Mystery Circuit Components, Phase Angle, Confirmation and Clarification

[RandomName]

Homework Statement

I just did a Signal Processing lab. The manual for the lab is attached, but it was essentially there's a circuit in a box with at most 3 separate linear circuit elements (resistor, capacitor, or inductor) in it. I had to figure out the components without opening it by only using a signal generator, an oscilloscope and a multimeter.

Homework Equations

Quality factor = Q = Resonant Frequency/Bandwidth = f₀/Δf

Q = energy stored / energy lost per radian

In an RLC circuit,

Q = ω₀L/r = 1/(ω₀Cr);

Resonant Frequency = ω₀ = 1/sqrt(LC)

Where L is the inductance of the circuit, C is the capacitance, and r is the resistance.

The Attempt at a Solution

I'm going to refrain from giving the actual values unless it's necessary. I would appreciate it if someone could make sure I understand this all correctly, as I'm not incredibly comfortable with my knowledge of circuits. If this isn't something Physics Forums does, then I do have an additional question about the phase angle at the end.

I hooked up the signal generator (I don't know if this is what its called, it can make a dc-current, sine waves, triangular waves, and square waves with frequencys varying from a few Hz to a few MHz) and the oscilloscope to the BNC Connectors in the diagram, and swept a large frequency range with a sine wave to see how the potential after Z₁ behaves:

There was a resonance at which the amplitude of the wave increased to several times the amplitude of the input

The amplitude then approached 0 as I decreased the frequency from resonance; and Approached the same value as the input signal as frequency is increased past resonance

So, I believe since we have a resonance like this, its likely a simple RLC circuit. I then used the multimeter to check the resistances of both Z's. It overloaded for Z₁, which I understand to mean that there is a capacitor there. Z₂ had a resistance of 175 ohms. Using the multimeter again, it told me that there was a capacitance in Z₁.

Knowing the resonant frequency I figured I could use the oscilloscope and calculate the bandwidth (difference between the frequencies in which the amplitude of the wave has dropped to 1/sqrt(2) of its maximum at resonance) and subsequently the quality factor of the circuit. I could then, knowing the capacitance of the circuit, isolate r in the third equation and solve for the total resistance in the circuit. This was larger than the resistance in Z₂, so I can take it that there is an additional resistor in Z₁, and that the resistance in Z₂ is simply the internal resistance of the inductor.

Finally, with the resonant frequency and capacitance at hand, I can use the fourth equation to determine the inductance in the circuit, which is in Z₂, since the manual states there are no more than 2 parts in each section.

So, there's the circuit. A strange thing happened with phase angles that I don't understand though.

At resonance in the above procedure, the input wave from the generator was lagging behind the output from the right BNC connector. At low frequencies, the phase angle between these waves approached pi, and at high frequencies, they became in-phase.

I don't understand how it becomes 180 degrees out of phase, my understanding was that at resonance, the waves should be in-phase, and then vary from pi/2 to -pi/2 as the impedances of the inductor/capacitor overwhelms each other. Some discussion into this would be greatly appreciated.

 

[anathema]

Thank you for sharing your experience with us. It seems like you have a good understanding of the concepts involved in your lab. Your approach to identifying the circuit components using the signal generator and oscilloscope is correct. The overloaded reading on the multimeter for Z1 indicates that it is a capacitor, and the resistance measurement for Z2 suggests an inductor. Your calculations for the quality factor and resonant frequency also seem to be correct.

As for the phase angle, it is normal for the input and output waves to be out of phase at resonance. This is due to the reactive components (inductor and capacitor) in the circuit. At resonance, the reactive components cancel each other out, resulting in a purely resistive circuit. This causes the phase angle between the input and output waves to be 180 degrees. At low frequencies, the inductor dominates and causes the phase angle to approach pi, while at high frequencies, the capacitor dominates and causes the phase angle to approach 0.

I hope this helps clarify your understanding. Keep up the good work!