RLC DC Transient over-damped response

[Jag1972]

Hello All,
I have a bit of a problem with a series RLC circuit with no supply voltage. In the process of actually solving the second order differential equation for the voltage across the capacitor I have lost a handle on what it actually represents in terms of the circuit.

I have attached the question as a word document.

If someone could help me relate my solution to the circuit I would be very great full.

Jag.

 

[jsgruszynski]

Hello All,
I have a bit of a problem with a series RLC circuit with no supply voltage. In the process of actually solving the second order differential equation for the voltage across the capacitor I have lost a handle on what it actually represents in terms of the circuit.

I have attached the question as a word document.

If someone could help me relate my solution to the circuit I would be very great full.

Jag.

The no-input case is the homogeneous solution or the "natural" solution. In cases when you have resonance, it's the "natural resonance" of the system. It's the response that dominated from a step or impulse input - the d/dt basically exists only instantaneously and there is no further "driven input" so you see the natural response by itself. Another analogy: tapping a bell - the hammer tap is an impulsive driven input and the remaining ringing is the natural response.

Applying a signal/input is the "driven" solution. You can drive a circuit at any frequency, for example, including off-resonance. There will be a response but it might not be particular interesting or spectacular. When you use an impedance analyzer, you are driving the DUT at various frequencies and observing the response. When you get the natural resonance, you'll get a particularly strong response.

So when you plot response vs. frequency you get sometime like on Figure 1-18 (page 1-13) ofhttp://cp.literature.agilent.com/litweb/pdf/5950-3000.pdf" . This is a resonance as seen in impedance/capacitance domain vs. frequency. A capacitor, because of parasitic inductance, will seem to suddenly stop being a capacitor due to resonance at a particular frequency - this resonance is the natural frequency of the RLC parasitic circuit.

This is the frequency domain representation of the natural response. Notice this curve contains "all frequencies". Then remember that the frequency response of an impulse function is a horizontal line in frequency space - "all frequencies". This is the relationship between frequency response, impulse response, convolution, Fourier transforms, natural frequency responses and homogeneous differential (time domain) equation homogeneous solutions. All connected intimately.

 

[Jag1972]

Thank you for the response much appreciated.