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Compact models and delay computation of sub-threshold interconnect circuits

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Abstract

Ultra-low power designs extensively exploit the sub-threshold region of operation of Complementary metal-oxide semiconductor (CMOS) circuits. Though sub-threshold circuit operation shows huge potential towards satisfying the ultra-low power requirement, increased crosstalk and delay have become serious design challenges particularly for sub-threshold interconnects. In this paper, novel analytical time-domain models governing the output voltage and crosstalk-induced delay of CMOS gates driving coupled resistive–capacitive interconnect in sub-threshold domain are presented. Subsequently, the transient analysis of simultaneously switching two and three coupled interconnects is carried out. It is demonstrated that the modeling of driver by linear resistance can lead to about 38 % average error in the estimation of propagation delay. The numerical results illustrate that the proposed model quite accurately estimates the performance of coupled on-chip interconnects. An average error of less than 7 % is observed in estimation of waveform shape and delay.

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Authors and Affiliations

  1. Electronics and Communication Engineering Department, National Institute of Technology (NIT) Hamirpur, Hamirpur, 177005, Himachal Pradesh, India

    Rohit Dhiman & Rajeevan Chandel

  2. Department of Electrical Engineering, Indian Institute of Technology (IIT) Ropar, Punjab, 140001, India

    Rohit Sharma

Authors
  1. Rohit Dhiman
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  2. Rohit Sharma
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  3. Rajeevan Chandel
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Correspondence to Rohit Dhiman.

Appendix 1: one or more transistors operate in the sub-linear region

Appendix 1: one or more transistors operate in the sub-linear region

For a three-line coupled system, it is assumed that Inv1 leaves the sub-saturation region first, followed by Inv2 and finally Inv3 entering the sub-linear region. The solutions of the output voltage when one or more active transistors in each CMOS inverter enter into the sub-linear region are now derived.

1.1 Inv1 operates in the sub-linear region

When Inv1 begins operating in the sub-linear region, the discharge currents of Inv2 and Inv3 are constant, being equal to \(B_{n2}\) and \(B_{n3}\), respectively. Therefore, the differential equations in a three-line coupled system become,

$$\left( {1 + R_{1} \gamma_{n1} } \right)C_{1c} \frac{{dV_{1} }}{dt} - C_{12} \frac{{dV_{2} }}{dt} = - \gamma_{n1} V_{1}$$
(72)
$$C_{2c} \frac{{dV_{2} }}{dt} - \left( {1 + R_{1} \gamma_{n1} } \right)C_{12} \frac{{dV_{1} }}{dt} - C_{23} \frac{{dV_{3} }}{dt} = - B_{n2}$$
(73)
$$C_{3c} \frac{{dV_{3} }}{dt} - C_{23} \frac{{dV_{2} }}{dt} = - B_{n3}$$
(74)

Applying a Laplace transform to (72), (73) and (74), following set of equations are produced,

$$\left[ {\left( {1 + R_{1} \gamma_{n1} } \right)C_{1c} s + \gamma_{n1} } \right]V_{1} \left( s \right) - C_{12} sV_{2} \left( s \right) = \left( {1 + R_{1} \gamma_{n1} } \right)C_{1c} V_{1} \left( {\tau_{nsat}^{1} } \right) - C_{12} V_{1} \left( {\tau_{nsat}^{1} } \right)$$
(75)
$$- \left( {1 + R_{1} \gamma_{n1} } \right)C_{12} V_{1} (s) + C_{2c} V_{2} (s) - C_{23} V_{3} (s) = - \frac{{B_{n2} }}{{s^{2} }} + \frac{{C_{2c} V_{2} \left( {\tau_{nsat}^{1} } \right)}}{s} - \frac{{\left( {1 + R_{1} \gamma_{n1} } \right)C_{12} V_{1} \left( {\tau_{nsat}^{1} } \right) + C_{23} V_{3} \left( {\tau_{nsat}^{1} } \right)}}{s}$$
(76)
$$- C_{23} V_{2} (s) + C_{3c} V_{3} (s) = - \frac{{B_{n3} }}{{s^{2} }} + \frac{{C_{3c} V_{3} \left( {\tau_{nsat}^{1} } \right) - C_{23} V_{2} \left( {\tau_{nsat}^{1} } \right)}}{s}$$
(77)

Let the various coefficients be defined as,

$$A_{1} = \left( {1 + R_{1} \gamma_{n1} } \right)C_{1c} s + \gamma_{n1}$$
(78)
$$B_{1} = - C_{12} s$$
(79)
$$D_{1} = \left( {1 + R_{1} \gamma_{n1} } \right)C_{1c} V_{1} \left( {\tau_{nsat}^{1} } \right) - C_{12} V_{1} \left( {\tau_{nsat}^{1} } \right)$$
(80)
$$A_{2} = - \left( {1 + R_{1} \gamma_{n1} } \right)C_{12}$$
(81)
$$B_{2} = C_{2c}$$
(82)
$$C_{2} = - C_{23}$$
(83)
$$D_{2} = - \frac{{B_{n2} }}{{s^{2} }} + \frac{{C_{2c} V_{2} \left( {\tau_{nsat}^{1} } \right)}}{s} - \frac{{\left( {1 + R_{1} \gamma_{n1} } \right)C_{12} V_{1} \left( {\tau_{nsat}^{1} } \right) + C_{23} V_{3} \left( {\tau_{nsat}^{1} } \right)}}{s}$$
(84)
$$B_{3} = - C_{23}$$
(85)
$$C_{3} = C_{3c}$$
(86)
$$D_{3} = - \frac{{B_{n3} }}{{s^{2} }} + \frac{{C_{3c} V_{3} \left( {\tau_{nsat}^{1} } \right) - C_{23} V_{2} \left( {\tau_{nsat}^{1} } \right)}}{s}$$
(87)

Eqs. (75–77), thus are of the type,

$$A_{1} V_{1} (s) + B_{1} V_{2} (s) = D_{1}$$
(88)
$$A_{2} V_{1} (s) + B_{2} V_{2} (s) + C_{2} V_{3} (s) = D_{2}$$
(89)
$$B_{3} V_{2} (s) + C_{3} V_{3} (s) = D_{3}$$
(90)

Eqs. (88–90) are solved to obtain,

$$V_{1} (s) = \frac{{D_{1} }}{{A_{1} }} + \frac{{B_{1} \left( {C_{3} \left( {A_{2} D_{1} - A_{1} D_{2} } \right) + A_{1} C_{2} D_{3} } \right)}}{{A_{1} \left( {\left( {A_{1} B_{2} - A_{2} B_{1} } \right)C_{3} - A_{1} B_{3} C_{2} } \right)}}$$
(91)
$$V_{2} (s) = - \frac{{C_{3} \left( {A_{2} D_{1} - A_{1} D_{2} } \right) + A_{1} C_{2} D_{3} }}{{\left( {A_{1} B_{2} - A_{2} B_{1} } \right)C_{3} - A_{1} B_{3} C_{2} }}$$
(92)
$$V_{3} (s) = \frac{{D_{3} }}{{A_{3} }} + \frac{{B_{3} \left( {C_{3} \left( {A_{2} D_{1} - A_{1} D_{2} } \right) + A_{1} C_{2} D_{3} } \right)}}{{C_{3} \left( {\left( {A_{1} B_{2} - A_{2} B_{1} } \right)C_{3} - A_{1} B_{3} C_{2} } \right)}}$$
(93)

1.2 Both Inv1 and Inv3 operate in the sub-linear region

When Inv1 and Inv3 operate in the sub-liner region, the discharge current of Inv2 is a constant i.e. dI2/dt = 0. Therefore, the differential Eqs. (72–74) become,

$$\left( {1 + R_{1} \gamma_{n1} } \right)C_{1c} \frac{{dV_{1} }}{dt} - C_{12} \frac{{dV_{2} }}{dt} = - \gamma_{n1} V_{1}$$
(94)
$$C_{2c} \frac{{dV_{2} }}{dt} - \left( {1 + R_{1} \gamma_{n1} } \right)C_{12} \frac{{dV_{1} }}{dt} - \left( {1 + R_{3} \gamma_{n3} } \right)C_{23} \frac{{dV_{3} }}{dt} = - B_{n2}$$
(95)
$$\left( {1 + R_{3} \gamma_{n3} } \right)C_{3c} \frac{{dV_{3} }}{dt} - C_{23} \frac{{dV_{2} }}{dt} = - \gamma_{n3} V_{3}$$
(96)

Applying a Laplace transform to (94), (95) and (96), a solution of the output voltages V1(s), V2(s) and V3(s) is produced. These expressions maintain the same formulation as (A.17)-(A.19). The various coefficients obtained are,

$$A_{1} = \left( {1 + R_{1} \gamma_{n1} } \right)C_{1c} s + \gamma_{n1}$$
(97)
$$B_{1} = - C_{12} s$$
(98)
$$D_{1} = \left( {1 + R_{1} \gamma_{n1} } \right)C_{1c} V_{1} \left( {\tau_{nsat}^{3} } \right) - C_{12} V_{2} \left( {\tau_{nsat}^{3} } \right)$$
(99)
$$A_{2} = - \left( {1 + R_{1} \gamma_{n1} } \right)C_{12}$$
(100)
$$B_{2} = C_{2c}$$
(101)
$$C_{2} = - \left( {1 + R_{3} \gamma_{n3} } \right)C_{23}$$
(102)
$$D_{2} = - \frac{{B_{n2} }}{{s^{2} }} + \frac{{C_{2c} V_{2} \left( {\tau_{nsat}^{3} } \right)}}{s} - \frac{{\left( {1 + R_{1} \gamma_{n1} } \right)C_{12} V_{1} \left( {\tau_{nsat}^{3} } \right) + \left( {1 + R_{3} \gamma_{n3} } \right)C_{23} V_{3} \left( {\tau_{nsat}^{3} } \right)}}{s}$$
(103)
$$B_{3} = - C_{23}$$
(104)
$$C_{3} = \left( {1 + R_{3} \gamma_{n3} } \right)C_{3c} s + \gamma_{n3}$$
(105)
$$D_{3} = \left( {1 + R_{3} \gamma_{n3} } \right)C_{3c} V_{3} \left( {\tau_{nsat}^{3} } \right) - C_{23} V_{2} \left( {\tau_{nsat}^{2} } \right)$$
(106)

1.3 Inv1, Inv2 and Inv3 all operate in the sub-linear region

When Inv1, Inv2 and Inv3 operate in the sub-liner region, the differential Eqs. (72–74) become,

$$\left( {1 + R_{1} \gamma_{n1} } \right)C_{1c} \frac{{dV_{1} }}{dt} - C_{12} \frac{{dV_{2} }}{dt} = - \gamma_{n1} V_{1}$$
(107)
$$\left( {1 + R_{2} \gamma_{n2} } \right)C_{2c} \frac{{dV_{2} }}{dt} - \left( {1 + R_{1} \gamma_{n1} } \right)C_{12} \frac{{dV_{1} }}{dt} - \left( {1 + R_{3} \gamma_{n3} } \right)C_{23} \frac{{dV_{3} }}{dt} = - \gamma_{n2} V_{2}$$
(108)
$$\left( {1 + R_{3} \gamma_{n3} } \right)C_{3c} \frac{{dV_{3} }}{dt} - C_{23} \frac{{dV_{2} }}{dt} = - \gamma_{n3} V_{3}$$
(109)

Applying a Laplace transform to (107)–(109), a solution of the output voltages V1(s), V2(s) and V3(s) is produced. The various coefficients obtained are,

$$A_{1} = \left( {1 + R_{1} \gamma_{n1} } \right)C_{1c} s + \gamma_{n1}$$
(110)
$$B_{1} = - C_{12} s$$
(111)
$$D_{1} = \left( {1 + R_{1} \gamma_{n1} } \right)C_{1c} V_{1} \left( {\tau_{nsat}^{2} } \right) - C_{12} V_{2} \left( {\tau_{nsat}^{2} } \right)$$
(112)
$$A_{2} = - \left( {1 + R_{1} \gamma_{n1} } \right)C_{12} s$$
(113)
$$B_{2} = \left( {1 + R_{2} \gamma_{n2} } \right)C_{2c} s + \gamma_{n2}$$
(114)
$$C_{2} = - \left( {1 + R_{3} \gamma_{n3} } \right)C_{23} s$$
(115)
$$D_{2} = \left( {1 + R_{2} \gamma_{n2} } \right)C_{2c} V_{2} \left( {\tau_{nsat}^{2} } \right) - \left( {1 + R_{1} \gamma_{n1} } \right)C_{12} V_{1} \left( {\tau_{nsat}^{2} } \right) - \left( {1 + R_{3} \gamma_{n3} } \right)C_{23} V_{3} \left( {\tau_{nsat}^{2} } \right)$$
(116)
$$B_{3} = - \left( {1 + R_{2} \gamma_{n2} } \right)C_{23} s$$
(117)
$$C_{3} = \left( {1 + R_{3} \gamma_{n3} } \right)C_{3c} s + \gamma_{n3}$$
(118)
$$D_{3} = \left( {1 + R_{3} \gamma_{n3} } \right)C_{3c} V_{3} \left( {\tau_{nsat}^{2} } \right) - \left( {1 + R_{2} \gamma_{n2} } \right)C_{23} V_{2} \left( {\tau_{nsat}^{2} } \right)$$
(119)

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Dhiman, R., Sharma, R. & Chandel, R. Compact models and delay computation of sub-threshold interconnect circuits. Analog Integr Circ Sig Process 84, 53–65 (2015). https://doi.org/10.1007/s10470-015-0557-4

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