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System Simulator >
Frequency Domain Analyses >
Single-Tone Frequency Domain Analysis

Single-Tone Frequency Domain Analysis

Single tone frequency-domain analysis is applicable only to electrical systems comprised of linear/nonlinear frequency-based elements. Systems containing functional time-based elements may not be analyzed in the frequency domain. Single-tone frequency analysis ignores any intermodulation products/harmonics generated by nonlinear elements in the system, such as amplifiers, mixers, and frequency multipliers. Frequency domain analysis is applicable to a general multi-channel nonlinear electrical topology like the one shown in the following figure:

Such a topology is typically comprised of multi-port linear passive elements and two-port linear/nonlinear active elements. In addition, it is assumed that the system has only one input port where a single-tone RF source is applied during analysis, such as those sources with swept parameters.

Single-tone frequency domain analysis is primarily based on evaluating the admittance matrix Y and the corresponding noise based correlation matrix J. The admittance matrix may then be used to solve for the system’s internal/external voltages using Kirchoff’s Current Law (KCL) linear formulation. The Y-parameter representation for two-port/multi-port elements may be easily obtained from the S-parameter characterization by means of commonly used transformation formulas [1].

For an RF source with a given input carrier frequency f_c and an available input power P₁ applied to the input port of the system, the Y-parameters and noise correlation matrices are calculated for each passive and active electrical element within the entire system. These matrix calculations are typically associated with each element’s single-tone input frequency.

In general, during single-tone frequency domain analysis, the frequency within the system may be translated due to the presence of mixers. It is assumed that this frequency translation (i.e., mixing) is linear in nature (i.e., does not generate any additional harmonics) to ensure the presence of a single-tone at each node within the system.

Analysis of harmonics and intermodulation products generated by mixers and nonlinear amplifiers will be discussed later in the section on Multi-tone Frequency Domain Analysis.

For an N-port linear electrical element, the NxN Y-parameters matrix relates the element’s voltages and currents using the following set of linear equations:

Using this NxN Y-parameter matrix, the passive noise correlation matrix may be derived as follows [2]:

where:
K = Boltzmann’s constant
T = Physical temperature in Kelvin
= Noise bandwidth (specified in the frequency domain solution setup dialog)

= Statistical average
* = Complex conjugate

For a two-port active or passive element (Figure 2), the 2x2 Y-parameter matrix relates the element’s voltages and currents using the following set of linear equations:

For an active linear or nonlinear electrical element, the Y-based active noise correlation matrix is given by [2]:

Where:

< . > = Statistical average

* = Complex conjugate

F_min, G₀, B₀, and R are the four spot noise parameters with defaults of 0dB, 1/50, 0, and 50Ω, respectively.

Based on the connections of the different elements within the system, the overall system Y-parameters matrix Y and noise correlation matrix J may be constructed from the individual matrices obtained for all the elements in the system. Assuming the system has a total number of M ports, the system’s Y and J matrices may be constructed in the following form:

is the System’s Y-parameters matrix.

is the System’s passive and active noise correlation matrix .

Where

e = Number of external ports in the system

i = Number of internal ports in the system
= Total number of ports in the system

The system’s Y-parameters and noise correlation matrices Y and J may be reduced to their equivalent Y-parameters and noise correlation matrices Y_eq and J_eq respectively [2]: