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Exercise:
A pulse wave is a periodical signal consisting of constant pulses. For a given period T a pulse wave is characterised by its amplitude V_ and the pulse duration tau. The duty cycle Dtau/T is the ratio of pulse duration and period. center includegraphicswidthtextwidth#image_path:pulswavno-fill-# center abcliste abc Derive a formal expression for the mean value of a pulse wave with duty cycle D. abc Derive a formal expression for the root mean square value of a pulse wave with duty cycle D. abcliste

Solution:
abcliste abc The mean value is given by V_m fracT _^T vt textdt fracT _^tau V_ textdt + fracV_ tauT V_ D abc The mean square sscVrms is given by sscV^rms fracT _^T v^t textdt fracT _^tau V_^ textdt + fracV_^ tauT V_^ D It follows for the root mean square sscVrms V_ sqrtD abcliste The figure below shows the mean and root mean square for a duty cycle DDO. The shaded area above the mean value is equal to the shaded area below for each cycle. Since the duty cycle is always less than the root mean square is always greater than the mean value. An animated version is linked to this exercise. center includegraphicswidthtextwidth#image_path:pulswav# center The graphical meaning of the root mean square can be seen in the figure below square of the signal. The shaded area above the square of the root mean square value is equal to the shaded area below for each cycle. center includegraphicswidthtextwidth#image_path:pulswavsquared-# center
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Exercise:
A pulse wave is a periodical signal consisting of constant pulses. For a given period T a pulse wave is characterised by its amplitude V_ and the pulse duration tau. The duty cycle Dtau/T is the ratio of pulse duration and period. center includegraphicswidthtextwidth#image_path:pulswavno-fill-# center abcliste abc Derive a formal expression for the mean value of a pulse wave with duty cycle D. abc Derive a formal expression for the root mean square value of a pulse wave with duty cycle D. abcliste

Solution:
abcliste abc The mean value is given by V_m fracT _^T vt textdt fracT _^tau V_ textdt + fracV_ tauT V_ D abc The mean square sscVrms is given by sscV^rms fracT _^T v^t textdt fracT _^tau V_^ textdt + fracV_^ tauT V_^ D It follows for the root mean square sscVrms V_ sqrtD abcliste The figure below shows the mean and root mean square for a duty cycle DDO. The shaded area above the mean value is equal to the shaded area below for each cycle. Since the duty cycle is always less than the root mean square is always greater than the mean value. An animated version is linked to this exercise. center includegraphicswidthtextwidth#image_path:pulswav# center The graphical meaning of the root mean square can be seen in the figure below square of the signal. The shaded area above the square of the root mean square value is equal to the shaded area below for each cycle. center includegraphicswidthtextwidth#image_path:pulswavsquared-# center
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Branches
Electrodynamics
Tags
ac circuit, pulse width, rms
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Difficulty
(2, default)
Points
0 (default)
Language
ENG (English)
Type
Calculative / Quantity
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Decoration
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