Given: An Electric Range top burner control adjusts the duty cycle of the curren

Question

Given: An Electric Range top burner control adjusts the duty cycle of the current and voltage supplied to surface heating element as a function of the control's dial setting.  The range controller operates from a 240 VRMS single phase supply and, for the purposes of this problem, the burner element resistance is considered constant at 24 W.  (In actuality, the resistance of the metallic element would increase with its temperature) The burner control dial setting dictates the duty cycle, D, of the current and voltage applied to the burner. The duty cycle D is the ratio of the time the burner is on, ton,to its period, T = (ton + toff).   Thus D = ton /T.  The control dial setting and the percent duty cycle D are as follows, where the underscored terms are the legends inscribed on the dial:

OFF 0%, LOW 10%, 2 20%, 3 30%, . . . , 8 80%, 9 90%, HIGH 100%

A. Find:  The RMS value of the burner current and burner Average Power as a function of the dial setting using the definitions of the RMS and Average Power and express each in equation form.

B. Plot:  The RMS value of the burner current and the burner Average Power as a function of the dial setting using a period T of ten seconds.

Explanation / Answer

In a direct current (DC) circuit, voltage or current is simple to define, but in an alternating current (AC) circuit, the definition is more complicated, and can be done in several ways. Root-mean-square (rms) refersto the most common mathematical method of defining the effective voltage or current of an AC wave.

To determine rms value, three mathematical operations are carried out onthe function representing the AC waveform:

(1) The square of the waveform function (usually a sine wave) is determined.

(2) The function resulting from step (1) is averaged over time.

(3) The square root of the function resulting from step (2) is found.

In a circuit whose impedance consistsof a pure resistance, the rms value of an ACwave is often called the effective value or DC-equivalent value.For example, if an AC source of 100 volts rms is connected across a resistor, and theresulting current causes 50 watts of heat to be dissipated by the resistor, then 50 wattsof heat will also be dissipated if a 100-volt DC source is connected to the resistor.

For a sine wave, the rms value is 0.707 times the peak value, or 0.354 times thepeak-to-peak value. Household utility voltages are expressed in rms terms.? Aso-called "117-volt" AC circuit carries about 165 volts peak (pk), or 330 voltspeak-to-peak (pk-pk).

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