AC Electric Circuits

Mixed-Frequency Signals

31 questions By Tony R. Kuphaldt

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Question 13 of 31

The Fourier series for a square wave is as follows:

v_square = 4
π V_m ( sinωt + 1
3 sin3 ωt + 1
5 sin5 ωt + 1
7 sin7 ωt … + 1
n + sinn ωt )

Where,

V_m = Peak amplitude of square wave

ω = Angular velocity of square wave (equal to 2 πf, where f is the fundamental frequency)

n = An odd integer

Electrically, we might represent a square-wave voltage source as a circle with a square-wave symbol inside, like this:

Knowing the Fourier series of this voltage, however, allows us to represent the same voltage source as a set of series-connected voltage sources, each with its own (sinusoidal) frequency. Draw the equivalent schematic for a 10 volt (peak), 200 Hz square-wave source in this manner showing only the first four harmonics, labeling each sinusoidal voltage source with its own RMS voltage value and frequency:

Hint: ω = 2 πf

Reveal answer

Notes:
To be honest, the four-harmonic equivalent circuit is a rather poor approximation for a square wave. The real purpose of this question, though, is to have students relate the sinusoidal terms of a common Fourier series (for a square wave) to a schematic diagram, translating between angular velocity and frequency, peak values and RMS values.

Please note that the voltage magnitudes shown in the answer are RMS and not peak! If you were to calculate peak sinusoid source values, you would obtain these results:

1st harmonic: [40/(π)] volts peak = 12.73 volts peak

3rd harmonic: [40/(3 π)] volts peak = 4.244 volts peak

5th harmonic: [40/(5 π)] volts peak = 2.546 volts peak

7th harmonic: [40/(7 π)] volts peak = 1.819 volts peak
Question 14 of 31

Suppose a non-sinusoidal voltage source is represented by the following Fourier series:

v(t) = 23.2 + 30 sin(377t) + 15.5 sin(1131t + 90) + 2.7 sin(1508t - 40)

Electrically, we might represent this non-sinusoidal voltage source as a circle, like this:

Knowing the Fourier series of this voltage, however, allows us to represent the same voltage source as a set of series-connected voltage sources, each with its own (sinusoidal) frequency. Draw the equivalent schematic in this manner, labeling each voltage source with its RMS voltage value, frequency (in Hz), and phase angle:

Hint: ω = 2 πf

Reveal answer

Notes:
The purpose of this question is to have students relate the sinusoidal terms of a particular Fourier series to a schematic diagram, translating between angular velocity and frequency, peak values and RMS values.
Question 15 of 31

Calculate the power dissipated by a 25 Ω resistor, when powered by a square-wave with a symmetrical amplitude of 100 volts and a frequency of 2 kHz:

Reveal answer

P_R = 400 watts

Notes:
To calculate this power figure, students have to determine the RMS value of the square wave. Thankfully, this is not difficult.