Compact Broadband Negative Group Delay Circuit with Flatness and Bandwidth Enhancement

Abstract

A novel compact broadband negative group delay (NGD) circuit with flatness and bandwidth enhancement is presented. The presented negative group delay circuit (NGDC) consists of a high-impedance transmission line connected by two resistors, which are linked together with two coupled lines and a low-impedance transmission line. The flatness of the NGD is enhanced by tuning the characteristic impedance of the transmission lines. In order to verify the method, a compact broadband NGDC with a size of 29.4 mm × 58.1 mm (0.14 λ_g × 0.29 λ_g) is designed, fabricated, and measured at the center frequency of 1.0 GHz. The measured results show that an NGD time of −0.49 ns at the center frequency is obtained with return loss and insertion loss of 35.0 dB and 18 dB, respectively. And, the flat-NGD bandwidth reaches 509 MHz (50.9%) over 0.766 to 1.275 GHz with 19% group-delay fluctuation.

1. Introduction

Recently, negative group delay circuits (NGDCs) have attracted more attention due to their potential applications. The NGDCs have been employed for amplifiers to achieve group delay (GD) equalization [1], to achieve the compensation of the beam squint in series-fed antenna arrays [2] and to improve the performance of signal anticipation [3], and kinds of filters were designed with the NGD characteristic [4,5,6]. In [7], the NGDCs were used in designing non-foster components. Furthermore, reconfigurable non-foster circuits with NGDCs were designed [8]. Besides, the compensation of GD [9] or the anticipation of the mechanical actuators’ signals [10] can be achieved by cascading the NGDCs and other components. In order to obtain high performance, some NGDCs based on new methods such as filter theory [11], signal interference techniques [12], lossy transmission lines [13] and split-ring resonators [14] have been presented. However, the GD near the center frequency is not flat and cannot satisfy wideband applications such as the broadband phase shifter [15].

In order to increase the flatness of the GD, several flat-NGDCs have been presented [16,17,18,19,20,21,22,23,24]. In accordance with filter theory, a 4.3% flat-NGD bandwidth of a NGDC with a ripple of 16% was achieved [16]. In [17], through bringing the two working frequencies of the dual-band NGDC closer, a flat-NGDC was designed. But the absolute flattening of GD cannot be achieved in theory. Afterwards, power dividers with flat-NGD characteristics were proposed in [18,19,20]. In addition, based on the transmission lines with step impedance or T-type structures, a flat NGD characteristic with an NGD time of −0.5 ns and −1.11 ns was achieved [21,22]. Nevertheless, the circuit size was not small enough. To achieve the size reduction, two NGDCs based on two TL structures and three coupled lines were presented, respectively [23,24]. However, the ideal port matching characteristics could not be obtained, and the circuit size needed to be further enhanced.

In this paper, a compact broadband NGDC is presented to achieve flat-NGD characteristics and ideal port matching. The presented NGDC consists of two couple lines connected by a transmission line and two resistors linked with a transmission line. The couple lines are utilized to improve the port matching characteristics and the two transmission lines are used to strengthen the flatness of the NGD response. And, the equivalent of the open-circuited coupled line is a Π structure [25]. Therefore, compared with the circuit in [23], the proposed circuit structure with the coupled lines is completely different. Furthermore, under the condition of the flat-NGD characteristic, the ideal port matching of the proposed circuit is realized. The circuit and design process are presented and discussed.

2. Circuit Structure and Design Theory

2.1. Even- and Odd-Mode Analysis

Figure 1 gives the configuration of the proposed compact broadband NGDC. It consists of two couple lines with even- and odd-mode characteristic impedance Z_0e, Z_0o and electrical length θ, which is connected by a transmission line with characteristic impedance Z₁ and electrical length θ₁, and two resistors R link, by transmission line, with characteristic impedance Z₂ and electrical length θ₂.

The even- and odd-mode equivalent circuits of the proposed NGDC is shown in Figure 2. And, the even- and odd-mode input impedance Z_{in_odd} and Z_{in_even} can be expressed as

$$Z_{\mathrm{in}\_\mathrm{odd}}=A_1+\mathrm{j}{A}_2,$$

(1)

$$Z_{\mathrm{in}\_\mathrm{even}}=A_3+\mathrm{j}{A}_4,$$

(2)

where A₁, A₂, A₃ and A₄ are given in Equations (3)–(6).

$$A_1={\displaystyle \frac{Z_2^{2}R}{{\left[Z_2+M_{\mathrm{odd}}\cot \left({\theta }_2/2\right)\right]}^2+R^2{\cot }^2\left({\theta }_2/2\right)}},$$

(3)

$$A_2={\displaystyle \frac{Z_2^2{M}_{\mathrm{odd}}+Z_2{M}_{\mathrm{odd}}^2\cot \left({\theta }_2/2\right)+Z_2{R}^2\cot \left({\theta }_2/2\right)}{{\left[Z_2+M_{\mathrm{odd}}\cot \left({\theta }_2/2\right)\right]}^2+R^2{\cot }^2\left({\theta }_2/2\right)}},$$

(4)

$$A_3={\displaystyle \frac{Z_2^{2}R}{{\left[Z_2-M_{\mathrm{even}}\tan \left({\theta }_2/2\right)\right]}^2+R^2{\tan }^2\left({\theta }_2/2\right)}},$$

(5)

$$A_4={\displaystyle \frac{Z_2^2{M}_{\mathrm{even}}-Z_2{M}_{\mathrm{even}}^2\tan \left({\theta }_2/2\right)-Z_2{R}^2\tan \left({\theta }_2/2\right)}{{\left[Z_2-M_{\mathrm{even}}\tan \left({\theta }_2/2\right)\right]}^2+R^2{\tan }^2\left({\theta }_2/2\right)}},$$

(6)

where the expression of M_odd and M_even are

$$M_{\mathrm{odd}}={\displaystyle \frac{Z_{0\mathrm{e}}-Z_{0\mathrm{o}}}{2}}\times {\displaystyle \frac{2Z_1\tan \left({\theta }_1/2\right)-2Z_{0\mathrm{o}}\cot \theta +\left(Z_{0\mathrm{e}}-Z_{0\mathrm{o}}\right)\tan \theta }{Z_{0\mathrm{e}}+Z_{0\mathrm{o}}-2Z_1\tan \left({\theta }_1/2\right)\tan \theta }}-Z_{0\mathrm{o}}\cot \theta ,$$

(7)

$$M_{\mathrm{even}}={\displaystyle \frac{Z_{0\mathrm{e}}-Z_{0\mathrm{o}}}{2}}\times {\displaystyle \frac{-2Z_1\cot \left({\theta }_1/2\right)-2Z_{0\mathrm{o}}\cot \theta +\left(Z_{0\mathrm{e}}-Z_{0\mathrm{o}}\right)\tan \theta }{Z_{0\mathrm{e}}+Z_{0\mathrm{o}}+2Z_1\cot \left({\theta }_1/2\right)\tan \theta }}-Z_{0\mathrm{o}}\cot \theta ,$$

(8)

And then the S-parameters of the presented compact broadband NGDC are expressed as

$$S_{11}={\displaystyle \frac{Z_{\mathrm{in}\_\mathrm{odd}}{Z}_{\mathrm{in}\_\mathrm{even}}-Z_0^2}{\left(Z_{\mathrm{in}\_\mathrm{odd}}+Z_0\right)\left(Z_{\mathrm{in}\_\mathrm{even}}+Z_0\right)}},$$

(9)

$$S_{21}={\displaystyle \frac{\left(Z_{\mathrm{in}\_\mathrm{even}}-Z_{\mathrm{in}\_\mathrm{odd}}\right)Z_0}{\left(Z_{\mathrm{in}\_\mathrm{odd}}+Z_0\right)\left(Z_{\mathrm{in}\_\mathrm{even}}+Z_0\right)}}={\displaystyle \frac{X_1+\mathrm{j}{X}_2}{X_3+\mathrm{j}{X}_4}},$$

(10)

where Z₀ is the port impedance. The unknown parameters in Equation (10) are

$$X_1=\left(A_3-A_1\right)Z_0,$$

(11)

$$X_2=\left(A_4-A_2\right)Z_0,$$

(12)

$$X_3=\left(A_1+Z_0\right)\left(A_3+Z_0\right)-A_2{A}_4,$$

(13)

$$X_4=A_2\left(A_3+Z_0\right)+A_4\left(A_1+Z_0\right),$$

(14)

Furthermore, the GD of the presented NGDC can be expressed as

$${\tau }_g=-{\displaystyle \frac{d\angle S_{21}}{d\omega }},$$

(15)

2.2. S-Parameters and NGD Analysis

In the proposed compact broadband NGDC, the electronic length is assumed as θ₁ = θ₂ = θ = π/2 at center frequency f₀. When the electronic lengths in Equations (1)–(15) are π/2, the simplification of S-parameters and GD at f₀ are

$$S_{11}\left(f_0\right)={\displaystyle \frac{A_{10}^2+A_{20}^2-Z_0^2}{\left(A_{10}+\mathrm{j}{A}_{20}+Z_0\right)\left(A_{10}-\mathrm{j}{A}_{20}+Z_0\right)}},$$

(16)

$$S_{21}\left(f_0\right)={\displaystyle \frac{2A_{20}{Z}_0}{A_{10}^2+A_{20}^2+Z_0^2+2A_{10}{Z}_0}},$$

(17)

$$GD\left(f_0\right)={\displaystyle \frac{{A^{\prime }}_{10}}{A_{20}}}+{\displaystyle \frac{2A_{10}{A^{\prime }}_{20}+2Z_0{A^{\prime }}_{20}-2{A^{\prime }}_{10}{A}_{20}}{A_{10}^2+A_{20}^2+Z_0^2+2A_{10}{Z}_0}},$$

(18)

where the unknown parameters are given as

$$A_{10}={\displaystyle \frac{Z_2^{2}R}{{\left(Z_2-P\right)}^2+R^2}},$$

(19)

$$A_{20}={\displaystyle \frac{-Z_2^{2}P+Z_2{P}^2+Z_2{R}^2}{{\left(Z_2-P\right)}^2+R^2}},$$

(20)

$${A^{\prime }}_{10}=-{\displaystyle \frac{\left[2\left(Z_2-P\right)\left(Q+P{\tau }_2\right)-2R^2{\tau }_2\right]Z_2^{2}R}{{\left[{\left(Z_2-P\right)}^2+R^2\right]}^2}},$$

(21)

$${A^{\prime }}_{20}={\displaystyle \frac{\left(Z_2^{2}Q-2Z_{2}PQ-Z_2{P}^2{\tau }_2-Z_2{R}^2{\tau }_2\right)\left[{\left(Z_2-P\right)}^2+R^2\right]-\left[2\left(Z_2-P\right)\left(Q+P{\tau }_2\right)-2R^2{\tau }_2\right]\left(-Z_2^{2}P+Z_2{P}^2+Z_2{R}^2\right)}{{\left[{\left(Z_2-P\right)}^2+R^2\right]}^2}},$$

(22)

$${\tau }_i={\theta }_i/2\mathsf{\pi }{f}_0(i=1,2),$$

(23)

where P and Q can be expressed as

$$P={\displaystyle \frac{{\left(Z_{0\mathrm{e}}-Z_{0\mathrm{o}}\right)}^2}{4Z_1}},$$

(24)

$$Q={\displaystyle \frac{\left(Z_{0\mathrm{e}}-Z_{0\mathrm{o}}\right)\tau }{2}}+{\displaystyle \frac{{\left(Z_{0\mathrm{e}}-Z_{0\mathrm{o}}\right)}^2\left(Z_{0\mathrm{e}}+Z_{0\mathrm{o}}\right)\tau }{8Z_1^2}}+{\displaystyle \frac{{\left(Z_{0\mathrm{e}}-Z_{0\mathrm{o}}\right)}^2{\tau }_1}{4Z_1}}+Z_{0\mathrm{o}}\tau ,$$

(25)

$$\tau =\theta /2\mathsf{\pi }{f}_0,$$

(26)

As can be seen from the expressions of S₁₁(f₀), S₂₁(f₀) and GD(f₀) in Equations (16)–(18), there are four variables Z₂, R, P and Q in them. So, if Z₂ is determined, the values of R, P and Q can be solved for. Once P and Q are known, there are only three variables, Z₁, Z_0e and Z_0o, in Equations (24) and (25). When Z₁ is determined, the values of Z_0e and Z_0o can also be solved for.

Therefore, it is a must to determine the desired values of the S₁₁(f₀), S₂₁(f₀) and GD(f₀) ahead and chose proper values of the free variables Z₁ and Z₂ during the design.

2.3. Free Variables Analysis

Based on the theory in Section 2.2, when S₁₁(f₀) = 0 for port matching, the values of S₂₁(f₀) and GD(f₀) can be selected according to different needs of designers. In this section, the GD and S-parameters, especially the flat-NGD bandwidth and flatness, of the free variables Z₁ and Z₂ at different values will be discussed.

Figure 3 gives the influences of Z₂ on the GD and S-parameters. It can be seen that when Z₂ increases from 140 Ω to 160 Ω, the ripple of τ_g × f₀ near the center frequency f₀ increases. Moreover, |S₁₁| and |S₂₁| change a little within the flat-NGD bandwidth. Consequently, Z₂ can be employed to additionally improve the flatness of GD.

The changes in circuit performances with different values for Z₁ are given in Figure 4. As can be seen, when Z₂ increases from 55 Ω to 65 Ω, the ripple of τ_g × f₀ near the center frequency f₀ decreases. Moreover, |S₁₁| and |S₂₁| remain unchanged within the flat-NGD bandwidth. Consequently, Z₁ can also be used to achieve the GD flatness enhancement.

Figure 5 shows the effect of different Z₁ and Z₂ couples on the entire circuit’s performance. It can be seen that the flat-NGD characteristic can be achieved by adjusting the value of Z₁ and Z₂. Meanwhile, it is also demonstrated that the design method can achieve different NGD values compared to Figure 3 and Figure 4. Besides, it also can be obtained from Figure 3, Figure 4 and Figure 5; once the |S₂₁| and τ_g × f₀ at f₀ are fixed, the maximum flat-NGD bandwidth is unchanged.

It is worthy to note the effect of different values for |S₂₁| on the entire circuit performance when the Z₁, Z₂ and NGD values are fixed at f₀. As can be seen in Figure 6, when |S₂₁| decreases from −16 dB to −18.5 dB, the ripple of τ_g × f₀ near the center frequency f₀ also decreases. Therefore, the NGD time, |S₂₁| and flat-NGD bandwidth need to be considered comprehensively.

For the discussion of the maximum flat-NGD bandwidth, Figure 7 gives the τ_g × f₀ and S-parameters under the same |S₂₁|with that in Figure 6. To obtain the maximum flat-NGD bandwidth, the values of Z₁ and Z₂ are changed in Figure 7. It can be seen that the maximum flat-NGD bandwidth increases with the |S₂₁| decreases.

For convenience, a simple design process of the proposed circuit is summarized as follows.

Step 1: Determine the desired |S₂₁| and NGD time at the center frequency f₀.

Step 2: Choose a proper Z₂ according to Figure 3 and Figure 5. Then, calculate R, P and Q using Equations (16)–(23).

Step 3: Select an appropriate Z₁ according to Figure 4 and Figure 5. Then, calculate Z_0e and Z_0o using Equations (24)–(26).

Step 4: If the flatness of GD is not enough after the first three steps, return to Step 2 and adjust Z₁ and Z₂ to other potential values.

Step 5: If the flatness of GD is not enough or the maximum flat-NGD bandwidth is not satisfied after the first four steps, return to Step 1 and adjust |S₂₁|.

3. Experimental Verification

To confirm the design concept, a compact broadband NGDC is designed at the center frequency f₀ = 1.0 GHz with GD = −0.5 ns. Using the design procedures in Section 2, the electrical parameters can be obtained as Z₁ = 59 Ω, Z₂ = 150 Ω, Z_0e = 74.6 Ω, Z_0o = 39.1 Ω, R = 52 Ω. Then the flat-NGDC is implemented on the substrate with a thickness of 1.0 mm, with a dielectric constant of 2.65 and a loss tangent of 0.003. The layout of the proposed compact broadband NGDC is given in Figure 8a. After optimizing by HFSS EM 2018 software, the final physical dimensions are w₁ = 2.0 mm, l₁₁ = 7.0 mm, l₁₂ = 23.0 mm, l₁₃ = 5.0 mm, w₂ = 0.2 mm, l₂₁ = 6.5 mm, l₂₂ = 17.8 mm, l₂₃ = 6.9 mm, w = 2.0 mm, s = 0.2 mm, l = 49.0 mm, w_p = 1.8 mm, l_p = 5.0 mm, with R = 50 Ω. The photo of the fabricated compact broadband NGDC is given in Figure 8b. The overall circuit dimension is 29.4 mm × 58.1 mm (0.14 λ_g × 0.29 λ_g, where λ_g is the guided wavelength of 50-Ω TLs at the center frequency). The measurements are completed by an AV3565A network analyzer manufactured in China Electronics Technology Group Corporation 41st Research Institute, Beijing, China.

Figure 9 shows the measured GD and S-parameters, along with the simulated and calculated results for comparison. It can be seen from Figure 9a that the calculated, simulated and measured flatness and GD (f₀) are basically consistent. And, the measured flat-NGD bandwidth is slightly offset directly from the simulated one. A relatively good consistency for calculated, simulated and measured |S₂₁| has been shown in Figure 9b. The calculated |S₁₁| and |S₂₂| at f₀ is minimized, which means the port is matching ideally. And, the simulated and measured |S₁₁|, |S₂₂| has a slight frequency shift, but good port matching characteristics are implemented within the flat-NGD bandwidth. The differences between the measured results and the simulated results are due to the inaccuracy of the substrate parameters.

The measured GD, |S₂₁| and|S₁₁| at f₀ are −0.48 ns, −18 dB and −35 dB, respectively. The measured NGD bandwidth of the presented NGDC is 569 MHz (56.9%) over 741 to 1310 MHz; meanwhile, the flat-NGD bandwidth is 509 MHz (50.9%) over 766 to 1275 MHz. And, the input/output return loss is better than 13 dB within the flat-NGD bandwidth. The insertion loss bandwidth, which is defined as the 3-dB variation from the center frequency |S₂₁|, is 439 MHz (43.9%) over 800 to 1239 MHz. The comparisons of the presented NGDC with precious works are given in

Table 1. Performance comparison between the proposed compact broadband NGDC and previous flat-NGD circuits.

Refs.	f0 (GHz)	NGD (ns)	GD Ripple (%)	Flat-NGD Bandwidth (%)	IL (dB)	Size (λg × λg)	FOM1/FOM2 (×10−3)
[18]	2.14	−0.49	16	14.7	16.6	0.73 × 1.01	23.2/21.5
[19]	2.14	−0.20	40	9.30	14.0	0.68 × 0.41	8.0/28.7
[20]	2.00	−0.88	25	9.50	33.1	0.41× 0.51	3.7/17.7
[21]	1.00	−0.50	16	46.3	18.8	0.24 × 0.34	26.4/325.9
[22]	2.14	−1.11	23	6.10	9.75	0.65 × 1.18	46.9/61.1
[23]	0.68	−0.49	10	31.8	15.3	0.20 × 0.38	26.8/352.2
[24]	0.94	−0.54	14	35.0	18.3	0.09 × 0.48	21.6/501.4
This Work	1.00	−0.48	19	50.9	18.0	0.14 × 0.29	30.8/757.6

. Compared with [21,23,24], they have a close center frequency and NGD time, but the presented NGDC has the largest flat-NGD bandwidth with a moderate IL. For a better comparison, the figure of merits (FOM₁ and FOM₂) have been used, which are defined as [22]

$${\mathrm{FOM}}_1=\left|{\tau }_{\mathrm{g}}\left(f_0\right)\right|\times {\mathrm{BW}}_{\mathrm{flat}-\mathrm{NGD}}\times \left|S_{21}\left(f_0\right)\right|,$$

(27)

$${\mathrm{FOM}}_2=\left|{\tau }_{\mathrm{g}}\left(f_0\right)\right|\times {\mathrm{BW}}_{\mathrm{flat}-\mathrm{NGD}}\times \left|S_{21}\left(f_0\right)\right|/{\mathrm{FOM}}_1,$$

(28)

It can be seen from Table 1 that the proposed NGDC has a larger FOM₁ and FOM₂ than [18,19,20,21,22,23,24] with a moderate GD ripple. Moreover, the circuit size of the proposed NGDC is the smallest.

4. Conclusions

In this paper, a compact broadband NGDC has been proposed, and the design equations at the center frequency have been simplified. The free variables (Z₁ and Z₂) used to achieve the enhancement of the GD flatness have been discussed. An NGDC with a size of 0.14 λ_g × 0.29 λ_g has been designed, simulated, fabricated and measured. A very good agreement is confirmed by simulation and measurement results. Compared to the existing flat-NGDCs, the presented NGDC has a larger FOM₁ and FOM₂ with moderate GD ripple and the smallest circuit size.