Answer :
To create a passive realization of the given function \(\frac{V_0}{V_1}=\frac{1}{s^6+3.8637s^5+7.4641s^4+9.141s^3+7.464s^2+3.8637s+1}\) using resistors and other passive elements, we need to design an analog filter that corresponds to the given transfer function. Here's a step-by-step explanation of how to approach this:
1. Understand the Given Transfer Function:
The given transfer function is:
[tex]\[ \frac{V_0}{V_1} = \frac{1}{s^6 + 3.8637s^5 + 7.4641s^4 + 9.141s^3 + 7.464s^2 + 3.8637s + 1} \][/tex]
This suggests that the system is a low-pass filter with a sixth-order polynomial in the denominator.
2. Poles and Zeros:
To design the filter, we need to implement the poles and zeros of the transfer function. Here, we have no zeros (numerator is 1) and the polynomial in the denominator suggests a sixth-order system with six poles.
3. Passive Realization Basics:
Passive elements in circuitry typically include resistors (R), capacitors (C), and inductors (L). Higher-order filters can be realized using combinations of these elements to create networks that achieve the desired transfer function.
4. Use of LC Ladder Network:
A common technique for realizing such filters is to use an LC ladder network. This involves a series-parallel arrangement of inductors and capacitors.
5. Synthesizing the LC Ladder Network:
a. Select the Prototype:
We start by selecting a normalized low-pass filter prototype and then scale to the desired frequency range.
b. Component Values:
The coefficients in the transfer function imply that the network must be carefully designed to control the frequency response. However, the direct derivation of component values generally requires complex synthesis techniques, such as using the Cauer or Foster form of network synthesis.
6. Realizing the Sixth-Order Polynomial:
For a sixth-order filter, we might use a chain of series and parallel LC sections. Each section will correspond to a pair of conjugate poles or a single real pole.
7. Design Example:
While deriving exact component values generally requires sophisticated tools and techniques, here’s a conceptual example of a typical LC ladder network for a sixth-order filter:
```
R1 -- L1 -- C1 -- L2 -- C2 -- R2
| | | |
---------- C3-----------C4-----------
| | | |
----------------------C5---------
```
Each branch would represent an inductive or capacitive element contributing to the overall impedance of the network. The exact values of these components would need to match the frequency and polynomial terms in the transfer function.
8. Design and Verification:
Using network synthesis software or solvers can calculate the exact values for the inductor and capacitor elements to match the polynomial exactly. These tools automate much of the design process for complex high-order filters.
9. Practical Implementation:
Once the values are determined, real inductors and capacitors are used in circuit design. Care should be taken to ensure practical component tolerances and limitations do not overly degrade the filter response.
10. Final Verification:
After construction, testing the realized filter with signal generators and network analyzers can ensure it meets the desired specifications. Tweaks and component value adjustments may be necessary to correct mismatches and achieve the exact desired response.
Implementing a sixth-order filter from scratch, particularly to match a specified set of polynomial coefficients, would often require iterative design and verification steps, possibly aided by computer-aided design software specialized for analog filter design.
1. Understand the Given Transfer Function:
The given transfer function is:
[tex]\[ \frac{V_0}{V_1} = \frac{1}{s^6 + 3.8637s^5 + 7.4641s^4 + 9.141s^3 + 7.464s^2 + 3.8637s + 1} \][/tex]
This suggests that the system is a low-pass filter with a sixth-order polynomial in the denominator.
2. Poles and Zeros:
To design the filter, we need to implement the poles and zeros of the transfer function. Here, we have no zeros (numerator is 1) and the polynomial in the denominator suggests a sixth-order system with six poles.
3. Passive Realization Basics:
Passive elements in circuitry typically include resistors (R), capacitors (C), and inductors (L). Higher-order filters can be realized using combinations of these elements to create networks that achieve the desired transfer function.
4. Use of LC Ladder Network:
A common technique for realizing such filters is to use an LC ladder network. This involves a series-parallel arrangement of inductors and capacitors.
5. Synthesizing the LC Ladder Network:
a. Select the Prototype:
We start by selecting a normalized low-pass filter prototype and then scale to the desired frequency range.
b. Component Values:
The coefficients in the transfer function imply that the network must be carefully designed to control the frequency response. However, the direct derivation of component values generally requires complex synthesis techniques, such as using the Cauer or Foster form of network synthesis.
6. Realizing the Sixth-Order Polynomial:
For a sixth-order filter, we might use a chain of series and parallel LC sections. Each section will correspond to a pair of conjugate poles or a single real pole.
7. Design Example:
While deriving exact component values generally requires sophisticated tools and techniques, here’s a conceptual example of a typical LC ladder network for a sixth-order filter:
```
R1 -- L1 -- C1 -- L2 -- C2 -- R2
| | | |
---------- C3-----------C4-----------
| | | |
----------------------C5---------
```
Each branch would represent an inductive or capacitive element contributing to the overall impedance of the network. The exact values of these components would need to match the frequency and polynomial terms in the transfer function.
8. Design and Verification:
Using network synthesis software or solvers can calculate the exact values for the inductor and capacitor elements to match the polynomial exactly. These tools automate much of the design process for complex high-order filters.
9. Practical Implementation:
Once the values are determined, real inductors and capacitors are used in circuit design. Care should be taken to ensure practical component tolerances and limitations do not overly degrade the filter response.
10. Final Verification:
After construction, testing the realized filter with signal generators and network analyzers can ensure it meets the desired specifications. Tweaks and component value adjustments may be necessary to correct mismatches and achieve the exact desired response.
Implementing a sixth-order filter from scratch, particularly to match a specified set of polynomial coefficients, would often require iterative design and verification steps, possibly aided by computer-aided design software specialized for analog filter design.
