Fourier Analysis is a method of analyzing complex periodic waveforms. It permits any nonsinusoidal period function to be resolved into sine or cosine waves, possibly an infinite number, and a DC component. This permits further analysis and allows you to determine the effect of combining the waveform with other signals.
Given the mathematical theorem of a Fourier series, the period function f(t) can be written as follows:
f(t) = A0 + A1cosωt + A2cos2ωt + … + B1sinωt + B2sin2ωt + …
A0 = The DC component of the original wave.
A1cosωt + B1sinωt = The fundamental component (has the same frequency and period as the original wave).
Ancosnωt + Bnsinnωt = The nth harmonic of the function.
A, B = The coefficients.
ω = 2¶/T = The fundamental angular frequency, or 2π times the frequency of the original periodic wave.
Each frequency component (or term) of the response is produced by the corresponding harmonic of the periodic waveform. Each term is considered a separate source. According to the principle of superposition, the total response is the sum of the responses produced by each term. Note that the amplitude of the harmonics decreases progressively as the order of the harmonics increases. This indicates that comparatively few terms yield a good approximation.
When Multisim performs Discrete Fourier Transform (DFT) calculations, only the second cycle of the fundamental component of a time-domain or transient response (extracted at the output node) is used. The first cycle is discarded for the settling time. The coefficient of each harmonic is calculated from the data gathered in the time domain, from the beginning of the cycle to time point t. That is set automatically and is a function of the fundamental frequency. This analysis requires a fundamental frequency matching the frequency of the AC source or the lowest common factor of multiple AC sources.
Fourier Analysis produces a graph of Fourier voltage component magnitudes and, optionally, phase components versus frequency. By default, the magnitude plot is a bargraph but may be displayed as a line graph.
The analysis also calculates Total Harmonic Distortion (THD) as a percentage. The THD is generated by notching out the fundamental frequency, taking the square root of the sum of the squares of each of the n harmonics, and then dividing this number by the magnitude of the notched out fundamental frequency:
THD = [ (Si=2Vi2)/V1] x 100%
where Vi is the magnitude of the ith harmonics.
Consider the triangle-wave generator shown in Figure 1. This circuit generates a triangular waveform with a frequency of about 1 kHz; the circuit was taken from . You will use Fourier Analysis to determine its frequency spectrum.
Figure 1. Triangle-wave generator.
Complete the following steps to configure and run a Fourier Analysis:
Table 1. Parameters used in Fourier Analysis.
Frequency resolution (Fundamental frequency)
Number of harmonics
Stop time for sampling (TSTOP)
Edit transient analysis
Display as bar graph
Degree of polynomial for interpolation
Note: In SPICE, the command that performs a Fourier Analysis has the following form:
.FOUR FREQUENCY OUTPUT_SPECIFICATION <OUTPUT_SPECIFICATION …>
Where .FOUR initializes a Fourier Analysis; FREQUENCY specifies the fundamental frequency of the transient waveform to be analyzed; OUTPUT_SPECIFICATION specifies the quantity to be reported as the result of the Fourier Analysis. Note that these parameters are similar to those defined in Table 1, however, in Multisim you do not have to worry about the SPICE syntax.
Figure 2. Analysis Parameters for the Fourier Analysis.
Figure 3. Transient Analysis window.
For more details on how to configure the Analysis Parameters tab refer to the Transient Analysis tutorial.
Figure 4. Output variable for the Fourier Analysis.
Figure 5. Fourier Analysis results.
As you can see, results are divided in two parts: the chart lists detailed information including the DC component and THD; the graph show the frequencies and their magnitudes. You can use the Cursors to take precise measurements.
 OP-AMP Circuits and Principles, Howard M. Berlin, Sams Publishing, 1991, ISBN 0-672-22767-3.
Collaborate with other users in our discussion forums
A valid service agreement may be required, and support options vary by country.