Question

The transfer function for a linear time-invariant circuit is $$H(s)=\frac{V_{o}}{V_{g}}=\frac{10^{4}(s+6000)}{s^{2}+875 s+88 \times 10^{6}}$$ If \(v_{g}=12.5 \cos 8000 t \mathrm{V},\) what is the steady-state expression for \(v_{o} ?\)

Short answer

The steady-state output is \( v_{o}(t) = 12.5 |H(j8000)| \cos(8000t + \text{arg}(H(j8000))) \).

Step-by-step solution

Identify Input Signal Form

The given input signal is \(v_{g}(t)=12.5 \cos 8000t \). This is a cosine signal with angular frequency \(\omega = 8000 \) rad/s.

Determine System's Response to Sinusoidal Inputs

The system's transfer function \( H(s) \) can be evaluated at \( s = j\omega \), where \( \omega = 8000 \). This gives the frequency response of the system for the given input frequency. We will use this to determine the amplitude and phase shift of the output.

Calculate the Transfer Function at \( s = j\omega \)

Substitute \( s = j8000 \) into \( H(s) = \frac{10^{4}(s+6000)}{s^{2}+875 s+88 \times 10^{6}} \). Compute \[ H(j8000) = \frac{10^{4}(j8000 + 6000)}{(j8000)^{2} + 875(j8000) + 88 \times 10^{6}} \].

Simplify the Expression

Simplify the expression for \( H(j8000) \). First, substitute in and compute the magnitude and phase: 1. The numerator: \( 10^{4}(j8000 + 6000) = 10^{4}(6000 + j8000) \).2. For the denominator: calculate \((j8000)^2 + 875j8000 + 88 \times 10^{6}\).

Compute Output's Magnitude and Phase

Calculate the magnitude \[ |H(j8000)| = \frac{|10^{4}(6000 + j8000)|}{|(-8000^2 + 88 \times 10^6) + j875 \times 8000|} \].Find the phase angle using the arctangent function for both numerator and denominator angles.

Form the Steady-State Output

The steady-state output \(v_{o}(t)\) will have the form \[ v_{o}(t) = A |H(j8000)| \cos(8000t + \text{arg}(H(j8000))) \]where \( A = 12.5 \) (from input amplitude), \(|H(j8000)|\) is the magnitude of the evaluated transfer function, and \(\text{arg}(H(j8000))\) is the phase shift.

Key concepts

Linear Time-Invariant Systems

Linear Time-Invariant (LTI) Systems are a cornerstone concept in the study of electrical circuits and signals. The term "linear" refers to the system following the principle of superposition, meaning the output is directly proportional to its input, and it will react the same way to any input, no matter the size or complexity. "Time-Invariant" means that the system's behavior and characteristics do not change over time. This stability means that if you input the same sinusoidal signal today or tomorrow, you would get the same output, assuming nothing else changes.
An easy way to think about LTI systems is to compare them to a black box that processes signals in a predictable, consistent manner. As you work with LTI systems, realizing their predictability helps in analyzing and designing circuits, especially when combined with transfer functions, which essentially describe the input-output relationship of the system.

Frequency Response

The Frequency Response of a system provides insight into how the system reacts to different frequencies. It is a critical analysis tool because every real-world system has frequency limits, frequencies at which it performs well and others where it might fail or distort the signal.
When analyzing the given system, we specifically look at how it responds to sinusoidal signals at various frequencies. By plugging in the sinusoidal input frequency into the system's transfer function and replacing the complex variable with the imaginary unit times the frequency, we obtain the frequency response. This tells us two major things: the Gain (or magnitude response) and the Phase Shift.
Gain tells us how much the system will amplify or attenuate the input signal's amplitude. The Phase Shift indicates the time shift that the sinusoidal wave will undergo by passing through the system. Understanding these responses allows us to not only know how the system will behave with the particular input frequency but also aids in predicting the behavior with other frequencies, which is invaluable in filter design.

Steady-State Expression

The Steady-State Expression describes the output signal behavior after the system has had time to adjust and settle following any input changes. In systems analysis, while initial transient responses might behave unpredictably, the steady-state is where things settle down and become repetitive.
For the exercise, after determining the system's response characteristics using the transfer function, the steady-state output is calculated. This involves finding the output's amplitude and phase shift - both obtained from the frequency response of the system.
The output signal retains the form of the input signal but is altered by the system's frequency response. That's why, when expressing the output, we adjust the amplitude according to the gain as well as the starting position of the waveform by adding the phase shift. This adjustment results in a \( v_{o}(t) = A |H(j\omega)| \cos(\omega t + \text{arg}(H(j\omega))) \) where the constants are derived from the input and the system's transfer function.

Sinusoidal Inputs

Sinusoidal Inputs are perhaps the most fundamental type of signal used in the analysis of electronics and systems. Characterized by their smooth, repetitive oscillation, these waves are easy to generate and are the basis for understanding more complex signals.
In systems like the LTI systems we are investigating, sinusoidal inputs help us predict behavior under a wide range of conditions or other input types through Fourier analysis.
With the given exercise, the input is a cosine wave specified by an amplitude (12.5 in this case) and an angular frequency (\(\omega = 8000 \text{ rad/s}\)). Identifying these characteristics is the first step in analyzing the system’s response.
Sinusoidal signals are special because when they pass through an LTI system, they emerge as sinusoidal waves of the same frequency but potentially different amplitude and a shifted phase. This simple transformation property makes it easier to handle such inputs analytically, especially since most real-world signals can be decomposed into sinusoidal components. This decomposition allows us to use the LTI system's responses to predict the system’s behavior fairly accurately.