This invention relates to a technique for implementing a dispersive group delay line for the electromagnetic signal.

Group delay has been a subject of interest in electromagnetic communications, wherein the transmission paths are required to have flat group delay in the pass-bands. For example, a band-pass filter based on conventional Chebyshev, Butterworth or elliptic method has a flat group delay in the pass-band and it has larger group delay near the edges of the pass-band. However, the larger group delay response outside of the pass-band is of no particular consequence in most cases. As a result, most of efforts focused on the flat group delay in the microwave components study. Unfortunately, electromagnetic communication channels suffer strong group delay variation in air or other transmission paths and the time domain waveforms become distorted when impulse signals are considered. The group delay line can be used to tame the distortion effect.

Dispersive delay lines using conventional all-pass technology experience small group delay time. A cascade connection of all-pass delay units improves the overall response in the sense of obtaining larger group delay time. However, it increases the circuit area as well as transmission losses. Although the surface acoustic wave devices are compact and provide large delays, their applications are limited to low-frequency and narrow-bandwidth applications. Therefore, there is a need for a technique for implementing a group delay line with larger frequency-sensitive delay time, low-loss response for wide-bandwidth applications.

Briefly, in accordance with the invention, a group-delay network is provided for tuning the propagation delay time of designated signal frequencies from the source to the output load. The basic cell of the group delay device comprises a main transmission path that is connected to source and output at two ends, a couple of pairs of unequal-length, parallel, open stubs, a couple of pairs of complementary slot lines. The pairs of unequal-length, parallel stubs are directly connected to the main transmission path, wherein one pair of stubs are different from another pair of stubs in the sense of electric length θ_i (i= 1, 2). In other words, two electric (and physical) lengths of stubs are different from each other, as shown in Fig. 1, and θ₁≠θ₂. The pair of stubs (2Z_1b, 2Z_1b) is referred to as unequal in length to the pair of stubs (2Z_2b, 2Z_2b). Two pairs of complementary slot lines are corresponding to the characteristics of two pairs of unequal-length, open stubs, respectively, which are omitted in Fig. 1. Z_S and Z_L in Fig. 1 are source and load impedances, respectively. Two pairs of unequal-length, parallel stubs are employed to generate an induced pass-band lying between two stop-bands. The maximum transmission coefficient in the induced pass-band, which is bounded by two transmission nulls in the frequency band, is determined by the following relationship $${\mathit{Z}}_{\mathit{1}\mathit{b}}\mathit{cot}{\mathit{\theta }}_{\mathit{1}}+{\mathit{Z}}_{\mathit{2}\mathit{b}}\mathit{cot}{\mathit{\theta }}_{\mathit{2}}=\mathbf{0.}$$ The maximum group delay G_d in the induced pass-band is $$G_d\approx 2T_o\frac{Z_o}{Z_{1b}{Z}_{2b}/\left(Z_{1b}+Z_{2b}\right)}\frac{1}{{{\delta }_o}^2},$$ where T_o is the propagation delay time for the signal traveling across one of unequal-length stubs, and δ_o is the normalized bandwidth of the induced pass-band.
In a preferred embodiment, the group delay is determined the propagation delay time of each unequal-length stubs, the normalized induced pass-band band-width and characteristic impedances of both main transmission path and unequal-length stubs.

In applications where group delays of certain bands of high-frequency signals are to be tuned, the present invention can be realized on a printed circuit board. For the main transmission path and two pairs of unequal-length, parallel stubs, each element is fabricated in changing the conductor strip width and length of the element. For the complementary slot line, the conductor is removed from the ground conductor plane to form the strip-like non-conductor strip. The complementary slot line is placed just beneath the corresponding stub, and the stub is separated from the complementary slot line with the insulating dielectric substrate.

• Fig. 1 shows the equivalent transmission line representation of the basic cell of a group delay line.
• Fig. 2 shows a schematic drawing of unequal-length stubs of basic cell in the top signal layer.
• Fig. 3 shows a schematic drawing of unequal-length complementary slot lines of basic cell in the bottom ground layer.
• Fig. 4 shows a three-dimension schematic drawing of basic cell of a group delay line.
• Fig. 5 shows the cascade connection of basic cells of the group delay line network in accordance with the present invention.

To appreciate the details of the present invention, a general understanding of transmission lines will prove helpful. In this regard, reference should be made to Fig. 1 for the basic cell of the group delay line, where (the drawing is a conventional, prior to art transmission line,) 2Z_ib (i=1, 2) is the characteristic impedance, β_ib is the propagation constant, and l_ib is the physical length of transmission line. The electric lengths of open stubs 2Z_1b and 2Z_2b are unequal , i.e., θ₁=β_1bl_1b≠β_2bl_2b=θ₂, or l_1b≠l_2b when β_lb=β_2b. The lengths L_1b and L_2b in Fig. 2 are used to represent l_ib and l_2b, respectively. In the following discussion, the equivalent characteristic impedance of parallel stubs (2Z_ib, 2Z_ib) is changed to Z_ib so as to simplify the mathematical representation.

Notice that a microstrip structure has a signal layer and a ground layer, while a stripline structure has a signal layer and two ground layers. The following discussion using transmission-line representation is suitable for both microstrip structure and stripline structure.

The input impedance Z_in,i looking from the main line Z₀ toward each of the open stub is $$Z_{\mathit{in},i}=-j{Z}_i\cot \left({\mathit{\beta }}_{\mathit{ib}}{\mathit{l}}_{\mathit{ib}}\right),\left(i=\mathit{1},\mathit{2}\right)\mathrm{.}$$ When one of the physical lengths l_ib is equal to a quarter guided wavelength, the input impedance Z_in,,_i is zero. As a result, a transmission zero occurs. When the open stub is smaller than a quarter guided wave-length, the open stub appears to be capacitive. On the other hand, if the open stub is larger than a quarter guided wave-length, it is inductive. When two parallel stubs with different physical lengths are implemented, two transmission zeros occur at two respective frequencies. At a frequency located between two transmission-zero frequencies, one Z_in,i (i=1,2) is inductive and another is capacitive. When Z_in,1 + Z_in,2 = 0, the total input impedance due to two parallel stubs is infinite, and a total transmission through the main line occurs. As a result, a pass-band is induced between two transmission nulls. The induced pass-band exhibits excessive group delay.

For the circuit shown in Fig. 1, the scattering parameter S₂₁ (or transmission coefficient) is as follows $$S_{21}=\left[\frac{2Z_{\mathit{in}}}{Z_{\mathit{in}}+Z_o}\right],$$ where $$Z_{\mathit{in}}=\left[\frac{1}{\frac{1}{Z_o}+\frac{1}{Z_{\mathit{in},1}}+\frac{1}{Z_{\mathit{in},2}}}\right]\mathrm{.}$$
Substituting both (3) and (5) into (4), we obtain the transmission coefficient S₂₁ $$S_{21}=\frac{1}{1+j\frac{Z_o\left(Z_{1b}\cot {\theta }_1+Z_{2b}{\theta }_2\right)}{2Z_{1b}{Z}_{2b}\cot {\theta }_1\cot {\theta }_2}},$$ where θ_i=β_ibl_ib(i = 1,2).

The complex scattering parameter S₂₁ can be expressed in the polar form as S₂₁=| S₂₁| ∠ S₂₁. ∠S₂₁ is the argument of S₂₁ and it is given as follows $$\angle S_{21}=-\mathrm{\Pi }-{\tan }^{-1}\left[\frac{Z_o\left(Z_{1b}\cot {\theta }_1+Z_{2b}\cot {\theta }_2\right)}{2Z_{1b}{Z}_{2b}\cot {\theta }_1\cot {\theta }_2}\right]\mathrm{.}$$
As stated in the above, an induced pass-band is lying between two transmission nulls caused by parallel stubs. The group delay G_d of the basic cell is defined as $$G_d=-\frac{d\angle S_{21}}{\mathit{d\varpi }},$$ where ω is the angular frequency of signal. The group delay G_d is determined by characteristic impedance Z_ib (i=1,2), and electrical length θ_i of transmission lines. Upon the substitution of (7) into (8), we obtain $$G_d=\frac{Z_o{Z}_{1b}{Z}_{2b}\left\{A-B\right\}}{2{Z_{1b}}^2{Z_{2b}}^2{\cot }^2{\theta }_1{\cot }^2{\theta }_2+2{Z_o}^2{\left(Z_{1b}\cot {\theta }_1+Z_{2b}\cot {\theta }_2\right)}^2},$$ where $$A=\left(Z_{1b}\cot {\theta }_1+Z_{2b}\cot {\theta }_2\right)\left[\left(\cot {\theta }_2+{\cot }^2{\theta }_1\cot {\theta }_2\right)T_1+\left(\cot {\theta }_1+{\cot }^2{\theta }_2\cot {\theta }_1\right)T_2\right],$$ and $$B=\left(Z_{1b}{T}_1+Z_{2b}{T}_2+Z_{1b}{T}_1{\cot }^2{\theta }_1+Z_{2b}{T}_2{\cot }^2{\theta }_2\right)\cot {\theta }_1\cot {\theta }_2\mathrm{.}$$
T₁ and T₂ in (9a) and (9b) are propagation delay time for signal traveling across lines l_1b and l_2b, respectively, i.e., dθ_i/dω=T_i (i=1,2). The maximum group delay occurs at the total transmission frequency. Substituting Z_1b cotθ₁+Z_2bcotθ₂
= 0 into (9), we obtain $$G_d=\frac{-Z_o}{2Z_{1b}{Z}_{2b}\cot {\theta }_1\cot {\theta }_2}\left[Z_{1b}\left(1+{\cot }^2{\theta }_1\right)T_1+Z_{2b}\left(1+{\cot }^2{\theta }_2\right)T_2\right]\mathrm{.}$$

To extract the physical insight regarding the maximum group delay of this dispersive transmission line, we further simplify its mathematical expressions. A transmission-zero frequency occurs when the physical length of a stub is a quarter guided wavelength. The electrical lengths of two stubs at the total-transmission frequency of induced pass-band can thus be set as follows $${\mathrm{\theta }}_{\mathit{1}}=\mathrm{\pi }/\mathrm{2}-{\mathrm{\delta }}_{\mathrm{1}},{\mathrm{\theta }}_{\mathit{2}}=\mathrm{\pi }/\mathrm{2}+{\mathrm{\delta }}_{\mathrm{2}}\mathrm{.}$$
δ_i (i=1,2) is the electrical length distance in radian between the electrical length at the total transmission frequency of induced pass-band and the electrical length at the transmission null frequency caused by the respective stub.
If it is assumed that δ₁ = δ₂ = δ, (10) is further simplified to the following $$G_d=\frac{Z_o\left[\left(Z_{1b}{T}_2+Z_{2b}{T}_1\right)\left(1+{\tan }^2\delta \right)\right]}{2Z_{1b}{Z}_{2b}{\tan }^2\delta }\mathrm{.}$$

For a narrow, induced pass-band, we have tanδ≈δ and tan²δ<<1 Under such a condition, the group delay G_d in (12) now becomes as follows $$G_d\approx \frac{Z_o}{2Z_{1b}{Z}_{2b}{\delta }^2}\left[Z_{1b}{T}_1+Z_{2b}{T}_2\right]\mathrm{.}$$
Notice that T_i (i=1, 2) is the propagation delay time for the signal traveling across the stub line. If we assume that δ₁=δ₂=δ_o/2 and T₁=T₂=T_o, (13) can be simplified further to the following $$G_{d,\mathit{narrowband}}\approx 2T_o\frac{Z_o}{Z_{1b}{Z}_{2b}/\left(Z_{1b}+Z_{2b}\right)}\frac{1}{{{\delta }_o}^2},$$
where T_o is the propagation delay time across a quarter guided wavelength and δ_o is the normalized bandwidth between two transmission nulls caused by two stubs.

As shown in Fig. 5, a cascade connection of the basic cells using segments Z₁, Z₂, ....., Z_n (n is a positive integer) to form a group delay line system.

The introduction of complementary slot lines is to transform the induced, band-limited pass-band to an all pass-band, which is |S₂₁|=1. L_1t and L_2t in Fig. 3 are the lengths of complementary slot lines 2Z_1t and 2Z_2t, respectively.

The three-dimension schematic drawing of basic cell of a group delay line in Fig. 4 is a three-layers structure, where (11) is the signal (top) layer, (12) is the insulating (middle) layer, and (13) is the conductor ground (bottom) layer.