### I. Introduction

*S*-parameter analysis [6–9], a lossy coupling matrix synthesis [10], and a filter analysis [11]. The design equations for negative group delay circuit (NGDC) using lumped RLC resonators are also available based on the specification of the signal attenuation and NGD [12, 13]. In this study, we derive the mathematical relations among signal attenuation, NGD, and bandwidth for a convenient and systematic design of the NGDC composed of RLC resonators. The effects of the NGDC are evaluated in the time and frequency domains using narrow- and wide-band pulse inputs. To verify the proposed design equations, we fabricate an NGDC composed of lumped elements at 1 GHz. The group delays of the fabricated NGDC are measured and compared with those obtained from the presented design equations.

### II. Analysis and Synthesis of a Group Delay Circuit

### 1. Analysis

*Z*

_{0}is the characteristic impedance of the host transmission line. Fig. 1(b) shows an equivalent circuit consisting of a series resonant circuit connected in parallel with a transmission line.

*Y*

_{0}is the characteristic admittance. Most passive NGD structures can be modeled on the basis of these equivalent circuits in Fig. 1(a) and (b).

*G*

_{0}is the conductance,

*ω*is the angular frequency,

*ω*

_{0}is the resonant angular frequency given by

*ω*

_{0}

*C*

_{0}) is the susceptance slope parameter. The last expression in (1) is an approximation based on the assumption that

*ω*is near

*ω*

_{0}.

*Z*

*can be approximated as*

_{NGD}*Z*

*is the sum of*

_{in}*Z*

*and*

_{NGD}*Z*

_{0}. The reflection coefficient

*S*

_{11}is obtained as

##### (3)

*S*

_{21}) of the NGDC unit cell can be obtained by the ratio of the transmitted voltage (

*S*

_{21}are represented by

*ω*

_{0}across the NGDC in Fig. 1(a) occurs without a phase delay. The phase for the NGDC is obtained as

*ω*

_{0}and zero when

*ω*is far away from

*ω*

_{0}. The group delay is another word for a signal envelope delay [14]. The frequency dependence in (10) prevents a pulse input from being entirely copied in advance of the input at the output terminal. The maximum NGD is expressed as

*G*

_{0}becomes smaller, the signal transmission becomes smaller, as implied by (8), but the effect of the NGD becomes larger, as shown by (11). The magnitude of the NGD is shown to be proportional to

*C*

_{0}as is the susceptance slope

*ω*

_{0}

*C*

_{0}.

*S*

_{21}when τ(

*ω*

_{0}) (11) is assumed to be −0.5, −1, and −5 ns and

*S*

_{21}(

*ω*

_{0})= 0.5 at 1 GHz. In the case of τ(

*ω*

_{0}) = −1 ns,

*Z*

_{0}= 50 Ω,

*G*

_{0}= 1/100 (1/Ω),

*C*

_{0}= 10 pF, and

*L*

_{0}= 2.53 nH. As the magnitude of τ(

*ω*

_{0}) becomes larger, the bandwidth of the NGD becomes smaller. This will result in more signal distortion if an input signal bandwidth is wider than the NGDC bandwidth. The positive phase slopes near 1 GHz shown in Fig. 2(b) lead to NGD s, as explained in (10).

*S*

_{21}and NGD τ.

### 2. Synthesis

*S*

_{21}(

*ω*

_{0}) and τ(

*ω*

_{0}) are desired for a particular NGD circuit design, (8) and (11) can be simultaneously solved for the design equations given by

*L*

_{0}is obtained with

*S*

_{21}(

*ω*

_{0}) and τ(

*ω*

_{0}) at 1 GHz using (13). Whereas

*G*

_{0}(12) depends only on

*S*

_{21}(

*ω*

_{0}),

*C*

_{0}and

*L*

_{0}depend on both

*S*

_{21}(

*ω*

_{0}) and τ(

*ω*

_{0}). Expressions (12)–(14) are actually the design equations to realize the specifically required values of

*S*

_{21}(

*ω*

_{0}) and τ(

*ω*

_{0}). With (12) and (13), the quality factor of the NGD circuit may be expressed as

*S*

_{21}(

*ω*

_{0}) (8) or τ(

*ω*

_{0}) (11) becomes small. Note that if the two among the three design parameters

*S*

_{21}(

*ω*

_{0}) (8), τ(

*ω*

_{0}) (11), and bandwidth (16) are specified, the rest is determined automatically. This relation is summarized in Table 1.

*ω*) (10) as a function of frequency for different

*S*

_{21}(

*ω*

_{0})’s of −6, −10, and −20 dB when τ(

*ω*

_{0}) is −1 ns at 1 GHz. As

*S*

_{21}(

*ω*

_{0}) becomes smaller, the bandwidth of τ(

*ω*) becomes larger; this is the important tradeoff feature of the presented NGDC. Fig. 4(b) shows

*S*

_{21}(

*ω*) as a function of frequency for the different

*S*

_{21}(

*ω*

_{0})’s of −10, −20, −30, and −40 dB when τ(

*ω*

_{0}) is fixed at −1 ns at 1 GHz. The symbols represent the calculated results using (5), and the lines represent the circuit-simulated results based on Fig. 1(a).

*ω*is near

*ω*

_{0}.

*S*

_{21}(

*ω*

_{0}) and τ(

*ω*

_{0}). The bandwidth increases as

*S*

_{21}(

*ω*

_{0}) decreases and |τ(

*ω*

_{0})| gets smaller.

*ω*

_{0}) = −0.5, −1, and −2 ns at 1 GHz. Based on these circuit values, the NGD circuits as shown in Fig. 1(a) can be easily realized.

*ω*

_{0}) = −1 ns and −2 ns, respectively. The rising/falling time and the width of the input pulse are 3 ns and 3 ns, respectively. The carrier frequency is again 1 GHz but not shown. As the magnitude of τ(

*ω*

_{0}) increases, the output appears more ahead of the input pulse, but its distortion compared with the input pulse becomes larger. In Fig. 8(b), we show the same when τ(

*ω*

_{0}) = −1, −2, and −5 ns. For this case, the rising/falling time and the width of the input pulse are 6 ns and 6 ns, respectively. The input pulse in Fig. 8(b) is slowly varying, and its spectrum should be narrower than that in Fig. 8(a). This leads to less distortion as demonstrated particularly in the case of τ(

*ω*

_{0}) = −2 ns.

*S*

_{21}(

*ω*

_{0}) = −20 dB and τ(

*ω*

_{0}) = −1 ns at 1 GHz. Fig. 1(b) is the photograph of the measurement setup. The permittivity of the microstrip transmission line with a thickness of 1.6 mm and the lumped element values are

*ɛ*

*= 2.2,*

_{r}*R*

_{0}= 900 Ω,

*C*

_{0}= 0.62 pF, and

*L*

_{0}= 41 nH using (12)–(14). The

*S*-parameters of the structure are measured using a network analyzer. The influence of the transmission line is de-embedded on the reference plane at the center of the structure.

*S*-parameters and group delays as a function of frequency in the case shown in Fig. 9. They all show to be in good agreement. The output time signals to the input pulses are similar to the dotted ones with τ(

*ω*

_{0}) = −1 ns in Fig. 8(a).

*N*of the identical unit cells increases, the signal transmission (17) decreases but the NGD (18) does not change much.

*Z*

_{0},

*G*

_{0},

*C*

_{0},

*L*

_{0}, and

*Z*

*in this study to*

_{in}*Y*

_{0},

*R*

_{0},

*L*

_{0},

*C*

_{0}, and

*Y*

*using the duality principle. The presented design method is applicable to other previous passive NGD circuits, not just the one demonstrated in this study.*

_{in}### III. Conclusion

*S*

_{21}(

*ω*

_{0}), τ(

*ω*

_{0}), and bandwidth. Some design examples are provided and analyzed in the time and frequency domains. The relations among

*S*

_{21}(

*ω*

_{0}), τ(

*ω*

_{0}), and the group delay bandwidth are explained using closed-form expressions. The circuit-/EM-simulated and the measured

*S*-parameters and group delays are all shown to be in good agreement. The presented NGDC design methods may be useful for many applications, such as filters, feed-forward amplifiers, array antennas, and non-Foster reactive elements, among others.