What is the formula for the steady state response?
2 Answers
The formula for the steady-state response of a system depends on the type of system and the input applied to it. In the context of electrical engineering, especially when dealing with linear time-invariant (LTI) systems, the steady-state response is typically derived after the transient effects have died out and the system has reached equilibrium.
### 1. **Steady-State Response in Sinusoidal Steady-State Analysis (AC Circuits)**
When dealing with AC circuits or systems under a sinusoidal input, the steady-state response is often found using **phasor analysis**. The general form of the steady-state response \( y_{ss}(t) \) to a sinusoidal input \( x(t) = X_m \cos(\omega t + \theta) \) is:
\[
y_{ss}(t) = |H(j\omega)| X_m \cos(\omega t + \theta + \angle H(j\omega))
\]
where:
- \( X_m \) is the amplitude of the input signal.
- \( \omega \) is the angular frequency of the input.
- \( \theta \) is the phase of the input.
- \( H(j\omega) \) is the frequency response (or transfer function) of the system evaluated at \( \omega \).
- \( |H(j\omega)| \) is the magnitude of the frequency response.
- \( \angle H(j\omega) \) is the phase angle of the frequency response.
### 2. **Steady-State Response in General LTI Systems**
For general LTI systems, the steady-state response to a given input can be obtained using the principle of superposition, convolution, or directly from the system's transfer function \( H(s) \).
#### a. **For a sinusoidal input** \( x(t) = X_m \cos(\omega t + \theta) \):
Given the input, the steady-state response can be found using the transfer function \( H(j\omega) \):
\[
y_{ss}(t) = \text{Re}\left\{ H(j\omega) \cdot X_m e^{j(\omega t + \theta)} \right\}
\]
This results in:
\[
y_{ss}(t) = X_m |H(j\omega)| \cos(\omega t + \theta + \angle H(j\omega))
\]
#### b. **For an arbitrary input** \( x(t) \):
If the input \( x(t) \) is not purely sinusoidal, but can be represented as a sum of sinusoids (using Fourier series or Fourier transform), the steady-state output can be obtained by summing the corresponding outputs for each frequency component.
#### c. **Using Convolution**:
The steady-state response can also be found using the convolution of the input signal \( x(t) \) with the impulse response \( h(t) \) of the system:
\[
y_{ss}(t) = \int_{-\infty}^{\infty} h(\tau) x(t-\tau) d\tau
\]
### 3. **Steady-State Response in Control Systems**
In control systems, especially for systems with transfer functions \( H(s) \), the steady-state response to a step input is often found using the final value theorem:
\[
y_{ss} = \lim_{s \to 0} s \cdot Y(s) = \lim_{s \to 0} s \cdot \left[ H(s) \cdot \frac{1}{s} \right]
\]
Where \( Y(s) \) is the Laplace transform of the output \( y(t) \).
### 4. **Example**
If the system's transfer function \( H(s) = \frac{2}{s + 1} \), and the input is a unit step \( x(t) = u(t) \), the steady-state response can be calculated as:
\[
y_{ss} = \lim_{s \to 0} s \cdot \left[ \frac{2}{s+1} \cdot \frac{1}{s} \right] = \lim_{s \to 0} \frac{2s}{s(s+1)} = \lim_{s \to 0} \frac{2}{s+1} = 2
\]
Thus, the steady-state response is \( y_{ss} = 2 \).
### Summary
- **For sinusoidal inputs**: Use phasor analysis and the system's transfer function \( H(j\omega) \) to find the magnitude and phase shift of the output.
- **For general inputs**: Use the convolution integral or Fourier methods to analyze the system's response.
- **For step inputs in control systems**: Use the final value theorem to determine the steady-state value.
The specific formula for the steady-state response depends on the nature of the input and the characteristics of the system being analyzed.
### 1. **Steady-State Response in Sinusoidal Steady-State Analysis (AC Circuits)**
When dealing with AC circuits or systems under a sinusoidal input, the steady-state response is often found using **phasor analysis**. The general form of the steady-state response \( y_{ss}(t) \) to a sinusoidal input \( x(t) = X_m \cos(\omega t + \theta) \) is:
\[
y_{ss}(t) = |H(j\omega)| X_m \cos(\omega t + \theta + \angle H(j\omega))
\]
where:
- \( X_m \) is the amplitude of the input signal.
- \( \omega \) is the angular frequency of the input.
- \( \theta \) is the phase of the input.
- \( H(j\omega) \) is the frequency response (or transfer function) of the system evaluated at \( \omega \).
- \( |H(j\omega)| \) is the magnitude of the frequency response.
- \( \angle H(j\omega) \) is the phase angle of the frequency response.
### 2. **Steady-State Response in General LTI Systems**
For general LTI systems, the steady-state response to a given input can be obtained using the principle of superposition, convolution, or directly from the system's transfer function \( H(s) \).
#### a. **For a sinusoidal input** \( x(t) = X_m \cos(\omega t + \theta) \):
Given the input, the steady-state response can be found using the transfer function \( H(j\omega) \):
\[
y_{ss}(t) = \text{Re}\left\{ H(j\omega) \cdot X_m e^{j(\omega t + \theta)} \right\}
\]
This results in:
\[
y_{ss}(t) = X_m |H(j\omega)| \cos(\omega t + \theta + \angle H(j\omega))
\]
#### b. **For an arbitrary input** \( x(t) \):
If the input \( x(t) \) is not purely sinusoidal, but can be represented as a sum of sinusoids (using Fourier series or Fourier transform), the steady-state output can be obtained by summing the corresponding outputs for each frequency component.
#### c. **Using Convolution**:
The steady-state response can also be found using the convolution of the input signal \( x(t) \) with the impulse response \( h(t) \) of the system:
\[
y_{ss}(t) = \int_{-\infty}^{\infty} h(\tau) x(t-\tau) d\tau
\]
### 3. **Steady-State Response in Control Systems**
In control systems, especially for systems with transfer functions \( H(s) \), the steady-state response to a step input is often found using the final value theorem:
\[
y_{ss} = \lim_{s \to 0} s \cdot Y(s) = \lim_{s \to 0} s \cdot \left[ H(s) \cdot \frac{1}{s} \right]
\]
Where \( Y(s) \) is the Laplace transform of the output \( y(t) \).
### 4. **Example**
If the system's transfer function \( H(s) = \frac{2}{s + 1} \), and the input is a unit step \( x(t) = u(t) \), the steady-state response can be calculated as:
\[
y_{ss} = \lim_{s \to 0} s \cdot \left[ \frac{2}{s+1} \cdot \frac{1}{s} \right] = \lim_{s \to 0} \frac{2s}{s(s+1)} = \lim_{s \to 0} \frac{2}{s+1} = 2
\]
Thus, the steady-state response is \( y_{ss} = 2 \).
### Summary
- **For sinusoidal inputs**: Use phasor analysis and the system's transfer function \( H(j\omega) \) to find the magnitude and phase shift of the output.
- **For general inputs**: Use the convolution integral or Fourier methods to analyze the system's response.
- **For step inputs in control systems**: Use the final value theorem to determine the steady-state value.
The specific formula for the steady-state response depends on the nature of the input and the characteristics of the system being analyzed.