,

9

Silicon Cochlea Building Blocks

This chapter goes into the details of some of the circuit blocks used in the silicon cochleas discussed in Chapter 4. It looks at some of the basic circuit structures used in the design of various one-dimensional (1D) and two-dimensional (2D) silicon cochleas. Nearly all silicon cochleas are built around second-order low-pass or band-pass filters. These filters are usually described as second-order sections.

9.1   Introduction

For the 1D cochlea, voltage-domain filters have been most commonly used, largely because of historical reasons – voltage-domain circuits were well understood – and partly because there was no compelling reason for using current-domain circuits. The first 2D cochlea (Watts et al. 1992) indeed was also designed using voltage mode second-order filters, but the use of current mode second-order filters greatly simplifies the implementation of the resistive grid, modeling the fluid in the 2D cochlea (see Chapter 4). Hence, later versions of the 2D cochlea have used current-domain (i.e., log-domain) filters.

In this chapter, we will first look at the voltage-domain second-order filters, followed by the log-domain second-order filter. Both the 1D and the 2D cochleas, whether implemented in the voltage or current domain, need exponentially decreasing currents to bias the filters, which we discuss in a separate section. As discussed in Chapter 4, the inner hair cells (IHC) stimulate the auditory nerve neurons and cause them to fire. At the end of this chapter an implementation of the IHC will be presented.

9.2   Voltage-Domain Second-Order Filter

The second-order filters of the 1D voltage-domain cochlea are built from three transconductance amplifiers. We will first introduce the transconductance amplifier and then discuss the second-order filters. We also give a detailed analysis of the operation of these circuits.

9.2.1   Transconductance Amplifier

The transconductance amplifier in its most basic version is shown in Figure 9.1a. When biased in weak inversion it has a hyperbolic tangent transfer function given by

with I_bias, V₊, and V_ as in Figure 9.1, and n is the weak inversion slope factor which depends on the technology and normally has a value somewhere between 1 and 2. The thermal voltage U_T is given by U_T = kT/q, with k the Boltzmann constant, T the temperature in Kelvin, and q the charge of an electron. U_T is about 25 mV at room temperature.

Figure 9.1 A basic transconductance amplifier: (a) schematic and (c) symbol, a transconductance amplifier with an enlarged linear range; (b) schematic and (d) symbol

The transconductance amplifier is biased in weak inversion when the current through the differential pair transistors is much smaller than the specific current of these transistors, that is,

where W is the channel width of the transistor, L the length, μ the mobility of the minority carriers, and C_ox the gate oxide capacitance per unit area. The specific current depends on process parameters, the geometry of the transistor, and temperature (Vittoz 1994). For small inputs (|V₊ − V_| < 60 mV) we can approximate the amplifier as a linear transconductance:

with the transconductance given by

This last equation shows that the transconductance of the amplifier is proportional to the bias current when the amplifier operates in weak inversion.

As we will see in Section 9.2.4, we will also need a transconductance amplifier with an enlarged linear range as shown in Figure 9.1b. The transfer function of this amplifier is given by

and

The linear range of this amplifier is thus enlarged by a factor n + 1, where n is about 1.5, and the transconductance is reduced by a factor n + 1 with respect to the normal transconductance amplifier with the same bias current. The transfer functions of both amplifiers are shown in Figure 9.2.

Figure 9.2 Transfer functions of the transconductance amplifier (heavy line) and the modified transconductance amplifier

Figure 9.3 A second-order low-pass filter

9.2.2   Second-Order Low-Pass Filter

A second-order low-pass filter can be made with three transconductance amplifiers (A₁, A₂, A₃ as shown in Figure 9.3) and two capacitors. The transfer function of this filter is given in the Laplace domain by

where s = jω, j² = −1, and ω is the angular frequency, the time constant τ = C/g_m when both A₁ and A₂ have a conductance g_τ and both capacitors have capacitance C. The quality factor Q of the filter can be expressed as

where g_Q is the conductance of the amplifier A₃. The gain and phase response of the filter described by Eq. (9.7) are shown in Figure 9.4 for two values of Q.

Figure 9.4 (a) Gain and (b) phase response of the second-order low-pass filter for Q = 1 (heavy line) and Q = 10 when 1/τ = 1000 s⁻¹

9.2.3   Stability of the Filter

From Eq. (9.8), it is clear that Q becomes infinite when g_Q = 2g_τ, and from Eq. (9.7) we can then see that when ω = 1/τ, the gain of the filter also becomes infinite and the filter will be unstable. This gives g_Q < 2g_τ as the small-signal stability limit of the filter. However, the filter of Figure 9.3 also has a large-signal stability limit, which puts a more stringent constraint on the value of Q. In order to obtain the transfer function of Eq. (9.7), we have treated the filter as a linear system. This approximation is only valid for small input signals. It has been shown by Mead (1989) that large transient input signals can create a sustained oscillation in this filter. During most of this oscillation, all three amplifiers are saturated so that their output is either plus or minus their bias current. We can therefore adopt a piece-wise linear approach to analyze this situation in which we treat the amplifiers as current sources. The following analysis is mostly adapted from Mead’s, but is more general since it does not assume that the amplitude of the oscillation is equal to the supply voltage.

When V_in suddenly increases by a large amount, the amplifier A₁ will saturate and will charge the capacitor at its output with its maximum output current I_τ. If, at the same time, V₁ is larger than V_out, A₃ will also charge the capacitor with its maximum output current I_Q and we can write for V₁:

where V_out ≪ V₁ ≪ V_in. V₁ will thus rise at its maximum rate. Once V₁ catches up with V_in, the output current of A1 changes sign and we write for V₁:

where V_out ≪ V_in ≪ V₁.

In order to have Q larger than one, I_Q has to be larger than I_τ so that in this case V₁ will continue to increase with a smaller slope until it reaches the positive power supply V_dd or until V_out catches up with V₁. As long as V₁ stays larger than V_out, we can write in our piece-wise linear approach for V_out:

where V_out ≪ V₁.

Once V_out catches up with V₁, the sign of the output of A₃ will change, and V₁ will start its steep descent, until V₁ goes below V_in, when the sign of the output of A₁ changes and V₁ descends more slowly to the negative power supply V_ss, or until V_out catches up again. Figure 9.5 sketches the behavior of the circuit according to the above equations. The thick line shows the evolution of V₁ and the thin line shows the same for V_out. Whenever V_out catches up with V₁, the change in both voltages will switch direction.

By comparing the voltages at which V_out catches up with V₁ at the start and the end of a single rise and fall cycle, we can determine the nature of the oscillation. If ΔV (see Figure 9.5) is positive, the amplitude of the oscillation will decrease during each period and the oscillation will cease after a certain time. The limit of stability is reached when ΔV becomes zero, so that the amplitude of the oscillation stays constant. For the rising part of the oscillation we write:

Figure 9.5 Piece-wise linear approximation of the waveform for V₁ (bold) and V_out. The slope of each line is indicated next to it

where V_L and V_H are as shown in Figure 9.5. Similarly, for the falling part, we write when ΔV equals zero:

Equations (9.12) and (9.13) can only be satisfied when V_L = V_H. Substituting V for V_L and V_H in either equation, and dividing by CV/I_τ yields

Rewriting this equation, we obtain the following solution:

This gives the critical value for large-signal stability of the low-pass filter of Figure 9.3. Since the conductance of the amplifiers is directly proportional to the bias currents, this large-signal stability condition also limits g_Q/g_τ to this value, and thus limits Q to a maximum value of 2.62 (see Eq. 9.8). The large-signal stability limit therefore severely limits the maximum quality factor of the filter when using the basic transconductance amplifier.

9.2.4   Stabilised Second-Order Low-Pass Filter

The circuit can be improved by using two wide-range transconductance amplifiers to implement A₁ and A₂ (Figure 9.6) and a basic transconductance amplifier for A₃ (Watts et al. 1992) (see Figure 9.1b). In this case we can write for the conductance ratio:

which ensures that g_Q/g_τ becomes 2, that is, Q becomes infinite, before I_Q/I_τ becomes 1.62, since n is larger than 1. Thus the filter is always large-signal stable whenever it is small-signal stable.

With the two wide-range transconductance amplifiers A₁ and A₂, and the one basic transconductance amplifier A₃, we have a second-order low-pass filter for which we can set the cutoff frequency and the quality factor using the bias currents of these amplifiers. By cascading these filters and biasing the amplifiers with exponentially decreasing currents, we can then create a model of the basilar membrane. The limited input linear range of the second-order LPF in Figure 9.6 can be increased by modifying the amplifiers A₁ and A₂ so that the inputs go to the well terminals instead of the gate terminals of the input differential pair transistors (Sarpeshkar et al. 1997).

9.2.5   Differentiation

The voltage V_out (Figure 9.6) at the output of each second-order stage in the cochlear filter cascade represents the displacement of a small section of the basilar membrane. However, since the stimulation of the inner hair cells in the biological cochlea is proportional to the velocity of the basilar membrane, the output of each second-order stage has to be differentiated. This can be done by creating a copy of the output current I_dif of amplifier A₂ at every stage as in Watts et al. (1992). Since the voltage on a capacitor is proportional to the integral of the current onto the capacitor, I_dif is effectively proportional to the basilar membrane velocity. Yet, with equal displacement amplitudes, velocity will be much larger for high frequencies than for low frequencies, yielding output signals with amplitudes that decrease from the beginning of the cochlea to the end. This can be corrected by normalizing I_dif to give equal amplitude at every output. A resistive line controlling the gain of the current mirrors that create the copies of I_dif at each stage is used for this purpose by Watts et al. (1992). However, this resistive line introduces an extra source of mismatch in the circuit.

Figure 9.6 Modified second-order low-pass filter

Figure 9.7 One section of the cochlear cascade, with differentiator Adapted from van Schaik et al. (1996). Reproduced with permission of MIT Press

An alternative solution which does not need normalization is to take the difference between V_out and V₁ (see Figure 9.7). We can rewrite Eq. (9.3) applied to A₂ as

or

This is equivalent to differentiating V_out, with 0 dB gain at the cutoff frequency for all stages. Figure 9.8 shows the gain and phase response of the filter after differentiation. We can see that a single band-pass filter only has a shallow high frequency cutoff slope of 20 dB per decade. In the filter cascade, however, the 40 dB per decade cut-off slopes of the individual low-pass filters will be accumulated (Figure 9.4). This can yield very steep high frequency cut-off slopes, as we will see in the measurements later on in Figure 9.12.

Figure 9.8 (a) Gain and (b) phase response of the second-order low-pass filter with differentiator for Q = 1 (heavy line) and Q = 10 when 1/τ = 1000 s⁻¹

A current output I_dif can be taken from the output of the section by adding an additional transconductance amplifier A₄ as shown in Figure 9.7, which ensure that:

where g_m is the transconductance of this amplifier.

9.3   Current-Domain Second-Order Filter

In this section, we introduce the log-domain circuits used in the implementation of the 2D silicon cochlea presented in Hamilton et al. (2008a, 2008c). While these circuits are not specific to the 2D silicon cochlea, they provide a good introduction to log-domain filters and to operating circuits in the current mode; a good counterpoint to the circuits presented in Section 9.2.

9.3.1   The Translinear Loop

Log-domain filters are based on translinear loops. The translinear loop is a fundamental concept in log-domain circuit design (Gilbert 1975). It is described in Gilbert (1990) as follows:

In a closed loop containing an even number of forward-biased junctions, arranged so that there are an equal number of clockwise-facing and counter-clockwise-facing polarities, the product of the current-densities in the clockwise direction is equal to the product of the current densities in the counter-clockwise direction.

This concept is based on the relationship

and, as such, is only valid when current through a device is an exponential function of the voltages at the terminals of the device, as is the case for BJTs and MOSFETs operating in the sub-threshold region. This concept can be best described when looking at the circuit in Figure 9.9.

In Figure 9.9, M₁ and M₃ comprise the counter-clockwise junctions while M₂ and M₄ comprise the clockwise junctions. Transistor M₅ is necessary to correctly bias the source

Figure 9.9 A translinear multiplier

voltage of both M₂ and M₃ and sink the current running through M₂ and M₃. Hence, given the definition above we can say:

or, defining the current through M₂ as the output of the circuit, we can write:

Equation (9.21) demonstrates the ease with which a multiplier circuit can be constructed when operating in the log-domain. Understanding the principle of translinear loops can also be an important tool when analyzing the function of other log-domain circuits.

Tau Cell

The tau cell (van Schaik and Jin 2003) is a basic building block representing a class of log-domain filters. A schematic of the tau cell is shown in Figure 9.10. The tau cell is designed for complete programmability via the time constant, τ, and the current feedback, A_iv. It can be used as a building block to create a number of more complex, higher order filters.

The tau cell is based on the principle of translinear loops and its core structure is identical to that of Figure 9.9. In Figure 9.10, the closed loop of gate-source junctions necessary to form a translinear loop is created by transistors M₁ to M₄ and hence,

The capacitor C introduces dynamics into the translinear loop resulting in a filter. The current through the capacitor is given by

Figure 9.10 The tau cell

If we assume that the gate capacitance of M₃ and M₄ is significantly smaller than C, that is, if we assume that the dynamics at the gate of M₃ is much faster than at source of M₃, we can simplify Eq. (9.24) with I_M3 = I₀ and using Eq. (9.23) to eliminate I_M2:

In addition, since M₃ and M₄ have the same gate voltage and assuming the voltage at the drain of M₄ is at least 100 mV larger than V_ref, we can write:

Equation (9.26) implies that

We can use this result to eliminate V_C from Eq. (9.25):

In the Laplace domain, the transfer function for a single tau cell can thus be written as

where τ = is the time constant, τ_i is the time constant of stage i, I_i−1 is the input current, I_i is the output current of stage i and I_i+1 = T_i+1I_i. If there is no next stage then T_i+1 = 0 and A_i = 0 by definition.

9.3.2   Second-Order Tau Cell Log-Domain Filter

A second-order low-pass filter can be realized by connecting two tau cells as illustrated in Figure 9.11. Here we see that the first cell has a feedback gain A₁, while the second tau cell has no feedback. The current feedback, A₁I₀I_i+1/I_i, can be implemented using the multiplier of Figure 9.9. A more optimized method to implement the current feedback is described in Hamilton et al. (2008b). The schematic view of the second-order low-pass filter is given in Figure 9.12. The general equation for the second-order low-pass filter in Figure 9.12 is given by

Figure 9.11 Second-order tau cell filter structure

Figure 9.12 Second-order tau cell low-pass filter

where τ is the time constant and Q is the quality factor.

In order to construct a band-pass filter similar to that described in Section 9.2.5, the output from the second tau cell in Figure 9.12, I_out2, must be subtracted from the output of the first tau cell in Figure 9.12, I_out1, such that

where I_out is the output from the band-pass filter. Figure 9.13 shows a band-pass filter structure that is described by Eq. (9.31).

9.4   Exponential Bias Generation

Since there is an exponential relationship between position along the basilar membrane and best frequency in the real cochlea, we will need to use filters with exponentially decreasing cut-off frequencies in our model. In all the silicon cochlear models mentioned in Chapter 4, the exponential dependency is obtained using a linear decreasing voltage on the gates of MOS transistors operating in weak-inversion. In weak-inversion, the drain current of a saturated nFET with its source tied to the bulk and its gate voltage referred to the same bulk can be expressed by

Figure 9.13 A second-order tau cell band-pass filter

with I_S as defined in Eq. (9.2) and V_T0 the threshold voltage of the transistor. This shows that the drain current depends exponentially on the gate voltage. A spatial voltage distribution which decreases linearly with distance is easily created using a resistive polysilicon line; if there is a voltage difference between the two ends of the line, the voltage on the line will decrease linearly all along its length. It is therefore possible to create a filter cascade with an exponentially decreasing cut-off frequency by biasing the amplifiers of Figure 9.6 using MOS transistors whose gates are connected by equal lengths of the polysilicon line (Lyon and Mead 1988). As we can see in Eq. (9.32), however, the drain current also depends exponentially on the threshold voltage and small variations in V_T0 will introduce large variations in the drain current. Because both the cut-off frequency and the quality factor of the filters are proportional to these drain currents, large parameter variations are generated by small V_T0 variations. A root mean square (RMS) mismatch of 12% in the drain current of two identical transistors with equal gate and source voltages is not exceptional (Vittoz 1985), even when sufficient precautions are taken. We can circumvent this problem by using CMOS compatible lateral bipolar transistors (CLBTs) as bias transistors. A CLBT is obtained if the drain or source junction of a MOS transistor is forward-biased in order to inject minority carriers into the local substrate. If the gate voltage is negative enough (for an n-channel device), then no current can flow at the surface and the operation is purely bipolar (Arreguit 1989; Vittoz 1983). Figure 9.14 shows the major flows of current carriers in this mode of operation, with the source, drain, and well terminals renamed emitter E, collector C, and base B.

Since there is no p+ buried layer to prevent injection to the substrate, this lateral npn bipolar transistor is combined with a vertical npn. The emitter current I_E is thus split into a base current I_B, a lateral collector current I_C, and a substrate collector current I_Sub. Therefore, the common-base current gain B = −I_C/I_E cannot be close to 1. However, due to the very small rate of recombination inside the well and to the high emitter efficiency, the common-emitter current gain E = I_C/I_B can be large. Maximum values of E and B are obtained in concentric structures using a minimum size emitter surrounded by the collector and a minimum lateral base width. For V_CE = V_BE − V_BC larger than a few hundred millivolts, this transistor is in active mode and the collector current is given, as for a normal bipolar transistor, by

Figure 9.14 Bipolar operation of the MOS transistor: carrier flows and symbol. Adapted from van Schaik et al. (1996). Reproduced with permission of MIT Press

where I_Sb is the specific current in bipolar mode, proportional to the cross section of the emitter-to-collector flow of carriers. Since I_C is independent of the MOS transistor threshold voltage V_T0, the main source of mismatch of distributed MOS current sources is suppressed, when CLBTs are used to create the current sources. A disadvantage of the CLBT is its low early voltage, that is, the device has a low output resistance. Therefore, it is preferable to use a cascode circuit as shown in Fig. 9.15a. This yields an output resistance several hundred times larger than that of the single CLBT; whereas the area penalty, in a layout as shown in Figure 9.15b, is acceptable (Arreguit 1989).

Another disadvantage of CLBTs, when biased using a resistive line, is their base current, which introduces an additional voltage drop on the resistive line. However, since the cutoff frequencies in the cochlea are controlled by the output current of the CLBTs and since these cut-off frequencies are relatively small (typically 20 kHz or less), the output current of the CLBTs will be small. If the common-emitter current gain E is much larger than 1, the base current of these CLBTs will be very small compared to the current flowing through the resistive line and the voltage error introduced by the small base currents will be negligible. Furthermore, since the cut-off frequencies of the cochlea will typically span two decades with an exponentially decreasing cut-off frequency from the beginning to the end, only the first few filters will have any noticeable influence on the current drawn from the resistive line.

Figure 9.15 CLBT cascode circuit (a) and its layout (b). Adapted from van Schaik et al. (1996). Reproduced with permission of MIT Press

9.5   The Inner Hair Cell Model

As shown in Section 9.2.5 and Figure 9.7, both the voltage-domain and the current-domain silicon cochleae typically have a current output signal for each section representing the velocity of the BM vibration. This current then serves as the input to an IHC circuit, modeling the transduction from vibration to a neural signal. There have been a number of IHC circuit models developed since the first, proposed in Lazzaro and Mead (1989). The more complex circuit models, such as the one presented in McEwan and van Schaik (2004), aim to reproduce the IHC behavior in great detail, including all the various time constants. Here we discuss the IHC model proposed in Chan et al. (2006) that models the two main properties of the IHC, namely, half-wave rectification and low-pass filtering. The circuit for this IHC model is shown in Figure 9.16.

In this circuit a current which represents the band-pass output of a cochlea section, I_c, is half-wave rectified by a current mirror. A DC offset can be added by using V_ioff to set the current I_off. This half wave-rectified current is, as a first approximation, given by

The output of the half-wave rectifier is passed through a first-order log-domain low-pass filter that is based on the tau cell of Figure 9.10, except that it uses pFETs rather than nFETS. The transfer function is nonetheless the same and is given in the Laplace domain by

Figure 9.16 The inner hair cell circuit

where G is given by

The time constant of the low-pass filter is given τ = , where C is modeled by an MOS capacitor. The cut-off is set around 1 kHz as in the biological IHC, modeling the reduction in phase-locking observed on real auditory nerves at frequencies greater than 1kHz. The two control signals V_ref and V_ref0 are slightly below V_dd to allow the two pFETs providing 2I₀ to operate in saturation. Any voltage difference between V_ref and V_ref0 will show up as a current gain depending exponentially on this difference as given in Eq. (9.36).

As shown in Chapter 4, the biological IHC exhibits adaptation to an ongoing stimulus. As a result it responds more strongly to the onset rather than the sustained part of a stimulus. This is facilitated by temporarily suppressing its response after the offset of stimulation. This adaptation can be directly modeled in a more complex IHC circuit, as in McEwan and van Schaik (2004), or it can be emulated by using the output of the IHC to stimulate a neuron with an adaptive threshold. In the second case it is actually the neuron that has a stronger response to the onset of a sound, rather than the IHC, but the end result is very similar. Silicon neurons are discussed in great detail in Chapter 7.

9.6   Discussion

In this chapter we have described circuits for the main building blocks of the silicon cochlea models described in Chapter 4. As discussed in Chapter 4, there are still many open issues in analog silicon cochlea design today; for example, whether the operating domain of these circuits should be current or voltage. The circuits in this chapter face the same design challenges usually of analog designs: noise, dynamic range, and mismatch to name but a few. While current-mode circuits are usually more compact than voltage-mode circuits and have a larger dynamic range, they are more susceptible to variations in threshold voltages. To reduce the mismatch in the fabricated devices, transistors have to be sized properly and calibration circuits are needed. These tradeoffs are taken by the designer in the various cochlea designs. While the design of silicon cochleas has come a long way since the first implementation by Lyon and Mead (1988), we still have a long way to go to match the remarkable performance of biological cochleas. Several research groups worldwide are actively researching ways to get closer to this goal.

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