05/27 - Lab 26: Signals with Multiple Frequency Components

Lab 26: Signals with Multiple Frequency Components

In this lab, we used Waveforms to create complex voltage signals which were applied to a simple RC circuit. The first signal composed of multiple sinusoidal waves of different frequencies, and the second signal was a sinusoidal wave with a time-varying frequency. The response of the circuit was measured in order to see how the "shape" of the input is affected. The following circuit was used in this lab:

The values used in the circuit were R = 1 kOhm, and C = 100 nF, where V_in(t) is the complex sinusoidal signal, and V_out is the voltage measured across the "load" resistor. The actual circuit is pictured below:

For the pre-lab, we calculated the magnitude response of the circuit, which is simply the ratio of the amplitude of the input sinusoid to the amplitude of the output sinusoid, in order to determine the zeros and poles of the circuit.

The function used for our first input sinusoid was a function containing multiple sine functions at various frequencies:

V_in(t) = 20[sin(1000πt) + sin(2000πt) + sin(20,000πt)]

The frequency of the overall sinusoid was adjusted throughout the experiment. We used frequencies of 500 Hz, 1000 Hz, and 10000 Hz. Pictured below is the oscilloscope depicting the input and output voltages for each case:

500 Hz

Yellow: Input Voltage  |  Blue: Output Voltage

1000 Hz

Yellow: Input Voltage  |  Blue: Output Voltage

10000 Hz

Yellow: Input Voltage  |  Blue: Output Voltage

The second input function used in this lab was a sinusoidal sweep function that contained a time-varying frequency. We did not need a mathematical equation for this, as Waveforms has a built-in sweep function. The input wave using this built-in feature looks like this:

Pictured below is the Waveforms oscilloscope containing the input voltage, as well as the voltage across the "load" resistor.

In Class Examples

1. For the circuit shown at right, calculate the gain I o (ω)/I i (ω) and its poles and zeros.

2. Given the circuit shown below and i(t)= I cos ωt amps, find the transfer function H(ω) = Vo / I and sketch the frequency response.

3. Given the circuit shown at right and vin = Vin cos ωt volts, find the transfer function H(ω ) = Vout / Vin where Vout is the voltage across the inductor and sketch the frequency response.