A method for generating time-domain equivalent macromodel of transmission lines based on delay extraction

Abstract

A method for generating a time-domain equivalent macro model of a transmission line based on delay extraction is used for frequency-domain and time-domain simulation of high-speed interconnects. The specific steps include: (a) based on the given unit length parameters of resistance R, inductance L, capacitance C, and conductance G, a frequency-domain mathematical model of the transmission line impedance form is established; and the poles and residues of the mth order in the low-frequency band are calculated, and the value of m is continuously adjusted according to the given error threshold δ until the error condition is met; (b) based on the pole residues calculated in step (a), an approximate expression of the impedance response in the high-frequency band is constructed, and the signal is discretized and converted into a pure delay term in the frequency domain to complete the explicit delay extraction; (c) based on the calculation results in steps (a) and (b), the frequency-domain state space model of the transmission line is obtained after optimization, and the corresponding time-domain equivalent circuit netlist is generated.

Description

Technical Field

[0001] The invention belongs to the field of integrated circuits, and in particular relates to a method for generating a time-domain equivalent macro model of a transmission line under computer-aided design. Background Art

[0002] Transmission lines are mainly used to connect devices and transmit signals. In order to accurately simulate the behavior of transmission lines, a series of effects brought by transmission lines must be considered: delay, crosstalk, ringing effect, etc. With the increase of system clock frequency and scale complexity of integrated circuits, the requirements for accurate simulation modeling of high-speed transmission lines are gradually increasing to ensure the reliability and signal integrity of the entire system. Unreliable transmission line design will have an adverse effect on the performance of the entire system. Traditional SPICE simulation tools can provide accurate simulation results for reference in the early stages of design, but the calculation cost is high.

[0003] The scale of transmission line systems is very large, with a large number of nodes and high analysis complexity. Traditional model reduction technology uses numerical methods to reduce the lumped parameter model of the transmission line into a mathematical model with a smaller dimension. While mathematically retaining the input-output relationship of the original system, it can also retain physical properties such as stability and passivity. Its representative works include model reduction methods based on singular value decomposition methods [1][2] and model reduction methods based on Krylov subspaces [3][4].

[0004] The order reduction method based on Krylov subspace mainly uses Krylov subspace iterative calculation to obtain an approximate reduced-order system through moment matching. The kth moment of the system is the coefficient of the kth derivative of the transfer function at the frequency point ω. Moment matching means that at a specific frequency point, the reduced-order system transfer function retains the moment of the original transfer function to achieve an approximate frequency response. For wide-bandwidth and long transmission lines, matching only at a single frequency point cannot achieve high-precision approximation of the frequency band far away from the matching frequency point, so a moment matching method at multiple frequency points is required. This also means that in actual use, high-order approximation or calculation of more poles is required to capture the delay effect of the transmission line. Especially for lossy transmission lines and frequency-varying parameter transmission lines, this order reduction method consumes a lot of computing resources [5].

[0005] The main idea of ​​the order reduction method based on singular value decomposition is to solve all the singular values ​​of the system, retain the larger singular values, and discard the smaller singular values ​​to achieve the effect of reducing the model dimension. However, solving the singular values ​​requires Cholesky decomposition of the observable and controllable matrix and solving the Gram matrix of the system. The computational complexity is high and the iterative calculation robustness is weak. It is not suitable for large-scale circuit systems and is often used to process high-order control system models.

[0006] In order to solve a series of problems in the traditional order reduction method, a transmission line model based on explicit delay extraction is proposed. By using this explicit delay extraction method to calculate the frequency domain transfer function of the transmission line, a low-order rational expression with pure delay terms in the frequency domain can be obtained, which reduces the computational complexity and saves computational costs. At the same time, the consumption of time domain simulation computing resources is also reduced accordingly. Summary of the invention

[0007] The purpose of the present invention is to provide an optimized time-domain equivalent macro model of a transmission line, and to provide a method for establishing a time-domain equivalent model of a transmission line network composed of resistors, inductors, capacitors, conductors, and power devices.

[0008] In order to achieve the above-mentioned effects, the specific technical solution adopted by the present invention is:

[0009] A method for generating a time-domain equivalent macro model of a transmission line based on delay extraction, characterized in that the method comprises the following steps:

[0010] (a) Based on the given unit length parameters of resistance R, inductance L, capacitance C, and conductance G, a frequency domain mathematical model of the transmission line impedance form is established; and the poles and residues of the mth order in the low frequency band are calculated. The denominator polynomial composed of these poles and the corresponding residues determines the low frequency band frequency response of the impedance transfer function matrix;

[0011] (b) Based on the pole residues calculated in step (a), an approximate expression of the high-frequency impedance response is constructed, and the signal is discretized and converted into a pure delay term in the frequency domain to complete the explicit delay extraction;

[0012] (c) Based on the calculation results in step (a) and step (b), the frequency domain state space model of the transmission line is obtained after optimization, and the corresponding time domain equivalent circuit netlist is generated.

[0013] Beneficial Effects

[0014] The poles and residues contained in the high-frequency response signal do not need to be obtained by solving linear equations, but are constructed based on the poles and residues of the low-frequency signal. This step saves a lot of frequency domain data calculation costs.

[0015] Compared with the traditional reduced-order model, the characteristic delay of the transmission line is explicitly extracted, replacing most of the pole polynomials, which greatly reduces the scale of the time domain equivalent model and is conducive to improving the efficiency of time domain simulation.

[0016] The state space model is generated by using the state space minimum realization method. This algorithm can generate the most optimized diagonal standard state space model: the elements in the matrix A only appear on the diagonal, and it is a sparse diagonal matrix.

[0017] The entire model is passive, stable, and synthesizable; it can handle transmission lines with lossless, lossy, and frequency-varying parameters. BRIEF DESCRIPTION OF THE DRAWINGS

[0018] Figure 1 It is a vector diagram of the input and output ports of the transmission line system

[0019] Figure 2 This is the schematic diagram of the input signal and external devices in Example 1

[0020] Figure 3 This is the calculation result of the first 20 poles in Example 1.

[0021] Figure 4 This is the result of the high frequency pole construction in Example 1.

[0022] Figure 5 It is the netlist composition and time domain simulation calculation consumption

[0023] Figure 6 This is the output voltage signal waveform in Example 1 DETAILED DESCRIPTION

[0024] In conjunction with the relevant drawings, the specific implementation methods adopted by the present invention will be discussed in detail herein.

[0025] A method for establishing a time domain equivalent model of a transmission line network, characterized in that a time domain equivalent circuit model of a transmission line composed of resistance, inductance, capacitance, conductance, and power supply devices is established, and the specific steps of the establishment method are:

[0026] (a) Determine the number of coupled transmission lines that constitute the transmission line network, and establish a frequency domain mathematical model of the transmission line network impedance form based on the known unit length parameters of resistance R, inductance L, capacitance C, and conductance G, which can be expressed as follows:

[0027]

[0028] Where: x represents the position coordinate of the transmission line; V(x,s), I(x,s) represent the voltage vector and current vector at the position x of the transmission line respectively. R(s), L(s), G(s), C(s) represent the resistance, inductance, conductance, and capacitance parameters of the transmission line in the frequency domain respectively.

[0029] Combining the two equations in the above formula, we can get the transfer function matrix in impedance form. For a system consisting of N transmission lines, its impedance transfer function matrix is ​​expressed as follows:

[0030]

[0031] Where V0(s) and I0(s) represent the voltage vector and current vector of the input port at coordinate 0, respectively, and both have dimensions of N. l (s), I l (s) are respectively represented as the voltage vector and current vector at the output port at coordinate l, both with dimension N; the impedance matrix Z(s) has dimension 2N×2N, and each sub-matrix block Z mn The dimension is N×N.

[0032] The first m-order poles and residues of each impedance sub-matrix are calculated respectively. The denominator polynomial composed of these poles and corresponding residues determines the low-frequency frequency response of the impedance transfer function matrix. In order to solve the poles and residues, the impedance matrix Z(s) is split into an infinite series expression, and the specific m-th order pole P can be obtained. k,m Calculation formula:

[0033] det[Z S ]=0

[0034]

[0035] Among them, det represents the value of the calculated determinant, and Y P , Z S It is expressed as follows:

[0036]

[0037] Corresponding residue R k,m The calculation formula is:

[0038]

[0039]

[0040] According to the error convergence judgment condition and the given error threshold δ, the value of the modulus m that actually needs to be calculated is determined.

[0041] The specific pole iteration calculation steps are:

[0042] (i) Input data: δ, R(s), L(s), G(s), C(s)

[0043] (ii) Let modulus m = 1 and calculate the pole according to formula (1.3), denoted by P k,1 ; and calculate the residue corresponding to the pole according to formula (1.5), denoted as R k,1 .

[0044] Let modulus m = 2, and calculate the pole according to formula (1.3), denoted as P k,2 ; and calculate the residue corresponding to the pole according to formula (1.5), denoted as R k,2 .

[0045] (iii) Let m = m + 1 and calculate the extreme point, denoted by P k,m ; Calculate the residue, denoted as R k,m .

[0046] (iv) Calculation error judgment conditions:

[0047]

[0048] If the above judgment conditions are met, then exit the loop; otherwise, jump to step (iii).

[0049] (v) Output data: the first m-order poles P k,m and residue R k,m .

[0050] Since then, the previous The low-frequency transfer function expression composed of polynomials of order:

[0051]

[0052]

[0053] (b) Based on the first m-order poles and residues calculated in step (a), a frequency domain response expression for the high frequency band is constructed, and the expression is explicitly delayed extracted to obtain a low-order rational expression with pure delay terms.

[0054] Sub-step 1 constructs a high-frequency signal expression based on the calculation results in step (a). Compared with step (a), it does not need to solve the linear equation, which consumes too much calculation cost. The reason is that the error in step (a) has converged to below the threshold δ, and can be fitted with the help of the periodicity of the high-frequency response signal.

[0055]

[0056]

[0057] In the above formula, the m-th order parameter satisfies the following formula:

[0058]

[0059]

[0060] In step 2, the high frequency band signal is explicitly delayed and extracted. The high frequency band signal is discretized to obtain a discrete spectrum. The spectrum is Laplace transformed to obtain a low-order polynomial with a pure delay term.

[0061] The specific steps for delayed extraction are as follows:

[0062] 1. For the time domain signal s d (t) Discretization

[0063] s n (t) = s d (t-nT k ) (1.10)

[0064] in,

[0065] 2. Perform Fourier transform on discrete signals to obtain discrete Fourier spectrum

[0066]

[0067] 3. Calculate the frequency response in the s-domain based on the discrete Fourier spectrum

[0068]

[0069] After the above delay extraction, the approximate calculation formula of the high-frequency signal is as follows:

[0070]

[0071]

[0072] (c) In step (a) and step (b), the response signal of the low frequency band and the response signal of the high frequency band are obtained respectively, wherein the low frequency band response is composed of the first m-order poles and the corresponding residues, and the high frequency band signal is composed of the expression with pure delay terms, and the sum of the two together constitutes the response signal in the required frequency band:

[0073] Z 11 =Z 22 =Z l11 +Z h11

[0074] Z 12 =Z 21 =Z l12 +Z h12 (1.14)

[0075] Based on the above expression, it is expanded into a state space model in the frequency domain, and the corresponding time domain equivalent circuit netlist is generated according to the state space model. 11 , Z 12 , Z 21 , Z 22 All of them are composed of polynomials. These sub-matrices are transformed into equivalent state space models, which are expressed as follows:

[0076]

[0077] Among them, the vector i(t) is the current signal at the transmission line port, x(t) is the internal state variable, and v(t) is the voltage signal at the transmission line port.

[0078] The transfer function matrix Z(s) represented by formula (1.11) does not contain repeated poles. Using the state space minimum implementation algorithm, a state space model containing state variables with minimum dimensions can be obtained. Based on this minimized state space model, an equivalent circuit netlist with the least number of nodes can be generated. The specific implementation steps are as follows:

[0079] (i) The transfer function expression is expressed as

[0080] Among them, p1,…p q is the pole of the transfer function, R i The extreme point p i The corresponding residue matrix has the same dimension as the transfer function matrix.

[0081] (ii) According to the residue matrix R corresponding to each pole i , and the rank r of the residue matrix i , respectively construct the pole p i The diagonal standard form substate space (A i ,B i ,C i ,D i ).matrix is a diagonal matrix with the poles p i Repeat r on the opposite diagonal i The matrix A is constructed i It is expressed as follows:

[0082]

[0083] Matrix B i , Ci together constitute the residue matrix R i Matrix B i Represents the input and output port mapping matrix, matrix C i Represents the residue matrix corresponding to the poles and satisfies the following formula:

[0084] R i =C i B i (1.17)

[0085] (iii) The sub-state space (A) corresponding to each pole i ,B i ,C i ,D i ) to obtain the diagonal standard state space minimum realization of the system transfer function matrix G(s). It can be expressed as:

[0086]

[0087] v(t)=[C1 … C i ]x(t) (1.18)

[0088] Based on the state-space model of minimization implementation, the generated netlist has fewer nodes and contains fewer devices than the general state-space model. The netlist can be used to perform fast time-domain simulation of the transmission line.

[0089] The state space model represented by formula (1.15) consists of two equations:

[0090] It is a state variable equation, which reflects the influence of the input of the transmission line system on the transfer of state variables at a certain moment. Matrix A is a diagonal matrix composed of the poles in the impedance matrix, and matrix B is the input selection matrix.

[0091] v(t) = Cx(t) + Di(t) is the input-output equation, which indicates the relationship between the output and input of the system at a certain moment and the state variables. Matrix C is a symmetric matrix composed of the residues in the impedance matrix, and matrix D is a direct connection input and output matrix.

[0092] The standard synthesis method of transmission lines synthesizes the two equations in the state space into two sub-networks, the input-output network and the state variable network, and connects the two independent networks through independent sources and controlled sources. The independent sources and controlled sources jointly bear the coupling relationship between the input and output vectors of each state variable.

[0093] The specific synthesis method is explained below using a single pole as an example.

[0094] The single-pole state-space network is represented as follows:

[0095]

[0096]

[0097] 1) State variable network synthesis

[0098] In the state space model, the state variable x(t) closely links the input signal and the output signal, which is usually realized by using energy storage elements. The value of the state variable can be expressed as the voltage value across the capacitor or the current in the inductor at a certain moment. In the transfer function in the form of an impedance matrix, since the input vector is the current at the port and the output vector is the voltage at the port, x(t) is synthesized as the current value i flowing through the inductor at time t. L , The integrated voltage value is the voltage across the unit inductor. It is recorded as:

[0099]

[0100] Where i = 1, ..., N, N is the number of state variables in the state space. For each state variable x in the equation i (t), are integrated into an independent state network representation according to the parameter matrix.

[0101] Taking formula (1.19) as an example, the state equation can be divided into the following four sub-equations:

[0102]

[0103] According to each sub-equation, a corresponding state variable x is synthesized based on Kirchhoff's voltage law. i (t) of the subnetwork.

[0104] Matrix A represents the transfer of each state variable of the system at time T: the diagonal elements reflect the variable x i The state transitions of the self are represented by resistors; the off-diagonal elements reflect the variables x j For variable x i The transfer of the input variables to the state variables is represented by the controlled sources. The matrix B represents the transfer of the input variables to the state variables, all represented by the controlled sources.

[0105] In the simulation tool HSPICE, H is used to represent a current-controlled voltage source. Component H_s1 represents the input i1 to the variable The contribution of the component H_d1 to the variable x3 is Contribution; component Vst_1 is a zero-value voltage source, which is used to sample the current value flowing through the unit inductor L_1 in real time, that is, the state variable x1.

[0106] 2) Input and output network synthesis

[0107] According to the input-output equation in formula (1.15), the output variable v(t) consists of two parts, one part is directly affected by the input variable i(t) through the matrix D, and the other part is contributed by the state variable x(t) through the matrix C.

[0108] Taking formula (1.19) as an example, the input-output equation can be divided into the following two sub-equations:

[0109]

[0110] Similarly, the current value x sampled by the zero-value voltage source Vst_i in each state network is i, multiplied by the coefficients in matrix C, to form the output contributed by the state variable; the input current sampled by Vio_1 in its own network is multiplied by the coefficients in matrix D to form the output directly contributed by the input current. Since all the input and output networks are source devices, in order to avoid simulation errors, a minimum value resistor R_1 is embedded in the input and output networks.

[0111] The device values ​​in the above netlists may be negative, but as long as the passivity of the impedance matrix is ​​guaranteed, negative device values ​​in the netlist are allowed.

[0112] The following is an analysis and verification with specific numerical examples:

[0113] Example 1: The transmission line system consists of a single conductor transmission line, and its input signal and external devices are composed of Figure 2 As shown in the figure, the input resistor R1 has a value of 30 ohms, and the terminal capacitor C2 has a capacitance of 1.5 pF. The input signal is a step signal with a rise time of 200 ps.

[0114] The unit length parameter is entered in scientific notation as:

[0115] R=8.5646

[0116] L = 4.7876e-009

[0117] C=2.24e-12

[0118] G=0 (1.23)

[0119] According to the above input parameters, specific numerical calculations are performed according to the implementation steps.

[0120] In step (a), the first 20 poles are calculated. At this point, the error convergence condition is met and the calculation is stopped. The calculation results of the first 20 poles are as follows: Figure 3 shown.

[0121] Step (b), construct the high frequency response pole in formula (1.13), the result is as follows Figure 4 As shown. And calculate the delay T k =2.07136e-09.

[0122] Step (c) converts the transfer function matrix in the form of an impedance matrix into a state space model, synthesizes it into a netlist, and performs time domain simulation verification in SPICE simulation software. The number of components in the netlist, as well as the simulation time and memory consumption are shown in Figure 2. Figure 5 As shown. Under the input signal drive, the terminal output voltage signal waveform is as follows Figure 6 shown.

Claims

1. A method for generating a time-domain equivalent macro model of a transmission line based on delay extraction, characterized in that: The method has the following steps: (a) Based on the given unit length parameters of resistance R, inductance L, capacitance C, and conductance G, a frequency domain mathematical model of the transmission line network impedance form is established; and the poles and residues of the mth order in the low frequency band are calculated. The denominator polynomial composed of these poles and the corresponding residues determines the low frequency band frequency response of the impedance transfer function matrix; (b) Based on the pole residues calculated in step (a), an approximate expression for the high-frequency impedance response is constructed. The signal of the impedance response in the high frequency band is discretized and converted into a pure delay term in the frequency domain to complete the explicit delay extraction; (c) based on the calculation results in step (a) and step (b), a frequency domain state space model of the transmission line is obtained after optimization, and a corresponding time domain equivalent circuit netlist is generated; The frequency domain mathematical model of the transmission line network impedance form is expressed as follows: Where: x represents the position coordinate of the transmission line; V(x,s), I(x,s) represent the voltage vector and current vector at the position x of the transmission line respectively; R(s), L(s), G(s), C(s) represent the resistance, inductance, conductance, and capacitance parameters of the transmission line in the frequency domain respectively; Combining the two equations in the above formula, we can get the transfer function matrix in impedance form; for a system consisting of N transmission lines, its impedance transfer function matrix is ​​expressed as follows: Where V0(s) and I0(s) represent the voltage vector and current vector of the input port at coordinate 0, respectively, and both have dimensions of N. l (s), I l (s) are respectively represented as the voltage vector and current vector at the output port at coordinate l, both with dimension N; the impedance transfer function matrix Z(s) has dimension 2N×2N, and each sub-matrix block Z mn (Z 11 ,Z 12 ,Z 21 ,Z 22 ) dimension is N×N; The low-frequency frequency response calculation method of the impedance transfer function matrix Z(s) represented by formula (1.2) is as follows: Solving for Poles and Residues Split the impedance transfer function matrix Z(s) into an infinite series expression to obtain the specific m-th order pole P k,m Calculation formula: det[Z S ]=0 Among them, det represents the value of the calculated determinant, and Y P , Z S It is expressed as follows: Corresponding residue R k,m The calculation formula is: According to the error convergence judgment condition and the given error threshold δ, the value of the modulus m that actually needs to be calculated is determined; The specific pole iteration calculation steps are: (i) Input data: δ, R(s), L(s), G(s), C(s); (ii) Let modulus m = 1 and calculate the pole according to formula (1.3), denoted by P k,1 ; and calculate the residue corresponding to the pole according to formula (1.5), denoted as P k,1 ; Let modulus m = 2, and calculate the pole according to formula (1.3), denoted as P k,2 ; and calculate the residue corresponding to the pole according to formula (1.5), denoted as R k,2 ; (iii) Let m = m + 1 and calculate the extreme point, denoted by P k,m ; Calculate the residue, denoted as R k,m ; (iv) Calculation error judgment conditions: If the above judgment conditions are met, then jump out of the loop; otherwise jump to step (iii); (v) Output data: the first m-order poles P k,m and residue R k,m ; Before The low-frequency transfer function expression composed of polynomials of order: The specific process of step (b) is as follows: With the help of the periodicity of the high-frequency response signal, we can get: In the above formula, the m-th order parameter satisfies the following formula: By performing explicit delay extraction on the high-frequency signal and discretizing the high-frequency signal, a discrete spectrum can be obtained, and a low-order polynomial with a pure delay term can be obtained after Laplace transformation of the spectrum. The specific steps for delayed extraction are as follows: S1. For time domain signal s d (t) Discretization s n (t)=s d (t-nT k ) (1.10) in, S2. Perform Fourier transform on discrete signals to obtain discrete Fourier spectrum S3. Calculate the frequency response in the s-domain based on the discrete Fourier spectrum After the above delay extraction, the approximate calculation formula of the high-frequency signal is as follows: The specific process of step (c) is as follows: The sum of the low-frequency response and the high-frequency response obtained in step (a) and step (b) together constitutes the response signal within the required frequency band: WITH 11 =Z 22 =Z l11 +Z h11 WITH 12 =Z 21 =Z l12 +Z h12 (1.14) Based on the above expression, it is expanded into a state space model in the frequency domain, and the corresponding time domain equivalent circuit netlist is generated according to the state space model; the sub-matrix block Z in formula (1.14) 11 , Z 12 , Z 21 , Z 22 All of them are composed of polynomials. These sub-matrices are transformed into equivalent state space models, which are expressed as follows: Among them, vector i(t) is the current signal at the transmission line port, x(t) is the internal state variable, and v(t) is the voltage signal at the transmission line port; The impedance transfer function matrix Z(s) represented by formula (1.2) does not contain repeated poles. Using the state space minimum realization algorithm, a state space model containing state variables with minimum dimension can be obtained. Based on the minimized state space model, an equivalent circuit netlist with the least number of nodes can be generated. The specific implementation steps are as follows: (i) The impedance transfer function matrix expression is expressed as Among them, p1,…p q is the pole of the transfer function, r i The extreme point p i The corresponding residue matrix has the same dimension as the transfer function matrix; (ii) According to the residue matrix R corresponding to each pole i , and the rank r of the residue matrix i , respectively construct the pole p i The diagonal standard form substate space (A i ,B i ,C i ,D i );matrix is a diagonal matrix with the poles p i Repeat r on the opposite diagonal i The matrix A is constructed i It is expressed as follows: Matrix B i , C i Together they form the residue matrix R i ; Matrix B i Represents the input and output port mapping matrix, matrix C i Represents the residue matrix corresponding to the poles and satisfies the following formula: R i =C i B i (1.17) (iii) The sub-state space (A) corresponding to each pole i ,B i ,C i ,D i ) is combined to obtain the diagonal standard form state space minimum realization of the system transfer function matrix G(s); it is expressed as: v(t)=[C1 … C i ]x(t) (1.18)。 2. A method for generating a time-domain equivalent macro model of a transmission line based on delay extraction as claimed in claim 1, characterized in that: The method for generating the state space model adopts a state space minimum realization method, and the algorithm can generate the most optimized diagonal standard state space model: the elements in the matrix A only appear on the diagonal, and it is a sparse diagonal matrix.

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