In the above derivation, the half-power frequencies were determined. These frequencies are the frequencies where the dissipated power is half the maximum, or where the rms voltage/current is obtained. The graphs of frequency versus voltage/current are bell-shaped, so there are actually two frequencies where the dissipated power is halved, which is given by the derived formulas above (the results of a quadratic). The distance between the two half-power frequencies is termed the bandwidth.
In this example, the objective was to determine the resonant frequency and half-power frequency of this series RLC circuit. The resonant frequency for any RLC circuit is given by 1 over root LC. Then, the half-power frequencies were found using the derived formulas above. Next, the bandwidth and quality factor were determined. The bandwidth was given as the difference between the upper and lower half-power frequencies. The quality factor is given as the quotient between the resonant frequency and the bandwidth. Lastly, the amplitude of the current at the resonant and two half-power frequencies was determined. This was found by dividing the voltage by the impedance. At the resonance frequency there is only resistor and no reactance. At the half-power frequencies the current is just the rms of the amplitude of the current.
In this problem, the objective was to find the resonant frequency, quality factor and the bandwidth of the parallel RLC circuit. The resonant frequency, as always, is given by 1 over the root of LC. The bandwidth is given as the difference between the upper and lower half-power frequencies, or the resonant frequency divided by the quality factor since it is greater than 10. Next, the quality factor is given by multiplying the resonant frequency by RC.
In this example, the inductance and capacitance needed to obtain a bandwidth of 5 kHz and a resonant frequency of 640 kHz for the above series RLC circuit was determined. The resistance was given as 10 kOhms. The inductance was found by dividing the resistance by the bandwidth, and the capacitance was found from the resonant frequency using the recently determined inductance.
In this example, the cutoff frequency of the above filter was determined. The transfer function was first obtained, and then the transfer function values at time 0 and time infinity were calculated. Since the transfer function of time 0 is 1 and the transfer function at time infinity is zero, it was deduced that the filter is low pass.
Summary:
No lab was done this lecture meeting. Resonance in series and parallel RLC circuits were determined. This included the resonant frequency, half-power frequencies, bandwidth, quality factors and transfer functions. Then, the four different passive filters were reviewed. These were the high pass, low pass, bandpass, and bandstop filters. The low pass filter is given by and RC circuit, with the capacitor providing the output voltage. A lowpass filter has a 1 transfer function at time 0 and a zero transfer function at infinite time. A highpass filter also consists of an RC circuit, but the resistor is the output voltage provider. A highpass filter has a transfer function of zero at time 0 and a transfer function of 1 at infinite time. Next, a bandpass filter consists of a series RLC circuit, with the resistor as the output voltage provider. The bandpass filter is given a transfer function of 1 in the bandwidth region, and zero everywhere else. Lastly the bandstop filter is the opposite of the bandpass; it has a transfer function of zero in the bandwidth and 1 everywhere else. It also consists of a series RLC circuit, but the output voltage is the inductor and capacitor.
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