Tool for Modeling and Analysis of Electronic Circuits and Systems. Technical Report

Transcription

1 Tool for Modeling and Analysis of Electronic Circuits and Systems Ons Lahiouel, Henda Aridhi, Mohamed H.Zaki, and Sofiène Tahar Department of Electrical and Computer Engineering, Concordia University, Montreal, Canada {lahiouel,h Technical Report October, 2011 Abstract Analog circuit models provided by the circuit simulator Simulation Program with Integrated Circuit Emphasis offer the most accurate behavior of electronic circuits and systems. However, those models as well as the program that performs the device characterization are not available for researchers who need to have access to a mathematical description to perform optimization and verification of circuits. Consequently, we need to automatically generate analog circuits mathematical models and provide means to change their model parameters and verify their effects. In this report we present a tool for modeling and analysis of analog circuit implemented in MATLAB Object Oriented. This tool takes as input a Netlist file and generates a circuit object that contains all the circuit elements, their attributes and their connectivity. This circuit object together with a set of library Mfiles that model devices and some user defined parameters and constraints are input to an equations formulation step that implements Modified Nodal Analysis formulation and generates MATLAB Mfiles, used for different circuit analysis. We show the application of our proposed tool on a Differential Amplifier and a Rambus ring oscillator. Key words : SPICE, Modified Nodal Analysis, Ordinary Differential Equation 1

2 Table of Contents 1 Introduction 3 2 Tool Description Circuit Object Generation Class Circuit Parser Implementation Flat Circuit Object Generation MNA Object Generation DC Analysis Preparation AC Analysis Preparation Class MNA Modified Nodal Analysis Implementation Equations Generation DC Equations Differential Algebraic Equation Transfer Function Applications Rambus Ring Oscillator Differential amplifier Conclusions 20 References 21 2

3 1 Introduction SPICE [4](Simulation Program with Integrated Circuit Emphasis) is the industrystandard way to model and verify circuit operation at the transistor level before manufacturing an integrated circuit. The models of individual device components, algebraic and differential equations that describe analog circuit behavior, are hard coded into the SPICE tool. In fact, analog circuits mathematical model are just a set of Differential Algebraic Equation (DAE) that can be transformed using certain techniques to Ordinary Differential Equations (ODE). The DC, transient, frequency or whatever type of analysis of these circuits consist in applying numerical methods to the set of equations. In this work a tool for modeling and analysis of analog circuit implemented in Matlab Object Oriented (MOO) [3] is described. The tool parses Netlist file, recognizes circuit components and generates a set of Ordinary Differential Equations through the application of Modified Nodal Analysis (MNA) [2]. In section 2 we illustrate our proposed methodology and we give an overview about the parser that creates a circuit object out of a SPICE Netlist file. Then, we detail the class MNA that is responsible for the generation of the mathematical models of the circuit in different domains. Finally, we provide the form of equations that we generate. In section 3 we show our experimental results on a Differential Amplifier [5], a Rambus ring oscillator [6] and in section 4 we present our conclusions. 2 Tool Description Figure 1 provides the main components of the proposed tool. The necessary steps to generate the circuit models are : Circuit object generation : it consists in reading the Netlist file content, recognizing the SPICE syntax (composed of elements and sub-circuits) and generating a compact version of the circuit object that is structurally equivalent to the Netlist description. After that, a flat circuit object is generated using the flattening function. MNA object generation : this step modifies the flat circuit object to generate the equivalent object for each type of analysis (DC, AC, noise analysis). The modified flat circuit object is input to the generation of MNA object that implements MNA formulation and generates the MNA object. Equation generation : this part takes the MNA object and generates MATLAB Mfiles containing the mathematical models, taht can be used for different circuit analysis. 2.1 Circuit Object Generation Class Circuit 3

4 Circuit Netlist Parsing Circuit Object Flattening Flat Circuit Object Class Cicuit Generation Preparation for DC Analysis Circuit Object prepared for DC Analysis Generation of MNA Object Preparation for AC Analysis Circuit Object prepared for AC Analysis Adding noise sources Circuit Object prepared for noise Analysis Class MNA Generation MNA Object for DC Analysis MNA object MNA Object for AC Analysis MNA Object for noise Analysis DC equations Generation F( x) = 0 ODEs Generation Transfer Function Generation Transfer Function Generation x & = f ( x, t ) H (s) H (s) Generation of Equations Fig. 1: Proposed tool functionalities The class circuit (Figure 2) is a MOO class used to define circuit, sub-circuits and elements objects as they share the same properties defined as follows : Name : This field contains the name of the circuit written in the first line of the input file. Node : This field is a cell array that contains the node of the circuit. Component : This field is a cell array of objects of type circuit. This property is empty in case of elements. Parameter : This field is a cell array that can include parameter of nonlinear components, parameter analysis, characteristic values, physical characteristics, the current model name, list of dependent sources parameters... 4

5 classdef circuit properties name node parameter component end methods function CIRCUIT=circuit(Netlist) end end end Parser Implementation Fig. 2: Class circuit definition The parser is a part of the circuit class constructor that reads the input netlist and outputs an object of type circuit, that we detail its property in the previous section. Netlist Statements line Subcircuit Definition GetSubcircuitComponent Elementary component GetElement Subcircuit call Independent voltage or current source Mosfet Diode... R,L,C GetX GetCV GetD GetM GetComponent Fig. 3: Parser function Figure 3 shows a description of the parser implementation. We have 5 types of processus for different circuit components as they are uniquely defined according to the SPICE syntax. In the following subsections we detail the get element functions GetElement function If the element encountering during the the parsing of the Netlist line statement is an elementary component that is different of a subcircuit definition, then the function GetElement refers to the Get element functions described in the following subsections GetComponent function In case of a resistor, an inductor or a capacitor, we can notice that their syntax in 5

6 SPICE is similar that is why a single function GetComponent deals with those type of element. The name of the element is stored in the name property. Both nodes n1 and n2 are placed in the node property. The characteristic value is placed in the parameter cell field of the component. [name n1 n2 value] R,L,C GetM function In case of a transistor, the function GetM stores the name of the Mosfet and its nodes respectively in the name and node property of the component. The property parameter is filled in with the name of the Mosfet model and all parameters name and value. [Mxxx nd ng ns nb mname L=length W=width AD=val AS=val PD=val PS=val ] M GetD function The function GeD puts the name and node of the diode respectively in the name and node property and puts the diode model name reference in the parameter cell filed of the diode component. The property parameter involves parameters name and value of the diode model. [Dxxx nplus nminus mname ] D GetCV function The GetCV function deals with independent sources : AC, DC, transient and mixed sources. The source name and node are stored in their respective properties. The cell parameter property is filled depending on the type of the source. We can find in it the DC source value, the transient source function and the magnitude and phase of the AC source. [name n+ n- DC= dcval tranfun AC=acmag acphase ] I,V GetCCVS, GetCCVS, GetCCVS, GetVCVS functions The function GetCCVS is called to treat polynomial and linear type of current controlled voltage source. For both types of CCVS, the GetCCVS function puts the connecting node and name of the controlled source in the name and node property of the component and determines whether it is a linear or polynomial one. In case of a linear CCVS, the function puts in the parameter cell, the name of voltage sources, through which controlling current flows and the transresistance which is the current-tovoltage conversion factor. In case of a polynomial CCVS, the function first determines the number of polynomial dimensions that represents the number of the controlling variables of the source and fills it in the parameter cell property. Then the names of voltage sources, through which controlling current flows, as well as the polynomial coefficient are stored in the parameter cell. [Hxxx n+ n- CCVS vn1 transresistance ] CCVS In case of a voltage controlled voltage source, the function GetElement refers to the GetVCVS function that works similarly to the GetCCVS as well as the treatment of 6

7 current controlled current source which is made by the GetCCCS function and Voltage Controlled Current Source done by the GetVCCS function. The difference between those functions consists in the way the MATLAB cell parameter is filled. [Exxx n+ n- VCVS in+ in- gain ] VCVS [Fxxx n+ n- CCCS vn1 gain ] CCCS [Gxxx n+ n- VCCS in+ in- transconductance ] VCCS GetSubcircuit function The GetSubcircuit function puts the element name of the subcircuit in the name property as well as nodes and then stores the name of the subcircuit in the MATLAB cell parameter property. [X n1 n2 n3... SubcircuitName ] GetSubcircuitComponent function Once the parsing of elementary component is finished, the constructor of the class object analyzes the input file to browse for the definition statements of subcircuits already referenced in the netlist. For each subcircuit definition, the parser calls the GetSubcircuitComponent function that creates for every subcircuit a component of type circuit. The function puts the subcircuit name and node respectively in the name and node property and then calls recursively the function GetElement which parses the subcircuit definition, collects component, which can be element or subcircuit, and stores them in the MATLAB cell component. There is no limit to the size or complexity of subcircuits, and subcircuits may contain other subcircuits. After creating the subcircuit component, the GetSubcircuitComponent function stores the object in the cell component of the circuit object..subckt SubName n1 n2 n3... circuit element lines ĖNDS Finally, the parser analyzes the netlist file to recognize all parameters concerning the predefined models and the types of the circuit analysis that we want to perform Flat Circuit Object Generation The function ExpandSubcircuit is applied to the circuit object in order to produce a flat one before proceeding with equation set up. This method instantiates subcircuit elements and insert them wherever the subcircuit is referenced. 2.2 MNA Object Generation DC Analysis Preparation The function DCPreparation prepares the circuit for DC analysis, we disable all the dynamic elements of the circuit object. So, we replace a capacitor by an open circuit 7

8 Compact Circuit Object ExpandSubcircuit Number of subcircuit 0 Flat Circuit Object Fig. 4: Flattening process (large resistor) since at DC, dv dv dt = 0, so that i = C dt = 0 and inductors are replaced by a short circuit because, at DC, dt di di = 0, so that v = L dt = 0. In term of implementation, to prepare the circuit object for DC analysis, is called AC Analysis Preparation Each non linear component of the circuit object is replaced by it s small-signal model around the operating point determined by the DC Analysis. This is accomplished by the function ACPreparation. First, this function parses the cell component and replaces each DC independent voltage source by a short circuit and each current source by an open circuit. Then, it replaces each component corresponding to a diode by a linear voltage controlled current source component whose transresistance is the first term of the linearization expression of the pn-diode element equation. Finally, it replaces each Mosfet component by it s small signal model Class MNA The general form of the MNA equations for linear and non linear circuits is : C d x + Y x + w (x) = Z (t) dt (1) x (0) = x 0 Consequently, the MNA class is composed of 3 properties Y, C, and Z. The matrix w is taken into consideration during the generation of the equations. This class has a singular constructor which takes the circuit object and constructs MNA matrices. The following sections detail the implementation of the parser and the Modified Nodal Analysis method. 8

9 2.2.4 Modified Nodal Analysis Implementation The class MNA has 3 properties : The matrix Y,C and the matrix Z. Each of them will be created by combining several individual sub-matrices. The constructor of the MNA class takes as input the circuit object and generates an MNA object corresponding to this circuit. The first step is to calculate the number of nodes n and the number of independent voltage source m to determine matrices size. Flat circuit object MNA class constructor MNA object Fig. 5: Generation process of MNA object In this section we detail each step performed to construct the MNA object from a flat circuit object (Figure 5) Construction of the Y matrix For resistive circuit, the property Y is (m + n) (m + n) and a combination of 4 matrices G, B, E and D (Figure 6). n m n G B E D m Fig. 6: Matrix Y composition for resistive circuit The conductance matrix G : is nxn and determined by the interconnections between the passive circuit elements (resistors) The B matrix : is nxm and determined by the connection of the voltage sources The E matrix : is mxn and is the transpose of the B matrix The D matrix : is mxm and contains only zero 9

10 Construction of the G matrix The G matrix is the conductance matrix which is the output of the function ConstructG. The function ConstructG initializes to zero the G matrix, then parses the cell component and for each resistor element, it calculates its conductance value. According to its nodes connections, it replaces the conductance value into the G matrix. In other words, the G matrix is formed according two rules : Each element in the diagonal matrix is equal to the sum of the conductance (one over the resistance) of each element connected to the corresponding node. So the first diagonal element is the sum of conductances connected to node 1, the second diagonal element is the sum of conductance s connected to node 2, and so on. The off diagonal elements are the negative conductance of the element connected to the pair of corresponding node. For example if the property node of the element contains 1 and 2, the resistor value goes into the G matrix at location (1,2) and locations (2,1). If an element is grounded, it will contribute only to one entry in the G matrix diagonal. If it is ungrounded (if both nodes are different from zero), it will contribute to four entries in the matrix, two diagonal entries (corresponding to the two nodes) and two off-diagonal entries Construction of the B matrix The B matrix is the output of the function CostructB. It s is an nxm matrix with only 0, 1 and -1 elements where n is the number of node and m the number of independent voltage source in the circuit object. Each location in the matrix corresponds to a particular independent voltage source element (first dimension) or a node (second dimension). If the first element of the node property, which is the positive terminal, corresponding to the ith independent voltage source component is connected to node k, then the element (i,k) in the B matrix is a 1. If the first element of the node property, which is the negative terminal, of the ith independent voltage source component is connected to node k, then the element (i,k) in the B matrix is a -1. Otherwise, elements of the B matrix are zero. If a voltage source is ungrounded, it will have two elements in the B matrix (a 1 and a -1 in the same column). If it is grounded it will only have one element in the matrix Construction of the E matrix The E matrix is the output of the function CostructE. It s is an mxn matrix with only 0, 1 and -1 elements where m is the number of independent source of voltage and n the number of node in the object circuit. The E matrix is the transpose of the B matrix Construction of the D matrix The D matrix is the output of the function CostructD. It s an mxm matrix that is composed entirely of zeros and where m is the number of independent voltage source. 10

11 Once the matrix Y is constructed, it will be updated according to the type of element present in the object cell component. If the cell component of the circuit object contains inductor, the function CaseInductor adds the element stamps corresponding to inductor into the Y matrix. Finally, if the cell component includes a dependent source element, a function is called for each type of controlled source to add the element stamp corresponding Construction of the Z Matrix The next step is to construct the Z vector. This is accomplished by the function ConstructZ. The Z matrix holds with independent voltage and current sources component and is developed as the combination of 2 smaller matrices I and V. The Z vector is (m+n) 1 where n is the number of nodes, and m is the number of independent voltage sources component. The I matrix is nx1 and contains the sum of the currents through the passive elements into the corresponding node (either zero, or the sum of independent current sources) Construction of the V Matrix The V matrix is mx1 and holds the values of the independent voltage sources which are stored in the parameter property. Each element of the V matrix is equal in value to the corresponding independent voltage source Construction of the I Matrix Each element of the I matrix corresponds to a particular node. The value of each element of I is determined by the sum of current sources into the corresponding node. If there are no current sources connected to the node, the value is zero. Once the Z matrix is constructed it will be updated whenever an element stamps is added to the Y matrix Construction of the C Matrix The final step consist in constructing the C matrix that stores all capacitor and inductor values. The function ConstrutC initializes to zero the C matrix, then parses the cell component and for each capacitor element it replaces the capacitor value into the C matrix according to the node property of the capacitor. The C matrix is formed according two rules : Each element in the diagonal matrix is equal to the sum of the capacitor value and the negative inductor value of each capacitor or inductor element connected to the corresponding node. So the first diagonal element is the sum of capacitor value and the negative inductor value connected to node 1, the second diagonal element is the sum of capacitor value and the negative inductor value connected to node 2, and so on. 11

12 The off diagonal elements are the negative capacitor value of the capacitor element connected to the pair of corresponding node. For example if the property node of the element contains 1 and 2, the capacitor value goes into the C matrix at location (1,2) and locations (2,1). If an element is grounded, it will only have contribute to one entry in the C matrix, at the appropriate location on the diagonal. If it is ungrounded (if both nodes are different from zero) it will contribute to four entries in the matrix, two diagonal entries (corresponding to the two nodes) and two off-diagonal entries Modified Nodal Analysis for AC Analysis If the user want to perform AC Analysis, the object created from the circuit object is called mna laplace and has two properties which are A and Z matrices. In this case, the A matrix is constructed in a different form of what we describe before. This difference is in the way we generate the G matrix. The G matrix will not only contain the resistor values but also the complex impedance of capacitor and inductor element stored in the cell component. The complex impedance are placed in the G matrix in the same way it was done for resistor. The complex impedance of a resistor, capacitor and inductor are respectively : 2.3 Equations Generation DC Equations Z R = R Z C = 1 sc = j wc Z L = sl = jwl When intending to run any type of DC Analysis, and before being able to set up the MNA equations seen above, the circuit object generated by the parser is prepared for running DC analysis. This circuit object is the input of the MNA class constructor. The general nonlinear MNA system for DC analysis as given in equation 2 : Y x + w (x) = Z (2) Therefore, the MNA object generated from the circuit object prepared for DC analysis involves the matrices Y and Z as properties. This MNA object is the input of the function GenerateDCequation which generates a MATLAB file containing DC equations as given in equation 3: F = 0 (3) F = Y x + w (x) Z The matrix w, containing the non linear model of the current crossing non linear element, is taken into consideration while writing the equation in the MATLAB file. 12

13 2.3.2 Differential Algebraic Equation If the user of the tool want to perform transient analysis, the circuit object created by the parser is the input of the MNA constructor. The general dynamic nonlinear MNA system as given in equation 4: C dx dt + Y x + w (x) = Z(t) (4) The MNA object created contains as property the matrices Y, C and Z. The function GenerateODE creates a MATLAB file containing the Ordinary Differential Equation as given in equation 5: ẋ = f (x, t) (5) f (x, t) = C 1 (Z (t) Ax w (x)) Transfer Function After preparing the circuit object for AC analysis, the constructor of the MNA generates from the circuit object an MNA object. The general MNA system for AC analysis as given in equation 6: Ax = Z (6) Where : A = (Y + sc) (7) The circuit is solved by a simple matrix manipulation as: x = A 1 Z (8) 13

14 3 Applications In this section, we present experimental results for a Rambus Ring-Oscillator [6], and a differential amplifier [5]. 3.1 Rambus Ring Oscillator In this section, we consider a Rambus ring oscillator made with an even number of stages (n=4), as shown in Figure 7. Each stage has two forward (labeled fwd) inverters connected by a pair of cross-coupling (labeled cc) inverters. Each inverter is single ended and is composed of a cascaded n-channel and p-channel transistors and a capacitance C connected to their drains. The oscillating conditions depend on the transistors sizes and the inial conditions[6]. fwd fwd fwd fwd x(1) x(2) x(3) x(4) cc cc cc x(5) x(6) x(7) x(8) fwd fwd fwd fwd Fig. 7: Rambus Ring-Oscillator Figure 8 shows the SPICE netlist file of the rambus ring oscillator. 14

15 ***Rambus Ring Oscillator*** VDD 9 0 DC=1.0 XI8 4 7 INV_CC_G1 XI9 2 6 INV_CC_G1 XI INV_CC_G1 XI INV_CC_G1 XI INV_CC_G1 XI INV_CC_G1 XI INV_CC_G1 XI INV_CC_G1 XI7 1 8 INV_FWD_G2 XI6 2 1 INV_FWD_G2 XI5 4 2 INV_FWD_G2 XI4 3 4 INV_FWD_G2 XI3 5 3 INV_FWD_G2 XI2 6 5 INV_FWD_G2 XI1 7 6 INV_FWD_G2 XI0 8 7 INV_FWD_G2.SUBCKT INV_FWD_G2 IN OUT C0 OUT 0 1e-12 M1 OUT IN 9 9 CMOSP L=100e-9 W= E-06 MULTI=1.0 M=1.0 M0 OUT IN 0 0 CMOSN L=100e-9 W= E-06 MULTI=1.0 M=1.0.ENDS INV_FWD_G2.SUBCKT INV_CC_G1 IN OUT M1 OUT IN 0 0 CMOSN L=100e-9 W= E-06 MULTI=1.0 M=1.0 AD=46E-15 M0 OUT IN 9 9 CMOSP L=100e-9 W= E-06 MULTI=1.0 M=1.0.ENDS INV_CC_G1.END Fig. 8: Circuit description of a rambus ring oscillator Rambus ring oscillator Transient response The differential model of the Rambus ring oscillator is given in Figure 9. However, the time integration of these model requires the specification of the initial condition x(0) = IC, and leads to the transient behavior shown in Figure 10. Where x is the vector of voltages at the output of each inverter, gnd is the ground voltage, V DD = 1.8 is the power supply voltage, fn are functions that model the n, p- channel transistor current based on their gate, drain and source voltages as well as their sizes. The user may use different trannsistor models or may change the transistor sizes and parameters by changing the MATLAB Mfile fn modeling the behavior of the n and p channel transistors. 15

16 function xdot=ode_rambus(t,x) %%xdot is the derevative of the vector,x %%x(1),is the voltage at the node,1 %%x(2),is the voltage at the node,2 %%x(3),is the voltage at the node,3 %%x(4),is the voltage at the node,4 %%x(5),is the voltage at the node,5 %%x(6),is the voltage at the node,6 %%x(7),is the voltage at the node,7 %%x(8),is the voltage at the node,8 %%x(10),is the current through the voltage source,vdd xdot(1)=1/1e-012*(-fn(x(5),x(1),gnd,gnd,1, P)-fn(x(5),x(1),VDD,VDD,-1, P )-fn(x(2),x(1),vdd,vdd,-1, P )- fn(x(2),x(1),gnd,gnd,1, P )); xdot(2)=1/1e-012*(-fn(x(6),x(2),gnd,gnd,1, P )-fn(x(6),x(2),vdd,vdd,-1, P )-fn(x(4),x(2),vdd,vdd,-1, P )- fn(x(4),x(2),gnd,gnd,1, P )); xdot(3)=1/1e-012*(-fn(x(8),x(3),gnd,gnd,1, P )-fn(x(8),x(3),vdd,vdd,-1, P )-fn(x(5),x(3),vdd,vdd,-1, P )- fn(x(5),x(3),gnd,gnd,1, P )); xdot(4)=1/1e-012*(-fn(x(7),x(4),gnd,gnd,1, P )-fn(x(7),x(4),vdd,vdd,-1, P )-fn(x(3),x(4),vdd,vdd,-1, P )- fn(x(3),x(4),gnd,gnd,1, P )); xdot(5)=1/1e-012*(-fn(x(1),x(5),gnd,gnd,1, P )-fn(x(1),x(5),vdd,vdd,-1, P )-fn(x(6),x(5),vdd,vdd,-1, P )- fun_mos(x(6),x(5),gnd,gnd,1, P )); xdot(6)=1/1e-012*(-fn(x(2),x(6),gnd,gnd,1, P )-fn(x(2),x(6),vdd,vdd,-1, P )-fn(x(7),x(6),vdd,vdd,-1, P )- fun_mos(x(7),x(6),gnd,gnd,1, P )); xdot(7)=1/1e-012*(-fn(x(4),x(7),gnd,gnd,1, P )-fn(x(4),x(7),vdd,vdd,-1, P )-fn(x(8),x(7),vdd,vdd,-1, P )- fun_mos(x(8),x(7),gnd,gnd,1, P )); xdot(8)=1/1e-012*(-fn(x(3),x(8),gnd,gnd,1, P )-fn(x(3),x(8),vdd,vdd,-1, P )-fn(x(1),x(8),vdd,vdd,-1, P )- fun_mos(x(1),x(8),gnd,gnd,1, P )); xdot(9)=0; xdot=xdot(:); Fig. 9: Rambus ring oscillator ODE s x(1)..x(8) [V] Time [s] x Differential amplifier Fig. 10: Rambus ring oscillator transient response In this section, we consider the differential amplifier circuit shown in Figure 11. It consists of a differential pair and a current mirror. The differential pair uses n-channel transistors and load resistors r1 and r2. The current mirror is composed of an ideal DC current source of 60uA and two NMOS transistors used to define the amount of current that flows into the two branches of the differential pair. The differential input 16

17 vdd=1.8v Idc=60uA c1=47.8ff vop vip + - r1=12k Ω r2=12k Ω w=4.4um l=200nm von vin + - c2=47.8ff w=100um l=4.5um Fig. 11: CMOS Differential Amplifier (vid = vip-vin) is applied between the gate terminals of the differential pair transistors which have the same size and properties. Their drains are connected to the capacitors c1 and c2. The netlist file of this circuit is shown in Figure 12. ***CMOS Differential Amplifier*** R1 VDD! VON 12E3 R2 VDD! VOP 12E3 I1 VDD! NET21 DC=60E-6 C1 VON VSS 4.780e-14 M=1.0 C2 VOP VSS 4.780e-14 M=1.0 M1 NET21 NET21 VSS VSS CMOSN L=4.5E-6 W=100E-6 AD= E-11 +AS= E-11 PD= E-04 PS= E-04 NRD= E-03 +NRS= E-03 M=1.0 M2 VOP VIP VTAIL VSS CMOSN L=200E-9 W=4.4E-6 AD= E-12 +AS= E-12 PD= E-06 PS= E-06 NRD= E-02 +NRS= E-02 M=1.0 M3 VTAIL NET21 VSS VSS CMOSN L=4.5E-6 W=100E-6 AD= E-11 +AS= E-11 PD= E-04 PS= E-04 NRD= E-03 +NRS= E-03 M=1.0 M4 VON VIN VTAIL VSS CMOSN L=200E-9 W=4.4E-6 AD= E-12 +AS= E-12 PD= E-06 PS= E-06 NRD= E-02 +NRS= E-02 M=1.0 V1 VIN VSS AC 1.0 SIN E-6 10E9 V2 VIP VSS AC 1.0 SIN E-6 10E9 V0 VDD! VSS DC=1.8.END Fig. 12: SPICE Netlist file of a CMOS differential pair DC, transient and frequency responses of the differential amplifier The DC characteristics of the differential amplifier are shown in Figure 11. The input differential voltage (vid) varies between -1.8 and +1.8 volts. The output differential voltage (vod = vop - von) varies between -0.4 and 0.4 volts and the regions where one 17

18 of the MOSFET is in saturation region and both the transistors are in non-saturation region are clearly seen. The netlist file of this circuit, algebraic equations for DC response, DAEs for the transient response, and the transfer function for the frequency response of the differential amplifier generated by our tool can be found in [?] vop von [V] vip vin [V] Fig. 13: DC transfer characteristic of the differential amplifier function xdot=ode(t,x) %%xdot is the derevative of the vector,x %%x(1),is the voltage at the node,1 %%x(2),is the voltage at the node,2 %%x(3),is the voltage at the node,3 %%x(4),is the voltage at the node,4 %%x(5),is the voltage at the node,5 %%x(6),is the voltage at the node,6 %%x(7),is the voltage at the node,7 %%x(8),is the current through the voltage source,vdd %%x(9),is the current through the voltage source,vin %%x(10),is the current through the voltage source,vinp xdot(1)=0; xdot(2)=1/4.78e-014*(-(8.3333e-005)*x(2)-( e-005)*x(4)-fn(x(1),x(2),x(3),0,1, P )); xdot(3)=0; xdot(4)=0; xdot(5)=0; xdot(6)=1/4.78e-014*(-( e-005)*x(4)-(8.3333e-005)*x(6)-fn(x(7),x(6),x(3),0,1, P )); xdot(7)=0; xdot(8)=0; xdot(9)=0; xdot(10)=0; xdot=xdot(:); Fig. 14: Ordinary Differential Equations of the differential amplifier Where x is the vector of voltages at each node and fn is a function that models current of the transistor. Figure 15 shows the transient time response of the differential amplifier. The differential input vid is a one volt peak-to-peak sinusoidal signal. The signal traces show 18

19 both the initial transient region and the final steady state transient response of the differential amplifier. Moreover, we observe that the differential output signal vod is out of phase with the differential input and has a gain of 1.52 V/V vid=vip vin vod=vop von vid,vod [V] Time [s] x 10 7 Fig. 15: Transient response of the the differential amplifier Figure 16 shows the frequency response of the differential amplifier. The top graph shows the magnitude response and the bottom graph shows the phase response. Bode Diagram Magnitude (db) Phase (deg) Frequency 10(rad/sec) Fig. 16: Frequency response of the differential amplifier 19

20 4 Conclusions In this report, we have described a Tool for Modeling and Analysis of Electronic Circuits and Systems that takes as input a SPICE Netlist file containing circuit features, information and structures and generates a mathematical description of analog circuit resulting from applying Kirchhoff s law and Branch Equations on the circuit, and modeling the current across non linear element such as Mosfet and Diode. Those analog circuit models are generated using a nonlinear set of equations formulation step based on MNA. We demonstrated the capabilities of our tool on two different circuits: a differential amplifier and a Rambus ring oscillator. Moreover, our tool can easily be integrated within a more larger set of tools already existing in MOO inside the host research laboratory or with existing public and commercial layout design and simulation tools such as Cadence Virtuoso, Synopsis HSPICE, Magic and Alliance. Besides, mathematical model generated by the tool can be of greater interest for any verification methodology such as noise analysis, model order reduction and process variation. 20

21 References [1] M. R. Greenstreet. Verifying vlsi circuits. In ATVA 09, pages 1 20, [2] F. N. Najm, Circuit Simulation, John Wiley & Sons, Inc., publication, [3] Object-Oriented Programming in MATLAB, October [4] HSPICE Simulation and Analysis User Guide, SYNOPSYS, October sa.pdf. [5] B. Razavi. Design of Analog CMOS Integrated Circuits. McGraw Hill, [6] M. R. Greenstreet. Verifying vlsi circuits. In ATVA 09, pages 1 20,