#### Abstract

This work presents a novel approach to design a maximally flat negative group delay (NGD) circuit based on microwave recursive filters. The proposed NGD circuit is realized by cascading stages of quarter-wavelength stepped-impedance transformer. It is shown that the given circuit can be designed to have any prescribed group delay by changing the characteristic impedance of the quarter-wave transformers (QWTs) cascaded with each other. The proposed approach provides a systematic method to synthesize NGD of arbitrary amount without including any discrete lumped component. For various prescribed NGD, the characteristic impedance of QWT has been tabulated for two and three stages of the circuit. The widths and lengths of microstrip transmission lines can be obtained from characteristic impedance and the frequency of operation of the transmission line. The results are verified in both simulation and measurement, showing a good agreement.

#### 1. Introduction

Following the classical paper published by Brillouin and Sommerfeld in 1960 [1], Garrett and McCumber were the first to analytically prove that group velocity of wave can be greater than the speed of light [2]. However, its first practical demonstration was performed after a decade by Chu and Wong [3]. At first, the concept of a wave having velocity greater than the speed of light (superluminal velocity [1]) seems to defy the causality. Nevertheless, there exist enough practical experiments showing that the concept of superluminal velocity follows the causal system definition [3–5]. Since the development of this concept, there has been a lot of work done by scientists to utilize this property and apply it to practical applications. This concept has been used to enhance the efficiency of the feed-forward amplifier, broadband and constant phase shifter, and shortening of delay lines [6–8]. By using NGD circuits, positive group delays introduced by the circuit components and electrical connections in electronic systems can be compensated. In NGD circuits, electromagnetic waves can have superluminal velocity in the region of anomalous dispersion such that the phase of the higher frequency component of the wave will move in advance with respect to the lower frequency components.

Although there have been a lot of efforts put towards generating NGD using both active and passive techniques, little work has been done towards generating the prescribed NGD at the desired frequency. Recently, a maximally flat NGD circuit based on transversal filter concept has been proposed to synthesize the desired group delay [9]. In this paper, we will demonstrate that a microwave recursive filter can also be used to generate NGD of predetermined values in the desired frequency band by using only distributed components [10]. It is noted that very recently distributed components-based NGD circuits have also been proposed; however, few systematic methods were provided to synthesize the desired NGD [11]. The contribution in this work expands the theory derived in [9] and provides a full analysis of a microwave recursive-filter based NGD circuitry.

To illustrate, multistage QWTs are used to generate NGD of predetermined values as shown in Figure 1. Essentially, it behaves as a recursive filter since the reflected waves will bounce back and forth among the impedance interfaces and form an infinite impulse response (IIR). Following this concept, we will show a comprehensive methodology to synthesize the desired NGD by treating the multistage QWT as an IIR filter. In addition, it is worth mentioning that there are no lumped components present, which makes this design easy to fabricate and to be scaled up to higher frequency. We will also demonstrate the synthesis procedure of a two-stage QWT with −0.5 ns, −1 ns, and −2 ns group delays. In addition, a branch-line coupler is used to transfer NGD from a one-port circuit to a two-port one, making the NGD circuit more applicable in practices.

#### 2. Theory and Formulae

A microstrip-line based transmission line structure is used to fabricate QWTs. As depicted in Figure 1, the QWTs have the characteristic impedance of , , with an identical electrical length of . The impedance of all QWTs has been normalized to the termination impedance of 50 ohms. The design flow is as follows. First, we need to obtain the characteristic impedance of each of the microstrip transmission line QWTs that can generate the desired group delay. Once we obtain the characteristic impedance, we can calculate the width and length of the transmission line QWT according to the synthesis procedure shown in [12]. In order to solve for the desired characteristic impedance, we first relate it to the transmission matrix or ABCD matrix parameters. The relation between the characteristic impedance of the transmission line and the transmission matrix parameters for the th stage of QWT is as follows: Furthermore, when cascading stages of QWT we will obtainwhere is the number of stages in the circuit, corresponds to the impedance of the th QWT, and is the electrical length of each stage of the transmission line which is equal to at the center frequency (, where and ). We can then represent the product of the above transmission matrix parameter for stage as follows:in which , , , and represent the transmission matrix parameters for the entire -stage quarter-wave transformer circuit. The reflection coefficient for the -stage circuit can be written as The phase of the reflection coefficient can be expressed asIn order to obtain the prescribed maximally flat group delay, we need to obtain the characteristic impedance of each of the transmission lines by solving the following equations: Here is the prescribed magnitude, and is the prescribed NGD with a unit of the sampling period . In order to have the maximally flat response, we set all the higher order derivatives shown in (8) to zero. To give a quantitative example, let us take the two-stage QWTs case and assume to be −4 and to be 0.04 at the center frequency 1 GHz. Since the sampling period of a 90-degree delay line at 1 GHz is 0.25 ns, we can generate group delay of −4 × 0.25 = −1 ns. Similarly, by changing to −8 we can obtain group delay of −2 ns and so on. It is noted that we terminate our device with standard 50 ohms load, and the characteristic impedance of the transmission lines obtained here is normalized to the load impedance. Table 1 shows the different values of normalized characteristic impedance for various group delays by setting equal to 0.04 derived from (6)–(8). After obtaining the characteristic impedance, we can easily calculate the dimension of QWT [12]. The width and length of the QWT corresponding to the characteristic impedance which we calculate from (6)–(8) for negative group delay of –0.5 ns, –1 ns, and –2 ns are tabulated in Table 2. The pictorial representation of width and length of the transmission line is shown in Figure 2.

#### 3. Simulation and Measurement

To validate our concept of generating the prescribed NGD using transversal filter methodology, we designed a NGD circuit consisting of two-stage quarter-wave transformer using microstrip-line structures on a printed circuit board (PCB). The substrate that we used is RO3010 from Rogers Corporation, with a dielectric constant of 10.2 and thickness of 25 mils. Normalized characteristic impedance of two transmission lines for −1 ns group delay is obtained from Table 1: that is, , . After that we fabricated the quarter-wave transformer according to this normalized characteristic impedance. Figure 3 shows the simulated results of three different NGD at 1 GHz for a two-stage quarter-wave transformer using the tabulated coefficients. In addition, it is worth mentioning that one can also design a three-stage transformer to generate prescribed NGD as shown in Figure 4 with the coefficients obtained from Table 1. For a given NGD, the bandwidth can be enhanced when more stages are cascaded.

In practical application, it is often desired to introduce the negative group delay between two ports. In fact, we can transfer the group delay from one port to another port by simply using a branch-line coupler. The schematic for the complete circuit is shown in Figure 5. The parameters of the two-port network can be obtained by solving the branch-line coupler with odd and even mode analysis [12]. The port reduction technique [13] can be applied to get the reflection coefficient of two-port negative group delay circuit. Figure 5 shows all the voltages that we are required to find out the matrix for the circuit. and are the incident and reflected voltage at the input port 1. is the reflected voltage at the output port 2. represents incident and represents reflected voltage at junction ; similarly represents incident and represents the reflected voltage at junction . We can calculate by the following procedure:The reflected voltage from the output port 2, , can be written in terms of reflected voltages and asTherefore, , and we know. Let us assume; this will give usSimilarly, will be obtained asThus, we can writeSince , we can put , which will give us . The reflection coefficient matrix for the two-port negative group delay circuit can thus be represented asFrom (7) and (14), the group delay at desired frequency becomesThe constant term in (15) represents the extra group delay caused by the coupler, which also agrees with the result in [13]. The prototype of the design including a branch-line coupler along with two-stage quarter-wave transformers is shown in Figure 6. Transmission lines , , , and are included in the prototype for interconnections, which will result in addition of group delay in the circuit. The effects can be nullified by fine-tuning the dimensions of the NGD circuit based on quarter-wave transformers. Table 3 indicates the length and width of the branch-line coupler and quarter-wave transformer as depicted in Figure 6.

Figure 7 depicts the comparison of group delays between the simulation and measurement in which we use the branch-line coupler. The simulated and measured results agree with each other very well. The resulting NGD of the entire circuit is around −1 ns at 1.1 GHz. Furthermore, the magnitude of corresponding is plotted in Figure 8, which indicates around 23 dB signal attenuation during the NGD region.

#### 4. Conclusion

In this paper, we present a systematic technique of generating prescribed NGD by simply using multistage quarter-wave transformers to form microwave recursive filters. By properly choosing the impedance of transformer, we can realize the desired NGD. A table providing the associated values for desired NGD is given and utilized to synthesize the desired NGD. In addition, a branch-line coupler is used to transfer the group delay from one-port circuit to a two-port NGD circuit. The proposed method is promising to be further used in high frequency circuitries to synthesize any desired amount of NGD in order to compensate the undesired excessive group delay such as in feed-forward amplifiers and envelop tracking power amplifier, which can improve the system overall efficiency.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.