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[Jay Barracato]

Today I was setting up a demo to show the effect of changing frequency of a sinusoidal input on the amplitude and phase of the output. The expectation was that increasing the frequency should decrease the amplitude of the output and change the phase of the output.


Technically speaking, this is a sinusoidal steady state system, so with a input of vi(t) = Vi cos (wt) we would expect an output in the form vo(t) = A Vi cos (wt + phi) where A is a constant that is less than 1 and phi is the change in phase. (Not needed to understand the photos but for a LR series circuit A = R/sqrt(R^2 + w^2*L^2) and phi= arctan(-wL/R). Since the output cannot precede the input in time, the phi is chosen so the phase is lagging.


For the model, I used an input with a amplitude of 10V, a 1.0mH inductor, and a 1K resistor. The multisim model compares frequencies of 100kHz, 1MHz, and 10 MHz. For the real build, I dialed this back to a range from 50kHz to 500kHz.


The frequency effect on the amplitude and phase was clearly visible.


The real world point I was trying to convey is that any load can be modeled by a combination of its resistance, capacitance, and inductance (i.e. its impedance) and when using sinusoidal inputs we need to pay attention to both the amplitude and phase.

The first photo is the multisim model. The second is the breadboard, and the final three are the oscilloscope measurements comparing the purple input with the yellow output.

[Tim Hite]

I should probably know this already. What are the variables w and t?

[Jay Barracato]

I should probably know this already. What are the variables w and t?

W (usually written as Omega) is the frequency in radians or 2pi(frequency in hz) and t is time in seconds.

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[Frank Koenig]

Try some lissajous figures as an alternative representation. Keep up the good work. --Frank

[Jay Barracato]

Try some lissajous figures as an alternative representation. Keep up the good work. --Frank

Had to look that one up. What Frank is referring to is treating the Input and Output functions of t as parametric equations and graphing them x vs y. Since the w is the same for both functions, the graph shows the phi as a repeating pattern. Since the ratio of the amplitudes is NOT =1, the pattern is an ellipse.


Photos go from relatively low (50Khz) to high (500 kHz) in frequency.

[Tim Hite]

so as y/x approaches 1 you would expect a circle with diameter = phi?

Had to look that one up. What Frank is referring to is treating the Input and Output functions of t as parametric equations and graphing them x vs y. Since the w is the same for both functions, the graph shows the phi as a repeating pattern. Since the ratio of the amplitudes is NOT =1, the pattern is an ellipse.


Photos go from relatively low (50Khz) to high (500 kHz) in frequency.

[Jay Barracato]

so as y/x approaches 1 you would expect a circle with diameter = phi?

I am still trying to suss this out, but according to Wikipedia, the circle is when phi is 90 degrees. The aspect ratio of the ellipse, as phi goes from 0 to 90 degrees goes from a line, through ellipses, to a circle.

So it looks to me like the radius of the circle would be sqrt((Acos(wt))^2+(Bcos(wt+pi/2)^2)).

In other words, as a polar plot, we substituted cos(x+pi/2) for sin(x).

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[Tim McCulloch]

I am still trying to suss this out, but according to Wikipedia, the circle is when phi is 90 degrees. The aspect ratio of the ellipse, as phi goes from 0 to 90 degrees goes from a line, through ellipses, to a circle.

So it looks to me like the radius of the circle would be sqrt((Acos(wt))^2+(Bcos(wt+pi/2)^2)).

In other words, as a polar plot, we substituted cos(x+pi/2) for sin(x).

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Don Davis discusses tuning with Lissajous figures in the first edition of Sound System Engineering.  I think that was not included in subsequent editions.

[Jay Barracato]

Don Davis discusses tuning with Lissajous figures in the first edition of Sound System Engineering.  I think that was not included in subsequent editions.

I can picture that, especially at the crossover frequency and the corner frequency of the filters.

I was already thinking of patching together a set of adapters to measure the filters with smaart. It would be interesting to look at them with the scope simultaneously.

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