On chip LC lowpass ladder filters are used for realization of pulse shaping filters in broadband communication systems [1] and delay lines in equalizers [2]. Pulse shaping filters are used in broadband communications systems for limiting rise times of pulses and eliminating high frequency noise. These filters usually have a 3dB bandwidth that is 75% of the data rate [1], [3]. Fig. 1 shows fifth and seventh order LC ladder Bessel filters for pulse shaping 10 Gb/s data. Conventional implementations of these filters on an integrated circuit, such as the one in [1] require a large area as the inductors have to be laid out sufficiently far from each other to avoid coupling.

The inductors in a lowpass LC ladder filter form a chain and can also be thought of as a single inductor with multiple taps. Realizing separate coils for a number of inductors occupies a larger area than realizing the total inductance using a single coil. This is due to positive mutual coupling between different parts of the coil. Since inductors are usually the largest components on integrated circuits, significant chip area can be saved by using a single inductor with multiple taps instead of separate inductors which need to be laid out far from each other. On the other hand, with a single tapped coil, mutual coupling between different inductors causes deviations in the pulse response and the frequency response which need to be corrected.

In the next section, we compare a single inductor with multiple taps to multiple spirals. In Section III we analyze the effect of coupling between different inductors in a ladder filter. Section IV deals with techniques to restore the pulse response of the filter to the ideal one in presence of mutual coupling. Example realization of a Bessel filter using the proposed techniques are shown in Section V. Detailed simulation results are shown in Section VI.

SECTION II

## SINGLE INDUCTOR WITH MULTIPLE TAPS

Fig. 2(a) shows a simplified layout of the three inductors used for the filter in Fig. 1(b). Fig. 2(b) shows the total inductance (2.26 nH) realized using a single spiral with the same metal width and spacing as in Fig. 2(a). The area occupied by the single inductor is less than half the area occupied by the three inductors^{1}.

Realizing the total inductance as a single spiral results in mutual coupling between different coils. For on chip spirals, the coupling coefficient between turns is 0.6 or less.

SECTION III

## EFFECT OF COUPLING BETWEEN INDUCTORS IN A LADDER FILTER

When the total inductance in a ladder is laid out as a single spiral, there will be mutual coupling between different inductors. This coupling is highest for adjacent inductors and reduces as one goes down the ladder. The coupling between adjacent coils and alternate coils are analyzed below.

The transfer function of the fifth order filter (Fig. 3(a)) with a mutual inductance between adjacent inductors *L*_{2} and *L*_{4} is giver by
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$${V_0(s)\over V_s(s)} = {1-s^2M_{24}C_3\over D_5(s)}\eqno{\hbox{(1)}}$$where *D*_{5}(*s*) is a fifth order polynomial in *s*. Because of mutual coupling between adjacent coils, real zeros are introduced at . These zeros cause an undershoot in the pulse response and a reduced attenuation in the stopband. This effect can also be seen by using the well known representation of two coupled coils in series using three uncoupled coils (Fig. 3(b)). The series resonance of *C*_{3} with the negative inductance −*M*_{24} creates “notches”, i.e., zeros, on the real axis of the *s* plane. In higher order filters, a pair of symmetric zeros are created on the real axis for every pair of adjacent coupled inductors.

To assess the effect of coupling between adjacent inductors, the filter in Fig. 3(a) is simulated with different coupling coefficients between the inductors L _{2} and *L*_{4}. In each case, the effective inductance values *L*_{2}+ *M*_{24} and *L*_{4}+*M*_{24} are adjusted to the desired inductance values shown in Fig. 1(a). This is equivalent to simulating the ideal Bessel ladder filter with uncoupled inductors and an extra negative inductance −*M*_{24} in series with *C*_{3}. Figs. 4 and 5 compare the step response and the frequency response for *k*_{24} = 0 and *k*_{24} = 0.3. The step response exhibits an undershoot and a slower rise than the ideal response. Fig. 6 shows the undershoot in the step response of 5^{th} and 7^{th} order Bessel filters as a function of the coupling coefficient. Fig. 7(b) shows the degradation in the attenuation of these filters at 20 GHz. This is representative of the stopband attenuation.

In filters with three or more inductors, there will also be mutual coupling between inductors that are not adjacent to each other. Consider the seventh order ladder filter in Fig. 1(b) with a mutual inductance between *L*_{2} and *L*_{6}. Its transfer function is given by
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$${V_0(s)\over V_s(s)} ={1-s^2(C_3+C_5)M_{26}-s^4C_3C_5L_4M_{26}\over D_7(s)}\eqno{\hbox{(2)}}$$where *D*_{7}(*s*) is a seventh order polynomial in *s*. Two pairs of zeros are created, one on the real axis and one on the imaginary axis. The imaginary axis zeros will be at a higher frequency than the real axis zeros. The effect of real axis zeros is as described earlier. The imaginary axis zeros create a notch in the frequency response. Because of the zeros, the attenuation in the stopband will be reduced. As alternate inductors couple more weakly than adjacent inductors, the zeros due to the former can be expected to be at higher frequencies than those due to the latter.

SECTION IV

## REDUCING THE EFFECT OF COUPLING

If the coupling is small, its effect can be ignored in certain applications. For instance, assume that a fifth order Bessel filter is used for pulse shaping. The step response of an ideal fifth order Bessel filter has an overshoot of about 1.1% (Fig. 6, *k* = 0). An undershoot of the same magnitude can therefore be reasonably be permitted without undermining the pulse shaping function. This indicates that, if the coupling coefficient is less than 0.2, the filter with coupled inductors can be used without modifications for pulse shaping.

In many cases, the coupling between alternate inductors (*L*_{2} and *L*_{6} is small enough to be ignored but coupling between adjacent inductors is not. A technique to eliminate the effect of coupling between adjacent coils is described below.

As shown in Fig. 3(b), two coupled inductors with one common node can be represented using two uncoupled inductors and a negative inductance connected to the common node. To cancel the effect of coupling, a positive inductor can be connected in series with the branch containing the series inductor as shown in Fig. 8(a). When the positive inductance exactly cancels the negative inductance, the circuit reduces to the one shown in Fig. 8(b) and the zeros (1) disappear. Even if the cancellation is not exact, the zeros will be moved to higher frequencies on the real axis (if the net inductance is negative) or on the imaginary axis (if the net inductance is positive) of the *s* plane. As a result, the degradation in high frequency attenuation and the undershoot or other deviations in the pulse response will be reduced.

In practice, when the mutual inductance to be cancelled is small (a few tenths of a nanohenry), it is not necessary to explicitly implement the positive inductance used for cancellation. The capacitor *C*_{3} in Fig. 8 is connected to the inductor somewhere along the spiral. The starting and ending nodes of the spiral will be close to each other near the periphery. The inductance of the interconnect used to connect *C*_{3} to a tap along the spiral helps cancel the negative inductance. To minimize coupling between the main spiral and the compensating inductance *L*_{c3}, the interconnect can be routed along the centerlines of the square spiral as much as possible.

SECTION V

## SEVENTH ORDER BESSEL FILTER REALIZATION

Fig. 9 shows the layout of a seventh order Bessel filter in a 0.18 μm CMOS process. The spiral is wound using 2.06 μm thick top level metal layer with a 6 μm width and 5 μm spacing between turns. An extra 5 μm gap is left after the first tap to reduce the coupling coefficient to about 0.3. The capacitors used are metal-metal capacitors. Simulation and optimization of the inductor are carried out using Fasthenry [5]. The desired and obtained inductances between taps and coupling coefficients are shown in Table I. The obtained coupling coefficients are close to the assumed values. The interconnect inductance in series with *C*_{3} is much smaller than the desired value for compensation. Therefore, residual undershoot can be expected in this filter. With a clearance of 30 μm on three sides, this filter occupies 48,300 μm^{2}, which is about 15% smaller than the placement area of three separate spirals in Fig. 2(a).

Fig. 2(a) does not show the complete layout. Interconnects and capacitors will occupy additional space. Therefore area saved using the proposed technique is expected to be more than 15%.

SECTION VI

## SIMULATED RESULTS

The filter in Fig. 9 is simulated including all the inductances, parasitic capacitances, and series resistances. The inductance of the spiral, the tap connections, and the ground line and the coupling coefficients between them are simulated using Fasthenry. The parasitic capacitances and resistances are extracted from the layout. The resulting circuit is simulated in the time and frequency domains. The realized filter has an additional loss due to series resistance of the spiral. For clarity, the pulse and frequency responses are normalized to the dc gain of the filter.

Fig. 10 shows the step response of the filter. It shows an undershoot of 1.3% and an overshoot of 0.3%. The residual undershoot and the slowing down in the step response is due to the still uncompensated zero caused mainly by mutual inductance between *L*_{2} and *L*_{4}. Fig. 11 shows the frequency response of the filter. The magnitude error is very small in the passband. The filter provides a high frequency attenuation better than 37 dB. There is a notch in the magnitude response. This is due to parasitic capacitances between the turns of the inductor which create parallel LC branches in the ladder.

Fig. 12 shows the eye diagrams at the output of the ideal and the realized filters. The output eye from the realized filter has a slightly greater closure and asymmetry.

A single spiral with multiple taps can be used to realize lowpass LC ladder filters with a reduced area. Use of a single spiral results in mutual coupling between inductors of the ladder. Coupling between adjacent inductors creates a pair of symmetric zeros on the real axis and coupling between alternate inductors creates two pairs of symmetric zeros on the real and imaginary axes. The real axis zeros due to coupling between adjacent coils can be eliminated using inductors in series with the capacitors. In practice, the interconnect inductance from the taps along the spiral to the capacitors can be used to to eliminate the zeros or to move them to higher frequencies, thus realizing a better approximation to the desired transfer function.

The proposed techniques also benefit conventional ladder filter realizations with uncoupled inductors. The inductors can be moved closer to each other and the effect of resultant coupling can be eliminated using inductors in series with the capacitors. These techniques are also applicable to filters using spirals on a printed circuit board.

Since inductors occupy a large area and require large gaps to reduce mutual coupling to negligible levels, using a single spiral or moving the spirals closer result in significant area savings.