Ahmed M. Alaa
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This work presents a novel RF on-chip inductor that overcomes the problems with conventional large-sized inductors, which cannot be placed in an integrated circuit (IC) for large inductance values. The proposed inductor depends mainly on the meminductor, a new passive circuit element that has memory properties. This new circuit element is argued to be common in the nanoscale, where the dynamical properties of electrons and ions are likely to depend on the history of the system. Being nonlinear, it is shown that, by a suitable arrangement of meminductors and through appropriate initial values for their inductances, we can achieve a linear nanoscale inductor.
The rapid growth of the wireless communication market has fueled the demand for low-cost radio systems on a chip (Yue and Wong, 1999). On-chip inductors are used extensively in radio-frequency (RF) circuits for frequency tuning and impedance transformation (Long and Copeland, 1997). Silicon IC technology has progressed to offer a device performance suitable for analog operations up to several gigahertz and, therefore, presents the potential for integrating radios on a chip (Yue and Wong, 1999; Long and Copeland, 1997; Di Ventra et al., 2009; Sai et al., 2007.)
For a typical inductance ranging from 1 to 20 nH, conventional silicon technologies can deliver quality factor values (Qs) of about five. However, as interconnect technology advances, the achievable Q is improving to above 10. Although on-chip inductors have Qs significantly lower than their discrete counterparts (typical Qs of about 50), they have been proven to be useful and essential in highly integrated RF systems (Yue and Wong, 1999).
The commonly used on-chip inductor is the spiral inductor; physical models for spiral inductors on silicon, which are suitable for circuit simulation and layout optimization, were introduced recently. Key issues related to inductor modeling are the skin effect and silicon substrate loss. However, having inductors as on-chip elements is still challenging, and the realizations of complex poles in RF circuits, such as active filters, utilize operational amplifiers instead of inductors. In addition, currently known on-chip inductors are limited by relatively small inductances, as the inductor’s size is directly proportional to the device’s inductance.
The hypothetical fourth passive circuit element, the memristor (Strukov et al., 2008; Pershin and Di Ventra, 2008), was recently realized by scientists of HP Labs through nanoscale titanium oxide films. This breakthrough in circuit analysis introducing a passive memory resistor opens the door for creating similar nanoscale memory capacitors and inductors (namely, memcapacitors and meminductors). We discuss methods of having a linear nanoscale inductor through meminductors and canceling its memory effect to obtain a linear small-sized inductor regardless of the inductance.
We now introduce the inductive class of memory devices. First, we define the flux: \[{\bf{\phi}}\left({\text{t}}\right) = \mathop{\int}\nolimits_{{-}\infty}\nolimits^{t}{{v}\left({t}\right){dt}}, \tag{1} \] where v(t) is the induced voltage on the inductor (equal to minus the electromotive force), and ${\phi}$ is the time-varying flux. We call the approach described by Sai et al. (2007) an nth-order current-controlled meminductive system: \[{\bf{\phi}}\left({\text{t}}\right) = {L}\left({\text{x},\text{I},\text{t}}\right){I}\left({\text{t}}\right) \tag{2} \] \[{\text{x}}^{\ast} = {f}\left({\text{x},\text{I},\text{t}}\right), \tag{3} \] where L is the meminductance.
The voltage across the inductor is the derivative of the flux; therefore, the voltage current relation is \[{\text{V}}_{\text{L}} = \frac{{d}{\phi}}{dt} = {\text{L}}\frac{dI}{dt} + {\text{I}}\frac{dL}{dt}{.} \tag{4} \]
Figure 1 is a graph showing the relationship between the current and flux for a meminductor. Another idealized characteristic shows two levels of inductances for a meminductor (two slopes); these two levels are switched between threshold current values, and, in Fig. 1, we note that we have an infinite number of slopes at both ends of the plot. The idealized plot has two slopes only, depending on the history of the device’s inductance.
Fig 1 The relationship between the current flowing through the device and the flux.
This means that, as the current flowing through the meminductor increases toward iT, the meminductance is on the high level LH until it reaches the threshold value; it then switches to the low slope until it reaches the negative threshold value and then returns to the high-inductance slope. This corresponds to a two-level inductor that has a history, and the level of inductance is decided by the amplitude of the current flowing through it (Fig. 2).
Fig 2 The idealized relationship between the current flowing through the device and the flux.
The high- and low-inductance levels are frequency dependent; they are assumed to be affected by the speed of the current alternation through the device. As the frequency increases, the meminductor tends to be a linear inductor (as it becomes an linear time invariant (LTI) system). This means that the two inductance levels are very close, and the two slopes coincide to a single straight line. Figure 3 shows the effect of the frequency on the inductance.
Fig 3 The relationship between the current flowing through the device and the flux with varied frequencies. The hysteresis is narrowed as the frequency increases.
We find that, as the frequency increases, the high- and low-inductance levels decrease. This makes the high-level slope approach the low-level one until they coincide to a single line representing an LTI conventional inductor. We assume, ideally, that the inductance is inversely proportional with the frequency. As the frequency increases, the inductance decreases, so, instantly on either the high or low slopes, we have a temporarily linear inductor, and the impedance is directly proportional to the frequency. Assuming that the rate of change of frequency is higher than that of inductance, we maintain the direct relationship between the impedance and the signal’s frequency.
The proposed inductor involves a combination of two ideal memristors with two high- and low-inductance levels. We aim to apply the concept of CMOS operation, which depends on the switching operation of two connected NMOS and PMOS transistors. Here, we desire that each meminductor operates in a slope that differs from that of the other, and, when one of them switches to the second slope, the other switches to the first slope. This would result in the effect that the combination of the two meminductors appears to have a constant inductance along the way.
To guarantee such dynamic operation, we should make the initial meminductors’ inductances differ from each other; i.e., one meminductor is initially in the low-inductance slope, and the other is in the high-inductance one; when switching, each meminductor will have the other inductance level, but the combination of the two meminductors would have an ideally constant inductance. The two meminductors should be in a series combination, as each meminductor’s history is current dependent; one should be initially hanged on the high-inductance slope and the other on the low-inductance slope.
This could be easily done by applying a current that exceeds the threshold current to one of the meminductors. Then, when this current is removed, the meminductor will go to the high-inductance state, and, when the current is cut, it will keep its high inductance, as nanoscale memory elements have nonvolatile memory properties. This meminductor (with a high-inductance state) is connected in series with a low-state meminductor, as shown in Fig. 4.
Fig 4 A simple circuit showing an interconnection of two meminductors forming a linearized inductor in the nanoscale.
Assume that the meminductor M1 is initially in the high-inductance state, and the other meminductor M2 is initially in the low-inductance state. Initially, the current flows through the series combination of the meminductors. When the current is below the threshold level, both meminductors are kept in their initial states, and the series combination’s inductance is equivalent to the summation of both. If the current exceeds the threshold level, both meminductors switch to a different slope. Therefore, M1 moves to the low-inductance state, and M2 changes to the high-inductance one. However, the summation is kept constant, and the source V sees a consistent inductance that is the sum of the high and low states, regardless of the meminductor holding the high state and the one holding the low state.
Figure 5 illustrates the switching behavior of the meminductors M1 and M2. We note that the magnetic permeability can adjust to slow the periodic variations of the current, while it cannot at high frequencies, so that this device behaves as a nonlinear inductor at low frequencies and a linear inductor at high ones. This applies to individual memristors, but the overall series combination has a fixed linear time-invariant inductance. Therefore, we say that the system is linearized. The nanoscale inductor’s physical model is still unreliable; however, it is expected that a physical realization of the device will be implemented in the nanoscale, similar to memristors (Di Ventra et al., 2008).
Fig 5 (a) The hysteresis characteristics of M1 and M2. The current slope of the inductance is shaded in gray. This plot is for a current below the threshold, where M1 is initially in a low state and M2 in a high state. (b) The plot for a current above the threshold.
The energy stored in the current-controlled meminductive system can, therefore, be calculated as (Chua, 1971) \[{\text{U}}_{\text{L}}\left({\text{t}}\right) = \mathop{\int}\nolimits_{to}\nolimits^{t}\left[{{L}\frac{dI}{dt} + {I}\frac{dL}{dt}}\right]{I}\left({\tau}\right){d}\tau{.} \tag{5} \]
When L is constant, we readily obtain the well-known expression for the energy UL = L I2/2. This applies only for a single inductor. The energy stored in the proposed inductor depends on the fact that there is always a meminductor with low inductance LL and another with high inductance LH; therefore, we can assume that L is a constant and equals the summation of the two slopes: \[{\text{U}}_{\text{L}}\left({\text{t}}\right) = \left({\text{L}}_{\text{H}} + {\text{L}}_{\text{L}}\right){\text{I}}^{2}{/}{2}{.} \tag{6} \]
The passivity criterion of the meminductive system is guaranteed when t = t0; this device is in its minimal energy state, and UL (t) ≥ 0 at any time. This is valid for a series interconnection of devices (Long and Copeland, 1997; Di Ventra et al., 2009).
The article discussed an initial proposal for a nanoscale inductive device that can be employed as an on-chip inductor for RF applications that require on-chip inductors for the purpose of resonance and impedance transformation. The inductor is a series interconnection of two ideal two-state meminductors. The connection involves one of the meminductors initially being in the high state and the other in the low state of meminductance. Therefore, we have a constant inductance seen on the terminal of the series connection as the states alternate between the two meminductors. The passivity criterion and energy storage were discussed.
Ahmed M. Alaa (ahmedalaa@ieee.org) is with the Department of Electronics and Communications Engineering, Cairo University, Giza Governorate, 12613, Egypt.
Digital Object Identifier 10.1109/MPOT.2014.2357972