# AC Circuit Analysis: LCR & Resonance

• DTskkaii
In summary, the conversation is about a student struggling with an AC question in their first year electrical engineering coursework. The question involves finding the resonant frequency and voltage required to break a glass in a simple AC LCR circuit. The student is looking for guidance and clarification on the steps and equations involved in solving the question. They have already solved the first question and are now trying to figure out the next steps. The conversation also touches on the relationship between VL, Vc, and Vs at resonance, and the energy dissipated in a resistor at different frequencies. The student is seeking further clarification and help on the next steps of the question.
DTskkaii
Hi, I am currently struggling with this AC question that is required for my first year electrical engineering coursework. I know sort of how to go about it all, but am having a really difficult time implementing the correct calculations. Any help would be appreciated!

Context
You are given a simple AC LCR circuit (setup in that order, in series, depicted graphically as a square). The AC Voltage source in the circuit replicates the input energy from the sound waves hitting the glass. The inductor and capacitor replicate the resonance characteristics of the glass, and the resistor replicates the energy being converted by the glass as it vibrates. When the energy converted in the resistor is over 200J/s the glass can no longer support vibration and shatters.

Given Values
Vs = ?
L = 4H
C = 3 x 10^(-10)F
R = 50Ohms

Questions
(1) Find the resonant frequency (rad/sec and Hz) of the sheet of glass.
(2)Find the magnitude of the voltage across the resistor in the equivalent circuit, required to break the glass, at the resonant frequency.
(3) Find the magnitude of the voltage source in the equivalent circuit, required to break the glass at resonant frequency. Compare the last two values found.
(4) Using the voltages found above, find the gain (in dB, gain = 20log(Vout/Vin), where Vout is across the resistor) vs the frequency for the following points: 0.01 x w0, 0.1 x w0, w0, 10 x w0, 100 x w0 (where w0 is the resonant frequency.
(5) The clients plan to use that is only accurate to within +-20% of the desired frequency.
- How much sound power do they need to be sure to break the glass?
- At the worst case scenario (20% away from resonant frequency ) find the voltage magnitude and phase shift across each component; source, inductor, capacitor and resistor.

Sorry for the wall of text, but I just wanted to get it all out so there wasn't any confusion or misinformation. Again, it would be greatly appreciated if anyone can offer guidance on this question in any degree of haste. To note, I do really want to learn the methodology, so please include some small descripts of the steps youre taking, if its necessary :)

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Just fyi, I've done the first question, and am struggling to figure out my next step.

Just to show I'm not trying to get answers without trying :)

I agree with your resonant frequency.
Do you know the relationship of VL, Vc and the supply voltage Vs at resonance?

Thankyou for the fast response.

Sorry if I'm a little slow on the theory, we were only taught the content last week.

I believe this is the point where angular frequencies come into play.?

At resonance VL = Vc (you could say that they 'cancel' out) so the supply voltage is the voltage across the resistance R

Ooh okay.
So in the instance of Q2 and Q3, specifically the 'compare' the two figures, they must be equal. Cool :)

Can you allude at all to the Q2 calculations please?
I'm trying to find what the next step is, and it seem I should be saying that angular frequency = 200w/s, and finding Zr, Zc, Zl; Zt, then using I=Vs/Vt, and then Vr=IZr to get the potential across the resistor. However, I am not given the Vs. (obviously, this would make the question void in the first place). I can't find where I have been taught another way to approach this.

You calculate the frequency from the fact that XL = Xc at resonance.
You should have equations for calculating XL and Xc that include frequency...

DTskkaii said:
Ooh okay.
So in the instance of Q2 and Q3, specifically the 'compare' the two figures, they must be equal. Cool :)

Can you allude at all to the Q2 calculations please?
I'm trying to find what the next step is, and it seem I should be saying that angular frequency = 200w/s, and finding Zr, Zc, Zl; Zt, then using I=Vs/Vt, and then Vr=IZr to get the potential across the resistor. However, I am not given the Vs. (obviously, this would make the question void in the first place). I can't find where I have been taught another way to approach this.

Under what condition will the glass break? What is the energy dissipated in a resistor w.r.t. the voltage across it? Does it depend upon frequency?

*removed for revision*

Will post again if I have another question :)

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*removed for revision

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## 1. What is an LCR circuit?

An LCR circuit is an electrical circuit that contains an inductor (L), a capacitor (C), and a resistor (R). These three components are connected in series or parallel and interact with each other to create a varying voltage and current.

## 2. How do you calculate impedance in an LCR circuit?

The impedance of an LCR circuit can be calculated using the formula Z = √(R² + (Xl - Xc)²), where R is the resistance, Xl is the inductive reactance, and Xc is the capacitive reactance. Impedance is measured in ohms (Ω).

## 3. What is resonance in an LCR circuit?

Resonance in an LCR circuit occurs when the inductive reactance and capacitive reactance are equal, canceling each other out. This causes the impedance to be at its minimum, resulting in maximum current flow. Resonance can be observed at a specific frequency, known as the resonant frequency.

## 4. How do you find the resonant frequency of an LCR circuit?

The resonant frequency of an LCR circuit can be calculated using the formula f = 1 / (2π√(LC)), where f is the resonant frequency, L is the inductance, and C is the capacitance. The resonant frequency is measured in hertz (Hz).

## 5. What are some practical applications of LCR circuits and resonance?

LCR circuits and resonance have many practical applications in electronics, such as in radio and television tuners, bandpass filters, and frequency-selective amplifiers. They are also used in medical equipment, such as MRI machines, and in power transmission systems to improve efficiency and reduce energy loss.

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