect to input pulse. On the other hand, ifτ_{g} is negative, the peak of the output pulse emerges prior to the peak of the input pulse entering the medium, and the medium is said to exhibit a negative GD [3]. However, this does not violate the causality because the initial transient pulse is still limited to the front velocity, which will never exceed the speed of light [4].The negative group delay (NGD) occurs at a certain range of frequency where the absorption or SA is maximum [1]. Therefore, bandstop structures are used to realize NGD circuits. Based on either series or shunt
RLC resonators, various kinds of microwave NGD circuits have been presented and demonstrated in the literature [513]. To overcome the limited feasibility problem of lumped elements in microwave frequencies, the NGD circuits using distributed elements are also presented [57,12]. However, the conventional NGD circuits presented in previous works exhibited excess SA up to 35 dB for a 8 ns GD, which can cause serious stability issues when an NGD circuit is integrated with RF/microwave systems. Therefore, for the same GD, the passband SA must be as small as possible.A few studies have been conducted about NGD networks with small SA. In [13], a composite NGD network with smaller SA was presented. However, this circuit requires parallel lumped elements (such as capacitors and inductors) between two transmission lines, making implementation difficult at microwave frequencies.
In this paper, a design of the transmission line NGD circuit with reduced SA and multiple pole GD characteristics is presented.
II. DESIGN THEORY
Fig. 1 shows the structure of the conventional and proposed 1pole NGD circuits that consist of resistor
R and transmission lines with characteristic impedances of Z_{1} and Z_{2} and electrical lengths of λ/4. Total ABCDparameters of the proposed circuit shown in Fig. 1(b) can be found as (2)where
f andf _{0} are operating and design center frequencies, respectively. TheS parameters of the proposed circuit can be found by using ABCD toS parameters conversion relationship [14], which is given as (3):where Z_{0} is reference port impedance. Furthermore,
S parameters and GD at thef _{0} can be obtained as (4).As seen from (4c), the maximum achievable GD depends on Z_{1}, Z_{2}, and
R . To better understand (4b) and (4c), the calculated maximum achievable GD and SA atf _{0} = 1.96 GHz according to Z_{1} andR are shown in Fig. 2. As seen from this figure, the SA is improved as Z_{2} increases. Therefore, high Z_{2} and low Z_{1} are necessary for reduced SA.Fig. 3 shows the simulation results of the 1pole NGD circuit. In this simulation, the maximum achievable GD at
f _{0} = 1.96 GHz is assumed to be 5 ns. As seen from this figure, the proposed circuit provides reduced SA as compared to the conventional circuit [6]. The SA of the proposed circuit is further reduced by making the value Z_{2} is high. However, the NGD bandwidth is reduced.The temperature dependence of
R is represented by the following relationship:where
δ ,R _{0}, ΔR , and ΔT are temperature coefficient, initial resistance, resistance variation, and temperature variation, respectively.Fig. 4 shows the performance degradation of the proposed 1pole NGD circuit, assuming the resistance variation of ±5%. As seen from this figure, the GD and SA (magnitude of
S _{21}) variations are approximately ±0.61 ns and ±0.68 dB from the reference values. These results indicate that the proposed NGD circuit is considerably less sensitive to the temperaturedependent resistance variation.The NGD bandwidth can be enhanced by connecting 1pole NGD circuits with the slightly different center frequencies (
f _{0i},i = 1, 2, 3, …) separated by λ/4 transmission lines with characteristic impedance of Z_{s} = 50 Ω, as shown in Fig. 1(c). Due to the differentf _{0} of 1pole NGD circuits, multiple pole GD characteristics can be obtained.Fig. 5(a) shows the simulated GD and
S _{21} magnitude results of 2pole and 3pole NGD circuits. In the case of a 2pole NGD circuit, two NGD circuits withf _{01} = 1.935 GHz andf _{02} = 1.984 GHz are cascaded. Similarly, for the 3pole NGD circuit, thef _{0i} are given asf _{01} = 1.912 GHz,f _{02} = 1.963 GHz, andf _{03} = 2.03 GHz. In both cases, the circuit element values of 1pole NGD circuits are given as Z_{1} = 30 Ω, Z_{2} = 90 Ω, andR = 1139 Ω. As seen from these figures, the NGD bandwidth is enhanced due to 2pole and 3pole characteristics. The phase characteristics of 2pole and 3pole NGD circuits are shown in Fig. 5(b). As seen from this figure, the phase slope ofS _{21} is positive over a certain range of frequency, which signifies the presence of NGD characteristics.III. SIMULATION AND EXPERIMENTAL RESULTS
For experimental validation of the proposed circuit, the design goal was to obtain a GD of 6 ns at
f _{0} = 1.96 GHz. For this purpose, a 2pole NGD circuit was designed and fabricated. For given specifications, the calculated circuit element values of a 2pole NGD circuit are given as Z_{1} = 30 Ω, Z_{2} = 90 Ω, Z_{s} = 50 Ω, andR = 1,140 Ω. Thef _{0}s are the same as presented in Section II. The circuit was fabricated using RT/Duroid 5880 of Rogers Inc. with a dielectric constant (ε_{r} ) of 2.2 and the thickness (h ) of 31 mils. The simulation was performed using ANSYS HFSS 2014. The layout of the fabricated circuit is shown in Fig. 6. The physical dimensions of the fabricated circuit are shown in Table 1 after the optimization.Fig. 7 shows the simulated and measured GD and magnitude results of the 2pole NGD circuit. From the measurement, the GD was determined as 5.80±0.45 ns over a bandwidth of 80 MHz. The maximum SA at
f _{0} = 1.962 GHz was 24.67 dB. The SA can be easily compensated using general purpose gain amplifiers [5]. A photograph of the fabricated circuit is also shown in Fig. 7. The simulated and measured phase characteristics are shown in Fig. 8. As seen in this figure, the slope of the phase is positive over a certain region. This positive phase slope characteristic can be used to cancel out the negative phase slope to obtain zero GD or a phasecompensated response.IV. CONCLUSION
This paper demonstrates the design of an NGD circuit with multiple pole GD characteristics and reduced signal attenuation. The multiple pole NGD circuit is obtained by the cascade connection of several 1pole circuits having slightly different frequencies. For the experimental verification, the 2pole NGD circuit was designed, fabricated, and measured. The proposed topology can reduce the number of gaincompensating amplifier stages and can help improve efficiency, outofband noise reduction, and stable operations when integrated into RF/microwave systems.

13. Choi H., Chaudhary G., Moon T., Jeong Y., Lim J., Kim C. D. 2011 "A design of composite negative group delay circuit with lower signal attenuation for performance improvement of power amplifier linearization techniques," [in Proceedings of IEEE MTTS International Microwave Symposium Digest (MTT)] P.14

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[Fig. 1.] Structure of transmission line negative group delay circuits: (a) conventional, (b) proposed 1pole circuit, and (c) proposed multiple pole circuit.

[Fig. 2.] Calculated group delay and signal attenuation (S21) of a 1pole negative group delay circuit according to R and Z1 with Z2 = 90 Ω and f0 = 1.96 GHz: (a) 3D plot, (b) group delay (GD) with respect to Z1 where the color bar represents R, (c) GD according to R with the color bar denoting Z1 and (d) signal attenuation (S21) with respect to R.

[Fig. 3.] Simulated results of 1pole negative group delay circuit with different values of Z2.

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[Fig. 4.] Performance degradation of the proposed 1pole negative group delay circuit assuming ±5% resistance variation from the reference value.

[Fig. 5.] Simulated results of 2pole and 3pole negative group delay circuits: (a) group delay/magnitude and (b) phase characteristics.

[Fig. 6.] Layout of the fabricated 2pole negative group delay circuit with physical dimensions.

[Table 1.] Physical dimensions of the 2pole negative group delay circuit (unit = mm)

[Fig. 7.] Simulated and measured group delay/magnitude results of the 2pole negative group delay circuit.

[Fig. 8.] Simulated and measured phase characteristics of a 2pole negative group delay circuit.