Day 30 6/2/16: Series and Parallel Resonance Circuits, and Passive Filters

LECTURE:

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. 

Day 28; 26 May 2016: Transfer Functions, Frequency Responses, and Signals with Multiple Frequency Components

We first learned about the frequency response, which provides information about the behavior of a circuit as a function of the frequency of the source provided. We then learned about transfer functions, which provides the frequency response of a circuit when they are plotted. The transfer function is frequency-dependent and given by the ratio of a phasor output to a phasor input. The tranfer function can be given by voltage gain, current gain, transfer impedance (v/i) and transfer admittance (i/v). We then did many problems for calculating the transfer function and using it to graph the frequency response.

Next, the decibel and logarithmic scales were reviewed. In logarithmic scale, it was found that the gain with respect to voltage or current is 20log(V2/V1) - 10log(R2/R1). When R1 = R2, it simplifies to 20log(V2/V1). Lastly, we did a lab titled "Signals with Multiple Frequency Components".

LECTURE:

In this example, the current gain of the circuit was calculated. The capacitor was assumed to be the output of this circuit. Using current divider, the current across the capacitor was found, then divided by the total input current to obtain the transfer function as current gain. The zeros were then found by setting the numerator equal to zero, and the poles were found by setting the denominator to zero.

The purpose of this problem was to find the transfer impedance of this circuit and to sketch the frequency response. Using nodal analysis the transfer function Vo/I was determined, where the output voltage is the voltage across the inductor. This transfer function was then used to graph the frequency response in MATLAB. This is shown below:

The code used to graph in MATLAB is shown below:

In the problem below, the transfer function as a voltage gain was determined using simple KVL, where the output voltage was the voltage across the inductor. The values of jw when the gain beta is zero was determined, along with the values at an infinite beta.

LAB:

Signals with Multiple Frequency Components:

Purpose:

The purpose of this lab was to find the frequency response of an electrical circuit and determine the effect of the circuit on input signals such as multiple sinusoidal waves of different frequencies in one input and a sinusoidal signal with a time-varying frequency as another input.The experimental response of the circuit to these input signals was then measured and compared to the expected. In addition,  a circuit’s effect on the “shape” of a signal applied via the response was determined.

Prelab:

The parallel RC circuit shown in the picture above was analyzed by determining the transfer function (voltage gain) at different frequencies. The output voltage was the voltage across the parallel resistor with the capacitor. 

Apparatus:

The apparatus was the circuit seen in the prelab, along with the analog discovery and the laptop with Waveforms software.

Procedure/Data:

First, the circuit above was constructed. Channel 1 was used to measure in input voltage and Channel 2 to measure the output voltage, or the voltage across the resistor in parallel with the capacitor. The waveform generator was then used to apply the following wave equation: 

20[sin(1000pi*t) + sin(2000pi*t) + sin(20,000pi*t)

The output and input voltages were then measured in the oscilloscope window, which is shown below.

Below is another sine function that was applied to the circuit. It involves a sinusoidal wave of equal frequency but constantly changing amplitudes. 

Lastly, a sinusoidal sweep was applied to the circuit. A sinusoidal sweep in a sine wave whose frequency increases with time, as shown below. The amplitudes also varies with each period and slightly throughout one period. The oscilloscope window is shown below:

Data Analysis:

Based on all three oscilloscope graphs obtained, it seemed that the circuit greatly reduced the noise in the input and made all of those weird sinusoidal inputs more like traditional sinusoidal waves with fairly constant amplitude and frequency, except for the sinusoidal sweep. Overall, though, the output has really been a very unintensified version of each input, making it more like a traditional sine wave. This data obtained is expected based on the results obtained from the prelab and the calculations seen above. As the frequency increases, the voltage gain decreases, which was seen in these oscilloscope windows, especially the sinudoidal sweep.