In electrical engineering, ladder networks refer to the series and parallel combinations of inductors and capacitors in a circuit. When these networks are infinite, the problem of finding the response at the terminal end becomes difficult to solve. In this article, we will explore techniques to solve this problem and generate transfer functions that relate input and output magnitudes.
Understanding the Problem
Before diving into the solution, let us examine the problem of solving the response at the terminal end of an infinite ladder network. The issue arises because the input signal goes through an infinite number of inductors and capacitors before reaching the output. To solve the problem, we need to break down the circuit into smaller blocks and solve each of them sequentially. The transfer function can then be obtained by combining these smaller blocks.
Using Transfer Functions
The transfer function of a circuit is defined as the ratio of the output to the input. It is denoted by H(s) where s is a complex variable representing frequency. The transfer function of an infinite ladder network can be used to solve real-life problems, such as determining the input voltage response for a step current input.
Solving the Input Impedance
One technique to solve the input impedance of an infinite ladder network is to use an iterative approach. In this approach, the value of the impedance for each node of the ladder is calculated and the process continues until a steady-state is reached. This technique works well for the input impedance since we are looking at the network input.
Example
Let us consider an infinite ladder network consisting of a series combination of a 1-ohm resistor and a 1-μF capacitor, as shown below.
~/\/\/\~
| |
V_in ---> ----- ----- ---> V_out
| |
\/\/\/\/
Using the iterative approach, we can calculate the impedance at each node. Let Z_n be the impedance at node n. We can then obtain the following iterative equations:
Z_1 = R_1 + 1/(jwC_1)
Z_2 = R_1 + 1/(jwC_1) + 1/(jwC_2) + Z_1
Z_3 = R_1 + 1/(jwC_1) + 1/(jwC_2) + Z_2
.
.
.
As n approaches infinity, the impedance at each node approaches a constant value, which is the input impedance of the entire network.
Solving for the Output
However, the iterative approach does not work for solving the response at the terminal end of the infinite ladder network. As we move further away from the input, the number of nodes in the circuit increases, making the calculation much more complex. Additionally, the iterative method does not take into account the effect of output impedance on the transfer function.
Using Cascade Connection Technique
Therefore, we need a different technique to solve for the output of an infinite ladder network. One such technique is the cascade connection method, which involves breaking the infinite ladder network into smaller, finite ladder networks, and then combining their transfer functions to obtain the overall transfer function of the circuit.
This method also takes into account the effect of output impedance on the transfer function, providing a more accurate solution. The cascade technique involves the following steps:
Step 1
Break the infinite ladder network into smaller, finite ladder networks of N elements each. The length of the network N depends on the complexity of the original circuit and the accuracy required in the solution.
Step 2
Calculate the transfer function of each finite ladder network using the iterative approach described earlier, and express it in terms of an impedance function. Let this function be Hn(s), where n represents the nth finite ladder network.
Step 3
Cascade the transfer functions of the finite ladder networks using the following formula:
Where Zn is the output impedance of the nth finite ladder network.
Example
Consider the infinite ladder network shown below.
~/\/\/\~ ~/\/\/\~
| | | |
V_in ---> ----- ----- ----- ---> V_out
| | | |
\/\/\/\/ \/\/\/\/
Let us break the circuit into two finite ladder networks, with each network consisting of two capacitors and one resistor. Using the iterative approach, we can calculate the transfer function of each ladder network as:
H1(s) = R/(1 + jsCR)
H2(s) = [(jsC)2 + R/(1 + jsCR)]/(jsC)2
where R is the resistor value, C is the capacitor value and j is the imaginary unit.
We can then use the cascade connection technique to obtain the overall transfer function by cascading the transfer functions of the two ladder networks as:
H(s) = H1(s) + H2(s)/Z1(s)
where Z1(s) is the output impedance of the first ladder network.
The output impedance of the first ladder network can be calculated as:
Z1(s) = R1 + 1/(jsC1) + 1/(jsC2)
Substituting the value of H1(s), H2(s) and Z1(s) in the above equation and simplifying, we can obtain the overall transfer function H(s).
Conclusion
Thus, we have examined techniques to solve the response at the terminal end of an infinite ladder network. While the iterative approach works well for solving the input impedance, the cascade connection technique is better suited for solving the output response. By breaking down the circuit into smaller, finite ladder networks and cascading their transfer functions, we can obtain an accurate solution that takes into account the effect of output impedance. These techniques are essential in solving real-world problems that involve infinite ladder networks, such as transmission lines and filter circuits.
How Can the Response at the Terminal End of an Infinite Ladder Network Be Solved?
In electrical engineering, ladder networks refer to the series and parallel combinations of inductors and capacitors in a circuit. When these networks are infinite, the problem of finding the response at the terminal end becomes difficult to solve. In this article, we will explore techniques to solve this problem and generate transfer functions that relate input and output magnitudes.
Understanding the Problem
Before diving into the solution, let us examine the problem of solving the response at the terminal end of an infinite ladder network. The issue arises because the input signal goes through an infinite number of inductors and capacitors before reaching the output. To solve the problem, we need to break down the circuit into smaller blocks and solve each of them sequentially. The transfer function can then be obtained by combining these smaller blocks.
Using Transfer Functions
The transfer function of a circuit is defined as the ratio of the output to the input. It is denoted by H(s) where s is a complex variable representing frequency. The transfer function of an infinite ladder network can be used to solve real-life problems, such as determining the input voltage response for a step current input.
Solving the Input Impedance
One technique to solve the input impedance of an infinite ladder network is to use an iterative approach. In this approach, the value of the impedance for each node of the ladder is calculated and the process continues until a steady-state is reached. This technique works well for the input impedance since we are looking at the network input.
Example
Let us consider an infinite ladder network consisting of a series combination of a 1-ohm resistor and a 1-μF capacitor, as shown below. ~/\/\/\~ | | V_in ---> ----- ----- ---> V_out | | \/\/\/\/ Using the iterative approach, we can calculate the impedance at each node. Let Z_n be the impedance at node n. We can then obtain the following iterative equations: Z_1 = R_1 + 1/(jwC_1) Z_2 = R_1 + 1/(jwC_1) + 1/(jwC_2) + Z_1 Z_3 = R_1 + 1/(jwC_1) + 1/(jwC_2) + Z_2 . . . As n approaches infinity, the impedance at each node approaches a constant value, which is the input impedance of the entire network.Solving for the Output
However, the iterative approach does not work for solving the response at the terminal end of the infinite ladder network. As we move further away from the input, the number of nodes in the circuit increases, making the calculation much more complex. Additionally, the iterative method does not take into account the effect of output impedance on the transfer function.
Using Cascade Connection Technique
Therefore, we need a different technique to solve for the output of an infinite ladder network. One such technique is the cascade connection method, which involves breaking the infinite ladder network into smaller, finite ladder networks, and then combining their transfer functions to obtain the overall transfer function of the circuit.
This method also takes into account the effect of output impedance on the transfer function, providing a more accurate solution. The cascade technique involves the following steps:
Step 1
Break the infinite ladder network into smaller, finite ladder networks of N elements each. The length of the network N depends on the complexity of the original circuit and the accuracy required in the solution.
Step 2
Calculate the transfer function of each finite ladder network using the iterative approach described earlier, and express it in terms of an impedance function. Let this function be Hn(s), where n represents the nth finite ladder network.
Step 3
Cascade the transfer functions of the finite ladder networks using the following formula:
Where Zn is the output impedance of the nth finite ladder network.
Example
Consider the infinite ladder network shown below. ~/\/\/\~ ~/\/\/\~ | | | | V_in ---> ----- ----- ----- ---> V_out | | | | \/\/\/\/ \/\/\/\/ Let us break the circuit into two finite ladder networks, with each network consisting of two capacitors and one resistor. Using the iterative approach, we can calculate the transfer function of each ladder network as: H1(s) = R/(1 + jsCR) H2(s) = [(jsC)2 + R/(1 + jsCR)]/(jsC)2 where R is the resistor value, C is the capacitor value and j is the imaginary unit. We can then use the cascade connection technique to obtain the overall transfer function by cascading the transfer functions of the two ladder networks as: H(s) = H1(s) + H2(s)/Z1(s) where Z1(s) is the output impedance of the first ladder network. The output impedance of the first ladder network can be calculated as: Z1(s) = R1 + 1/(jsC1) + 1/(jsC2) Substituting the value of H1(s), H2(s) and Z1(s) in the above equation and simplifying, we can obtain the overall transfer function H(s).Conclusion
Thus, we have examined techniques to solve the response at the terminal end of an infinite ladder network. While the iterative approach works well for solving the input impedance, the cascade connection technique is better suited for solving the output response. By breaking down the circuit into smaller, finite ladder networks and cascading their transfer functions, we can obtain an accurate solution that takes into account the effect of output impedance. These techniques are essential in solving real-world problems that involve infinite ladder networks, such as transmission lines and filter circuits.