# LC Filter Design Tool Description

LC Filter Design Tool is a web-based application for lumped LC filter synthesis. It is feature rich, user-friendly and available for free from any desktop or mobile device.

### Features

• One-click synthesis from a given specifications
• Chebyshev, Elliptic, Butterworth, Bessel or Legendre filter types
• Multiple topologies
• Arbitrary input and output impedances
• Option to use closest capacitor and inductor standard values

### Introduction

The RF filter is a two-port linear device used to attenuate certain unwanted frequencies of a signal while passing other wanted ones. The frequency band over which the filter passes through is called the passband, and the frequency band it rejects is called the stopband. The filter frequency response is classified according to its passband and stopband boundaries. The most common ones are:

• Low-pass filter - passes frequencies below a cutoff frequency fc while attenuating frequencies above it.
• High-pass filter - passes frequencies above a cutoff frequency fc while attenuating frequencies below it.
• Band-pass filter - passes frequencies between a lower and upper cutoff frequency fl and fh while attenuating all other frequencies.
• Band-stop filter - attenuates frequencies between a lower and upper cutoff frequency fl and fh while passing all other frequencies.

Along having frequency selectivity, the RF filter is expected to have minimal influence on the pass band phase and amplitude response and maintain good impedance match at each port.

### S-parameters

The passive RF filter is a linear device with matched ports, which is typically described in the frequency domain; it is therefore convenient to model its response using s-parameters. An overview on s-parameters is available in Wikipedia. In the present context, the passive RF filters s-parameters consists of a two-by-two complex and frequency dependent matrix, $\mathbf{S}= \begin{bmatrix} S_{11} & S_{12} \\ S_{21} & S_{22} \end{bmatrix}$

Because the device is passive (and non-magnetic) it is also reciprocal, meaning $$S_{21} = S_{12}$$ and only three parameters are needed to describe the filter response, $$S_{11}$$, $$S_{21}$$, and $$S_{22}$$. The magnitude and phase of these correspond to several frequency dependent measures important for filter analysis:

Parameter Relation
Insertion Loss (dB)$$IL = -20log_{10}(|S_{21}|)$$
Input Return Loss (dB)$$RL_{in} = -20log_{10}(|S_{11}|)$$
Output Return Loss (dB)$$RL_{out} = -20log_{10}(|S_{22}|)$$
Phase (rad)$$\phi = arg(S_{21})$$
Group Delay (sec)$$\tau_{d} = -\frac{1}{2\pi}\frac{d\phi}{df}$$

### Standard Values

By default, filters are synthesized with exact components values and show ideal frequency response. However, actual mass-produced components have values limited to a set of standard values and bounded below by some minimum; moreover they are subject to manufacturing tolerances and temperature variations. Consequently, when implementing the design, actual component values differ from the nominal values and may negatively impact the filter performances.

To model the filter sensitivity to these variations, the user can limit the capacitance and inductance to E series of preferred values, and also set their minimum values.

#### E Series

Each E series values are equally spaced on a logarithmic scale, such that the relative variation/tolerance from the exact value is constant. Below is the list of available E series values and their corresponding tolerances. For more information on E series see Wikipedia.

E6 values (20% tolerance)

1.0, 1.5, 2.2, 3.3, 4.7, 6.8

E12 values (10% tolerance)

1.0, 1.2, 1.5, 1.8, 2.2, 2.7, 3.3, 3.9, 4.7, 5.6, 6.8, 8.2

E24 values (5% tolerance)

1.0, 1.1, 1.2, 1.3, 1.5, 1.6, 1.8, 2.0, 2.2, 2.4, 2.7, 3.0, 3.3, 3.6, 3.9, 4.3, 4.7, 5.1, 5.6, 6.2, 6.8, 7.5, 8.2, 9.1

E48 values (2% tolerance)

1.00, 1.05, 1.10, 1.15, 1.21, 1.27, 1.33, 1.40, 1.47, 1.54, 1.62, 1.69, 1.78, 1.87, 1.96, 2.05, 2.15, 2.26, 2.37, 2.49, 2.61, 2.74, 2.87, 3.01, 3.16, 3.32, 3.48, 3.65, 3.83, 4.02, 4.22, 4.42, 4.64, 4.87, 5.11, 5.36, 5.62, 5.90, 6.19, 6.49, 6.81, 7.15, 7.50, 7.87, 8.25, 8.66, 9.09, 9.53

E96 values (1% tolerance)

1.00, 1.02, 1.05, 1.07, 1.10, 1.13, 1.15, 1.18, 1.21, 1.24, 1.27, 1.30, 1.33, 1.37, 1.40, 1.43, 1.47, 1.50, 1.54, 1.58, 1.62, 1.65, 1.69, 1.74, 1.78, 1.82, 1.87, 1.91, 1.96, 2.00, 2.05, 2.10, 2.15, 2.21, 2.26, 2.32, 2.37, 2.43, 2.49, 2.55, 2.61, 2.67, 2.74, 2.80, 2.87, 2.94, 3.01, 3.09, 3.16, 3.24, 3.32, 3.40, 3.48, 3.57, 3.65, 3.74, 3.83, 3.92, 4.02, 4.12, 4.22, 4.32, 4.42, 4.53, 4.64, 4.75, 4.87, 4.99, 5.11, 5.23, 5.36, 5.49, 5.62, 5.76, 5.90, 6.04, 6.19, 6.34, 6.49, 6.65, 6.81, 6.98, 7.15, 7.32, 7.50, 7.68, 7.87, 8.06, 8.25, 8.45, 8.66, 8.87, 9.09, 9.31, 9.53, 9.76