A recent Idea for Design showed a graphical technique for determining the output of an RC filter driven by a pulse-width modulation (PWM) pulse train.1 It requires the manipulation of infinite series with a limit. Therefore, it does not yield steady-state results with confidence.
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A better approach uses the concept of continuity of states and steady-state “wrap-around” to eliminate this shortcoming, since current and voltage values in a real circuit cannot change instantaneously. Current and voltage are variables with continuous values in time, from moment to moment. For a circuit structure that is switched periodically among several states repeatedly, the end state of one structure serves as the starting state of the next.
Using the same RC filter, periodic-input pulse train, and designations as the referenced Idea for Design, Equation 1 shows the filter output when the driving source VA is non-zero:
When the driving source drops to zero, the output is given by:
In Equation 1 and Equation 2, the two starting conditions V0a and V0b are yet to be determined. A gating function with unit steps is also used. At steady state and at the switching boundaries, t = DT and t = T, so Equations 3a and 3b must hold:
This indicates that the end state of active duration with non-zero driving source must act as the starting state, V0b, of the inactive duration with zero drive. Similarly, the end state of the inactive interval must return to the same starting state, V0a, of the active segment. Using these facts and Equations 3a and 3b, we get:
In other words, cyclic starting states are actually known functions of driving source VA, duty cycle D, pulse period T, and time constant RC = τ. The steady state output in one cycle can be written as:
where the multiple-cycle output is given by:
Using the example of the previous Idea for Design, with VA = 1, D = 0.6, T = 20 μs, and τ = 50T, produces the steady-state results of the figure. The output dc level (Equation 7) is obtained by taking the average of Equation 5:and confirms Equation 16 of the previous approach.
The approach presented here gives the true steady-state output in compact, closed form with a high degree of confidence. The technique can be extended to other second- and higher-order circuits that are switched periodically among multiple states.2, 3
1. “Graphically Determine The Output Signal Level Of An RC Filter,” Electronic Design, Oct. 3, 2013.
2. Switch-mode Power Converters, Keng Wu, Elsevier, 2005, ISBN 13:978-0-12-088795.
3. Power Rectifiers, Inverters, and Converters, Keng Wu, Lulu.com, 2008, ISBN 978-1-4357-2023-7.
Keng C. Wu has a BS from Chiaotung University, Taiwan, and an MS from Northwestern University, Evanston, Ill. He has published four books and holds seven U.S. patents. He can be reached at [email protected].