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When an electromagnetic signal transmits through a coaxial cable, it propagates at speed determined by the dielectrics of insulator between the cooper core wire and the metallic shield. However, we demonstrate here that, once the shielding layer of the coaxial cable is cut into two parts leaving a small gap, while the copper core wire is still perfectly connected, a remarkable transmission delay immediately appears in the system. We have revealed by both computational simulation and experiments that, when the gap spacing between two parts of the shielding layer is small, this delay is mostly determined by the overall geometrical parameters of the conductive boundary which connects two parts of the cut shielding layer. A reduced analytic formula for the transmission delay related with geometrical parameters, which is based on an inductive model of the transmission system, matches well with the fitted formula of the simulated delay. This above structure is analog to the situation that an interconnect is between two inter-modules in a circuit. The results suggest that for high speed circuits and systems, parasitic inductance should be taken into full consideration, and compact conductive packaging is favorable for reducing transmission delay of inter-modules, therefore enhancing the performance of the system.

For modern high-electron-mobility transistors (HEMTs) based on III-V semiconductors e.g. GaAs, GaN, InAs and InSb [1-6], the high-frequency performance is mainly determined by the effective mass of charge carrier and energy gap. However, when these transistors are integrated into circuits or systems, parasitic effects induced by the device structure, interconnects, contacts, and packaging also affect the high-frequency performance of devices [7-11]. Historically, the parasitic impedance of interconnects is neglected and treated as a short. With the scaling down of integrated circuits, the capacitance of interconnects is comparable to the gate capacitance, and the resistance of interconnects increases dramatically, thus RC model for interconnect has been developed [

When high-speed devices, e.g., HEMTs, are operated in the frequency range of 0.5 - 1 THz [1,2,4,16], for instance, its characteristic response time is 1 - 2 picoseconds (ps). In a system consisting of many individual high-speed devices, the characteristic time of the system is then not only determined by individual devices themselves, but also affected by the transmission of electromagnetic (EM) signals between individual devices. The critical length of interconnects is a few tens of microns in 2010 [

However, we have given a clear picture in this paper that an inductive delay related with boundary conditions will occur and should be taken into account in some situation. To demonstrate this inductive delay, we have constructed a testing system where the shielding layer of a coaxial cable is cut into two parts, leaving a ringshaped small gap, where the copper core wire is still perfectly connected and the two shielding layers are connected and covered by a conductive cylinder. The two parts of the cut coaxial cable represent two individual devices and the copper core wire bridging the gap represents an interconnect. We have revealed by both computational simulation and experiments that, a remarkable delay occurs, and this delay is mostly determined by the overall geometrical parameters of the conductive cylinder when the gap spacing (corresponding to the length of interconnects) is small. Analysis of the results suggests that this delay is inductive.

In this paper of Section 2, we firstly construct a simulated structure of cylinder using high frequency structure simulator (HFSS), and determine the simulated frequency and the criteria of the scattering parameters (S parameters) convergence which are needed for HFSS using finite element method in frequency domain. In Section 3, we scan the structure parameters of the cylinder boundary conditions, and get the delay versus them in Section 3.1, as well as their fitting formula in Section 3.2. Experiments are also done and shown in Section 3.3 to support our simulation. At last we do the experimental analysis by reducing an analytic formula upon the parasitic inductance model in Section 3.4, and it fits well with the fitting formula of the above section, which shows the validity of our model. In Section 4 we conclude our works and show it significance in field of packaging.

To quantitatively calculate the delay in such a transmission situation, we have simplified the situation by choosing a system with cylindrical symmetry. The simulated system for this work is shown in

To precisely measure the delay for EM signals propagate ing from Port 1 to Port 2 as shown in

these experiments, the pulses have typical rise time and fall time around 2.5 ns, and a duration time around 13 ns. The frequency spectrum of such a pulse up to 400 MHz is shown in

We have applied the HFSS to perform the situation. The HFSS uses the finite element method to mesh the simulated structure and a criterion is needed to decide the convergence. We use both maximum deltas of the magnitude of S parameters (Mag S) and the phase of S parameters (Phase S) as the criteria for the matrix convergence because calculating of group delay is acquired.

A proper criterion of Mag S should be small enough so as to obtain sufficient accuracy, but not too small to avoid unnecessary long computational time. We set the criteria as 1 × 10^{–}^{3} and 5 × 10^{–3}, respectively. ^{–4}. The relative deviations are all less than 0.016% below 100 MHz showing a good convergence. Therefore we set the final criterion as 1 × 10^{–3}, which is proven to be small enough for a desired accuracy of the simulations.

The criterion of Phase S is more closely related to calculation of the group delay. At a specific frequency f, shift of phase ΔΦ, is given as:

where Δt is defined as the group delay. In our previous experimental setup for measuring the delay in transmission of pulsed signals, the time resolution of the measurement system is about 10 ps [^{–9} f. For simplicity and higher accuracy, we set the criterion of Phase S as 1 × 10^{–9} f, which corresponds to a time resolution about 3 ps.

By using the set criteria, we have systematically simulated the group delay of sine waves of varied parameters of frequency f, gap spacing d, height H and radius R of the cylinder.

The group delay of sine waves of varied f with fixed d, H and R have been simulated.

Although we have used pulsed signals instead of high frequency excitations in the simulation, and after Fourier transformation of the pulses the cut-off frequency of the spectra is set at only 100 MHz (see

The calculated delay is found dependent on the gap spacing d.

Finally, we set d = 0.1 mm so that we can ignore the in-

fluence of gap spacing for more quantitative simulations on varied H and R. The height H is assigned to vary from 2 mm to 60 mm with step of 2 mm. The radius R is assigned to vary from 2 mm to 30 mm with step of 1 mm.

The results are plotted in _{0} and height H at fixed R, but the correlation between the delay τ_{0} and radius R at fixed H is nonlinear.

The slope of τ_{0} versus H is related with R. So the fitting formula should have the form,

where K(R) is the slope at specific R, and Intercept is the interception. _{0} versus H at some fixed values of R at 2.0, 5.0, 10.0 and 30.0 mm, respectively. The four straight lines can fit well with these data separately in the figure.

Next, by using similar formula, we try to fit all the simulated data shown in

This fitting matches the simulated data perfectly, where the maximum absolute deviation in τ_{0} is only ±5 ps. Here we find the interception of 716.7 ps is very close to the transmission time of 716.1 ps along a perfect coaxial cable of 150 mm, which is equal to the total length of the two cables shown in _{c}, which gives

For modern high speed electronic systems working at 100 - 1000 GHz, their characteristic time scale is at 1 - 10 ps. The simulated transmission delay is in order of 10 ps for boundary dimension of 10 mm, as shown in

We also have done experiments to measure the time delay using this structure in our previous work [

Now we try to give a phenomenological explanation of the results. We assume the boundary condition around the gap has induced an additional inductance. Because the interception equals to the transmission time of the cable, the electromagnetic wave should transmit to one end of the gap, and then transmit through the cylinder, and finally pass through the other end of the gap. From the perspective of current, the current path is schematically highlighted with arrowed dotted lines shown

inductance formula for a coaxial cable, we calculate the inductance in our simulated structure as:

where R is the radius of the cylinder, and R_{0} is the outer radius of the coaxial cable, and μ is the space permeability with value of 4 × 10^{7} H/m. The equivalent circuit of the fixture is shown in

where T is the transmission coefficient, Z_{0}_{ }= (50 + jωL)Ω is the impedance seen from port 2, and Z_{L}_{ }= 50 Ω is the impedance seen from port 1. Assume ωL 100, then

and

For ωL 100, (ωL/100) is very small. Equation (9) can be approximated,

The group delay can be obtained from the formula,

Taking (10) and (6) into (11), it gets

Taking μ = 4 × 10^{7} H/m and R_{0}_{ }= 1.5 mm, it can be deduced

We can see the deduced Equation (13) and the fitted Equation (5) are very similar to each other. This confirms that the delay time is resulted from the parasitic inductance of the additional cylinder shielding around the gap. The parasitic inductance can be calculated using the inductance formula of a coaxial cable.

By using HFSS simulation, this work clearly reveals the origin of a boundary dependent delay in transmission of an electromagnetic signal between two coaxial cables with a small open gap in their shielding layers. For a metallic boundary with cylindrical symmetry, we have obtained an analytical formula for this delay, which matches well with the simulation results, showing that this delay has an inductive nature, although for the first glance one may assume this delay is introduced by parasitic capacitance of the system. As the performance of high-speed electronic systems relies on both the processing speed of individual devices and the transmission delay time between inter-modules, it resembles the cask effect that large transmission delay between inter-modules can limit the system performance dramatically. On the one hand, reducing the dimension size of the boundary along the transmission axis can reduce the delay effectively. On the other hand, reducing the dimension size of the boundary conditions can increase the parasitic capacitance of the system which can also limit the system performance. This can be helpful in the comprehensive consideration of parasitic inductance and parasitic capacitance [

We thank W. Q. Sun in Peking University for valuable discussions. This work was supported by MOST of China (Grant No 2011DFA51450 and 2011CB933002), and NSF of China (Grants 11074010).