|
Frequency
Response Measurements
Derive
Frequency Response From Step Response
LAB
in PDF format
Filters,
amplifiers, and control systems are usually characterized by their frequency
response functions. These functions are usually shown in graphical form as plots of log
amplitude vs. log frequency called Bode plots.
Oscilloscopes are primarily time domain measuring instruments.
They represent acquired waveforms as a time series, plotting signal
amplitude as a function of time. Utilizing
the mathematical capabilities available in modern digital oscilloscopes it is
possible to derive the frequency response function of a circuit based on the
measured time response to a step
function.
An
example of this measurement and analysis is shown in figure 1.
A 1 kHz square wave is applied to a low pass filter and the output of the
filter is acquired and displayed in the top trace (Ch 3).
The frequency response function is the Fourier transform of the circuits
impulse response. The impulse
response can be derived from the measured step response by differentiating the
step response. This step is
performed in trace A in figure 1.
Figure
1
– Transforming the measured step response of a filter into the frequency
response.
To increase
the dynamic range of this measurement and improve signal/noise ratio the impulse
response is averaged as shown in trace B.
The Fast
Fourier Transform (FFT) is used to convert the impulse response into the
frequency response function. Trace
C, not shown applies the FFT to trace B. Trace
D, the FFT Average function, provides averaging in the frequency domain for
further improvement in dynamic range. Note
that number of points used in the calculations is user selectable.
In this example the transform size is set to 1000 points yielding a 500
point frequency spectrum. LeCroy
oscilloscopes support FFT calculations with transform sizes of up to 4 Mpoints,
dependent on the options installed.
Trace D, is
the frequency response function shown as a plot of log amplitude (power
spectrum) vs. linear frequency. Relative
time cursors have been setup to measure the 3 dB point of the low pass filter as
33.4 MHz.
This data
can be converted into a classic Bode plot by saving the frequency spectrum to
floppy disk in spreadsheet format and plotting it in Log-Log format using a
spreadsheet, such as Microsoft Excel.
Figure
2 shows the data from trace D in figure 1, re-plotted Log – Log format using
an Excel spreadsheet.
Figure
2
|
;)
;)
