Low pass filter transfer function

On this page the passive low pass filter design is explained. After studying this explanation you should be able to choose the right components for your application, obtain a transfer function of a low pass filter and to draw the bode plot.
For some application we want to have a filter which only passes the low frequency components of an input signal.
Suppose the load is placed in parallel with the capacitor, as shown in the figure below.
The first Kirchhoff rule implies that the total current flowing through the resistor is equal to the sum of the currents flowing through the load and the capacitor.
With increasing frequency it becomes easier for the current to travel through the capacitor instead of through the load.
Accordingly, less current will pass through the load. The high frequency components of the incoming signal are attenuated by this circuit, which implies that this circuit acts as a low pass filter.
Another way to obtain a low pass filter is by using an inductor and a resistor.
The circuit in the figure below, for example, could also be used to attenuate the high frequency part of incoming signals.
Although it still incorporates a resistor it is placed parallel to the load. Unlike in the previous low pass filter not all the current delivered to the load has to pass the resistor.
This increases the efficiency.

In the equation below the functioning of the circuit is explained in a more mathematical way, leading to the transfer function of a first order low pass filter.
The impedance of the resistor is written as Z_{1} and the impedance of the capacitor is written as Z_{2}.

The corner frequency of this RC low pass filter is inversely proportional to the RC value:

Note that the dimension of w is in radians per second; if the frequency in Hertz is desired one should divide it by 2π:

Given the properties of the components, R = 5 kΩ and C = 0.1 µF, a corner frequency of 318 Hertz is obtained. This corresponds to the bode plot of the low pass filter transfer function:

In the filter above, all the power delivered to the load has to travel through the resistor. As is known a resistor is a dissipating component. In low power filters like signal filters this might not be a drawback, but in high power filters energy dissipation is undesirable because of the efficiency aspect and the accompanying heat production.

The associated transfer function of this RL low pass filter is given by:

The resistor and inductor values of the components can be chosen in such a way that the same corner frequency is obtained as in the resistor-capacitor filter. The bode plot is then equal to the one shown before.

The first mentioned filter type is called a RC low pass filter, while the second type is known as a RL low pass filter. Component values like the resistance, capacitance and inductance can be chosen in such a way that both filter types have the same transfer function.