What is the maximum pass band gain of the 6th Order Butterworth LPF (Using Sallen Key Topology)Butterworth Polynomials

1. Break Down the Problem

To determine the maximum pass band gain of a 6th Order Butterworth Low-Pass Filter (LPF) using the Sallen-Key topology, we need to consider the properties of Butterworth filters, including their gain characteristics.

2. Relevant Concepts

• The Butterworth filter is characterized by maximally flat frequency response in the passband.
• The maximum gain $G$ of a Butterworth LPF is typically set to 1 (or 0 dB) for normalized filters.

3. Analysis and Detail

• For a 6th Order Butterworth LPF, the filter can be implemented using cascaded second-order sections since the 6th order can be achieved using three second-order Sallen-Key stages.
• Each Sallen-Key stage for a second-order filter can contribute to the total gain, but in standard designs, the passband gain can be set to unity.
• The transfer function of a Butterworth filter exhibits $G = 1$ at frequencies below the cutoff frequency $\omega_c$.

4. Verify and Summarize

Upon analysis, the maximum passband gain of a standard 6th Order Butterworth LPF using the Sallen-Key topology is indeed 1 (0 dB) when appropriately designed for normalization.

Final Answer

The maximum pass band gain of a 6th Order Butterworth LPF using Sallen-Key topology is $G = 1$ (0 dB).