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.