A causal LTI system has the transfer function H(s) given below: 100000(s + 1) (s +1000) (s + 100000) H(s) = 1) Sketch the Bode magnitude plot for this system. 2) Determine a differential equation that relates the input of this system to the output of this system. 3) Suppose that the input to the system is r(t) = u(t). As t → ∞, does the output approach a steady value? If so, what is this value? If not, explain what the output looks like as t approaches ∞o.
A causal LTI system has the transfer function H(s) given below: 100000(s + 1) (s +1000) (s + 100000) H(s) = 1) Sketch the Bode magnitude plot for this system. 2) Determine a differential equation that relates the input of this system to the output of this system. 3) Suppose that the input to the system is r(t) = u(t). As t → ∞, does the output approach a steady value? If so, what is this value? If not, explain what the output looks like as t approaches ∞o.
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![**Title: Analysis of a Causal LTI System**
**Transfer Function:**
A causal Linear Time-Invariant (LTI) system is characterized by the transfer function \( H(s) \) given below:
\[
H(s) = \frac{100000(s + 1)}{(s + 1000)(s + 100000)}
\]
**Tasks:**
1. **Sketch the Bode Magnitude Plot:**
- The Bode magnitude plot represents the gain (magnitude) of the transfer function \( H(s) \) as a function of frequency (in logarithmic scale).
- To draw the Bode plot, identify the poles and zeros from the transfer function. Here, the system has one zero at \( s = -1 \) and two poles at \( s = -1000 \) and \( s = -100000 \).
- The plot will show:
- A rise in magnitude near the zero frequency \( s = -1 \).
- A drop near each of the pole frequencies \( s = -1000 \) and \( s = -100000 \).
2. **Determine the Differential Equation:**
- The transfer function can also be expressed in a time-domain differential equation form relating the input \( x(t) \) to the output \( y(t) \).
- The form is derived by expressing and equating the coefficients of \( s \) using inverse Laplace transforms.
3. **Steady-State Output Analysis:**
- Given the input to the system is \( x(t) = u(t) \) (a unit step function):
- Investigate whether the output \( y(t) \) approaches a steady value as \( t \to \infty \).
- Analyze the poles of \( H(s) \) to determine stability and steady-state behavior. Poles in the left-half of the complex plane indicate a stable system where the output converges to a constant value as \( t \to \infty \).
This analysis provides insights into the behavior of the LTI system over varying frequency ranges and time, crucial for understanding system dynamics in engineering applications.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F39a3543a-bde3-4119-be75-304b57c879c6%2F79f1254f-342d-4bd4-a877-d811ae1d553b%2Fuhpxirk_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Analysis of a Causal LTI System**
**Transfer Function:**
A causal Linear Time-Invariant (LTI) system is characterized by the transfer function \( H(s) \) given below:
\[
H(s) = \frac{100000(s + 1)}{(s + 1000)(s + 100000)}
\]
**Tasks:**
1. **Sketch the Bode Magnitude Plot:**
- The Bode magnitude plot represents the gain (magnitude) of the transfer function \( H(s) \) as a function of frequency (in logarithmic scale).
- To draw the Bode plot, identify the poles and zeros from the transfer function. Here, the system has one zero at \( s = -1 \) and two poles at \( s = -1000 \) and \( s = -100000 \).
- The plot will show:
- A rise in magnitude near the zero frequency \( s = -1 \).
- A drop near each of the pole frequencies \( s = -1000 \) and \( s = -100000 \).
2. **Determine the Differential Equation:**
- The transfer function can also be expressed in a time-domain differential equation form relating the input \( x(t) \) to the output \( y(t) \).
- The form is derived by expressing and equating the coefficients of \( s \) using inverse Laplace transforms.
3. **Steady-State Output Analysis:**
- Given the input to the system is \( x(t) = u(t) \) (a unit step function):
- Investigate whether the output \( y(t) \) approaches a steady value as \( t \to \infty \).
- Analyze the poles of \( H(s) \) to determine stability and steady-state behavior. Poles in the left-half of the complex plane indicate a stable system where the output converges to a constant value as \( t \to \infty \).
This analysis provides insights into the behavior of the LTI system over varying frequency ranges and time, crucial for understanding system dynamics in engineering applications.
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