For the network of Fig. 79 : a. Determine the mathematical...

For the network of Fig. 79 : a. Determine the mathematical expression for the angle by which $V_{o}$ leads $V_{i}$. b. Determine the phase angle at $f=100 \mathrm{~Hz}, 1 \mathrm{kHz}, 2 \mathrm{kHz}, 5 \mathrm{kHz}$, and $10 \mathrm{kHz}$, and plot the resulting curve for the frequency range of $100 \mathrm{~Hz}$ to $10 \mathrm{kHz}$. c. Determine the break frequency. d. Sketch the frequency response of $\theta$ for the frequency spectrum of part (b) and compare results.

For the network of Fig. 79 :
a. Determine the mathematical expression for the angle by which $V_{o}$ leads $V_{i}$.
b. Determine the phase angle at $f=100 \mathrm{~Hz}, 1 \mathrm{kHz}, 2 \mathrm{kHz}, 5 \mathrm{kHz}$, and $10 \mathrm{kHz}$, and plot the resulting curve for the frequency range of $100 \mathrm{~Hz}$ to $10 \mathrm{kHz}$.
c. Determine the break frequency.
d. Sketch the frequency response of $\theta$ for the frequency spectrum of part (b) and compare results.

Key Concepts

Frequency Response

Frequency response describes how the amplitude and phase of the output signal vary with frequency. It is a critical concept in analyzing and designing circuits, where the behavior over a range of frequencies is visualized typically through Bode plots. This helps in evaluating the system's performance in applications such as filtering.

Break Frequency

The break frequency, also known as the cutoff frequency, is the frequency at which the response of the circuit starts to noticeably deviate from its low-frequency behavior, often characterizing a -3 dB point in magnitude response. It is a key parameter in filter design, marking the transition from passband to stopband.

Phase Angle

The phase angle represents the time shift between the output and input signals. It is calculated as the argument of the complex transfer function and indicates whether the output signal leads or lags the input signal, which is crucial for understanding the timing and stability in circuits.

Transfer Function

A transfer function is a mathematical expression that characterizes the relationship between the output and input of a system in the frequency domain. It encapsulates both the magnitude and phase relationships, allowing us to understand how signals are modified by components such as resistors, capacitors, and inductors in a network.

Transcript

00:01 So a system that mimics a mass on a spring with damping is the rlc siri circuit.

00:09 And here we're going to investigate what it means for this circuit to resonate.

00:15 First of all, we can draw on a phaser diagram.

00:20 And remember, what we are doing with the x -axis is we are looking at things that are in phase with the driving.

00:31 Ac driving, so whether that's a cosine or a sign, that x -axis will represent the cosine, for example.

00:39 And the y -axis will represent the sign.

00:44 So the two phases are 90 degrees apart.

00:49 And if we draw the generalized resistances on the phaser diagram, the resistor has reactants, if you well that lies along the axis that goes with the driving term, which means that the current in that resistor is always in phase with the drive through the resistor.

01:18 On the y -axis, we show the inductive reactants, sorry, xl, and that is equal to omega -l.

01:34 The capacitive reactants, the however, points downwards.

01:40 So 1 over omega -c is equal to the capacitive reactants.

01:46 And the currents in both of those either lead or lag the voltage source by 90 degrees.

01:56 So what we mean by resonance is a couple things.

02:01 The first step is to recognize that you can have the capacitive reactants equal the inductive reactants.

02:11 And that will fix the resonance frequency.

02:19 Okay, when those two things are equal to each other.

02:25 And then we can actually solve for what that frequency is.

02:29 One over omega -c is the capacitive reactants.

02:33 Omega -l is the inductive reactants.

02:37 And solving for omega, it turns out that the resonance frequency is one over the square root of l times c.

02:51 And it's also equal to 2 pi times the resonance frequency in hertz, omega being angular frequency.

03:05 So for this circuit, we can actually calculate what that resonance frequency is.

03:10 We have 13 millie henry, 13 times 10 to the minus 3, in si units and 0 .2 milli -faird...

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