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A new computational technique for distortion analysis of nonlinear circuits is presented. The new technique is applicable to the same class of circuits, namely, weakly nonlinear and time-varying circuits, as the periodic Volterra series. However, unlike the Volterra series, it does not require the computation of the second and third derivatives of device models. The new method is computationally efficient compared with a complete multitone nonlinear steady-state analysis such as harmonic balance. Moreover, the new technique naturally allows computing and characterizing the contributions of individual circuit components to the overall circuit distortion. This paper presents the theory of the new technique, a discussion of the numerical aspects, and numerical results.

RF circuits are generally designed to be linear with respect to the signal path. However, the desired signal may be weakly distorted due to nonlinearities of the circuit components. Analyzing this nonlinear distortion is an important problem in the design of RF circuits [

The traditional approach to measuring distortion is to apply one or more pure test tones to the circuit’s input and determine harmonics or intermodulation products at the output [

Consequently, distortion characteristics can be obtained using quasiperiodic steady-state nonlinear analysis such as harmonic balance technique [

For distortion analysis of time-invariant weakly nonlinear circuits, the approach based on Volterra series is more efficient [

The essential disadvantage of this approach is the necessity to compute power series coefficients for each nonlinearity in the circuit. This requires that the simulator has explicitly coded second- and third-order derivatives of the device models. With modern device models, which include nonlinear dependencies in several variables as well as different model behavior in different regions of device operation, computation of high-order derivatives is extremely difficult, if not impossible. Also this requirement limits the introduction of new models that may prevent the distortion analysis with a wide class of behavioral models.

In comparison with conventional nonlinear distortion analysis based on computing distortion about the DC operating point, distortion analysis of communication circuits often requires determining the distortion about a periodically time-varying operating point. In this case, the input signal is considered as a small excursion about the periodically varying operating point, and the distortion characteristics can be obtained from the simulation of weakly nonlinear behavior with respect to the input signal [

Thus, the introduction of an efficient special-purpose distortion mode into circuit simulators is a current challenge in RF and microwave CAD engineering.

We focus on the following practical problems for distortion analysis. First, in order to avoid the analytic computation of second and third derivatives of device models, we need an alternative to the Volterra series technique that retains the same overall accuracy. Second, we need to handle the problem of distortion of periodic time-varying circuits. The third practical problem we wish to address is the computation of the individual contributions of each component of the circuits to the overall distortion. This of course has important applications in guiding circuit design. The Volterra series technique supports such computation, so we need to retain this capability.

This paper discusses a new approach for the distortion analysis [

It is important to note that the proposed approach does not require computing high-order derivatives of device model nonlinearities, and thus there is no need to code the second and third derivatives for all the device models. This approach provides basically the same order of accuracy as the Volterra series, due to properties of the simplified Newton’s method. The technique can be readily implemented in a general-purpose circuit simulator.

A similar approach to distortion analysis has been proposed [

The paper is organized as follows. Section

The distortion analysis is intended to provide a measure of the distortion products when one or more pure sinusoids are applied to the input of the circuit. In RF circuits, there may be an extra periodic excitation that determines the periodically varying operating point.

In this case, the nonlinear circuit can be described by a system of differential-algebraic equations [

Equation (

The purpose of the nonlinear distortion analysis is the evaluation of the deviation of the circuit behavior from the desired linear behavior. There are effective methods to provide this evaluation, such as Volterra series [

These methods present the approximate solution of (

Each deviation is computed by solving the linear system in the frequency domain. The matrices of linear systems are obtained using linear AC analysis. The rhs vectors (equivalent current sources) are obtained by the computation of polynomials of previous-order deviations. The polynomials coefficients are defined by Taylor expansion of voltage-current and voltage-charge characteristics of circuit devices. The accuracy of the approximate solution (

Usually it is assumed that nonlinearities are sufficiently small to neglect terms higher than third order. Thus, the analysis leads to successively solving the set of linear systems for each order beginning from the first order. The first-order solution

The main advantage of Volterra series in the comparison with complete nonlinear analysis is the reduction of computational efforts. This results in a much smaller dimension of the small signal linear systems in the comparison with the dimension of the linear system solved at the iteration step of nonlinear analysis.

The essential disadvantage of the Volterra series is the necessity of computing high-order power series coefficients for each nonlinearity in component models. This requirement limits the introduction of new device models because it leads to the very complex practical problem of analytically deriving high-order derivatives of the models.

Below, we discuss the new approach that provides the similar accuracy (

Firstly, we can point out that any computational method to determine steady-state solution of (

Here, we apply the simplified Newton’s method [

The numerical scheme contains one Newton step and (

Using the estimate

Thus, the simplified Newton’s method provides the same accuracy order as the Volterra series method (

To obtain corrections

One shortcoming of the approach is connected with the possible growth of numerical noise in (

The first-order Taylor expansion of

Taking into account that

Subtracting (

If step 2 is performed by (

Expression (

We now apply the simplified Newton’s method described in Section

Solving the nonlinear system (

The vector of unknowns contains all harmonics of the linear combinations with integer coefficients of all fundamentals

The number of complex unknowns in (

Now, we consider the approximate solution of (

The initial step (

After determining the initial guess

The number of unknowns in each system (

The entries of the matrix

Thus, each step of the simplified Newton’s iterative process includes two numerical procedures:

evaluation of the full rhs vector

solving the linear systems (

The evaluation of the full rhs vector for the modified step 3 (

Before proceeding with the distortion analysis, it is first necessary to determine the harmonics that must be considered in the computations.

The number of large signal harmonics is determined during the solving of the single-tone HB problem (

In the case of two small excitations, frequency sets (

Note that at the last step, it is sufficient to determine only those harmonics of (

Small signal systems at step 2 are solved only for second-order harmonics (

The computational procedure includes the following steps.

Solve the system (

Save in memory the solution plus the conductance and capacitance matrices obtained at the last Newton step

This step coincides with the standard periodic small signal analysis [

(a) Determine the first-order solution in time domain by applying inverse Fourier transform

(b) Compute time domain vectors of circuit charges, currents, and admittance matrices,

(c) Compute the residual vector in the frequency domain by applying the Fourier transform

(d) Decompose the vector

(e) Determine corrections of solution by solving the systems

(a) Transform the second-order correction to time domain by applying inverse Fourier transform

(b) Compute vector (

(c) Decompose vector

(d) Determine corrections by solving systems

The conversion of signals from time to frequency domain and vise versa is performed using the multidimensional fast Fourier transform, which results in some aliasing error due to the finite number of sample points. Defining this number in accordance with the Nyquist frequency (

The method for distortion analysis of weakly nonlinear circuits described by (

In this case, the initial step of the analysis is determining the vector of DC solution. Thus, the vector

The linear-centric approach for distortion analysis of time-varying and weakly nonlinear circuits has been described in [

The linear-centric technique [

It can be mentioned that this computational procedure is equivalent in practice to two first steps of our approach. However, as it follows from (

The following simple illustrative example demonstrates that the linear-centric model can lead to incorrect results due to limitations of number of solved linear problems.

Let a nonlinear resistor with voltage-current relation

The first-order correction is determined from the linear system obtained by linearization (

Thus, we see that the voltage distortion does not contain third harmonics. Here,

For this simple example, it is easy to obtain the solution of (

This function contains third-order terms in its Taylor expansion (

An important problem in nonlinear distortion analysis is the computation of the contributions of each nonlinear component to the total circuit distortion. This information allows designers to determine which circuit elements are responsible for the distortion, thus providing guidance in meeting the required specifications.

Numerical characterization of contributions is to be achieved if one can evaluate the distortion metric in the vector form

total distortion is the sum of all components of the vector

However, for third-order distortion, this approach is incomplete because various nonlinearities can interact, and this interaction prevents different contributions from being associated with only one nonlinearity [

Consider the two-stage amplifier shown in Figure

Two-stage amplifier.

The output signal of the amplifier is obtained by

One can see that the expression for the second-order distortion

In contrast, the expression for the third-order distortion contains not only terms corresponding to each nonlinearity (

Hence, the influences of circuit nonlinearities on the third-order distortion can be presented as a matrix of contributions

The meaning of the nondiagonal term

Here, the matrix

The amplifier with feedback loop.

The transfer function

The similar generation of third-order distortions by the mixing of linear and second-order signals is present in any nonlinear circuit (as can be proved by Volterra series theory). So the third-order distortion contributions of the circuit can be characterized by the square matrix

The diagonal entry

The following properties of the contribution matrix exist.

The sum of all matrix entries is equal to the total third-order distortion

If parameters of the

The sum of entries of

If the circuit contains a group of nonlinearities

Thus, the contribution matrix allows one to obtain various information on the influence of circuit nonlinearities upon the output nonlinear distortion.

The utility of the nondiagonal contribution entries can be illustrated by the following way. If the main contribution is defined by diagonal entry, then the overall circuit distortion can be improved by reducing of corresponding nonlinearity. But if such entry is nondiagonal one

For the determination of individual contributions, the functional description of circuit model

In accordance with (

For the representations (

Here, we exploit the fact that from (

Each term in the residual vector of (

To obtain third-order distortion as the sum of individual contributions, we use a modified expression (

Thus, the third-order distortion is

Usually the distortion at the output node is of interest. The components of vectors (

The algorithm for obtaining individual contributions is based on expressions (

The computations can be performed in the same numerical framework presented in Section

Let

We compare our new approach for the periodic distortion analysis based on HB method (PDHB) with complete multitone HB analysis. Numerical experiments are performed for two MOSFET mixers. The first circuit is the single-balanced mixer containing 3 MOS transistors. The first mixer has input parameters

The computed dependencies of the output power of the frequency component (

Output power of frequency component (

Output power of frequency component (

The computational efficiency of the PDHB approach for the two mixer circuits can be seen from Table

Circuit | Linear system order | CPU time, sec | |||
---|---|---|---|---|---|

Number | HB | PDHB | HB | PDHB | |

1 | 30 | 91140 | 1860 | 360 | 13 |

2 | 50 | 79380 | 1500 | 399 | 15 |

The first example is the two-stage bipolar amplifier shown in Figure

Two-stage bipolar amplifier.

Third harmonic and its contributions.

The second example is an OpAmp UA741 [

Contributions to the third harmonic.

A new approach for nonlinear distortion analysis has been presented. This approach does not require the computation of second and third derivatives of the semiconductor device models and, thus, has an important advantage over the Volterra series-based distortion analysis. The new technique is applicable to the class of weakly nonlinear and periodic time-varying circuits.

A computational algorithm for periodic distortion analysis of communication circuits has been developed. Three steps of the algorithm provide the same order of accuracy that three steps of the distortion procedure based on Volterra series methods.

In comparison with complete nonlinear steady-state analysis (such as harmonic balance), the approach provides significant reducing of computational efforts.

In addition, the new technique allows computing the contributions of individual circuit components to the distortion. A matrix form for the characterization of third-order individual distortion contributions has been proposed. The numerical procedures for computing individual contributions have been developed, which provide designers with valuable information on the influence of various nonlinearities upon the total distortion.