First Order Active Low Pass Filters Transfer Function

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The transfer function is also known as systems function or network function of the control system. It is expressed as a mathematical function. When used in frequency domain analysis of a system, it utilizes transform methods like Laplace transform. Filters work on signal frequency.

Thus, analytical and graphical descriptions using the frequency domain are the most potent instruments to describe the conduct of filters. Thus, frequency domain equations and curves of gain vs. frequency and phase vs. frequency are commonly used.

It requires their mathematical description in terms of the system’s transfer function to study the frequency domain of networks. The voltage transfer function is the proportion of the Laplace transforms of the output and input signals for a particular scheme as shown below.

Block -Diagram -of - Transfer -Function — Block Diagram of a Transfer Function

$H(s)=\frac{V_{0}(s)}{V_{i}(s)}$

Where V0(s) and Vi(s) are the output and input voltages and s is the complex Laplace transform variable.

A single-pole low pass filter is designed for low-frequency applications by connecting a resistor and a capacitor as shown below.

Transfer - function - circuit — Transfer function circuit

The transfer function of the above circuit can be given as

$H(s)= \frac{1/sC}{R+1/sC} =\frac{1}{RcS+1} = \frac{1/RC}{s+1/RC}$

Now, in the above equation, the laplace constants are replaced with its equivalent value in frequency domain.

$|H(\omega)|= \frac{1/RC}{j\omega+1/RC}$

The absolute value of transfer function is defined as magnitude or volatge gain and it can be represented as shown below

$|H(\omega)|= |\frac{1}{RcS+1}|=|\frac{1}{jRC\omega+1}|= \frac{1}{\sqrt{1+RC(\omega)^{2}}}$

The frequency at which the resistance is equal to the impedance of the capacitor, it is said to be at critical frequency which is given as below.

$\begin{align*} \omega = \omega_{c} when R = \frac{1}{\omega_{c}} \end{align*}$

Therefore,

${\omega_{c}}= \frac{1}{RC}$

After proper substituion into this equation, we get

$Gain = |H(\omega)= \frac{1}{\sqrt{1+(\omega/\omega_{c})^{2}}} = \frac{1}{\sqrt{1+(f/f_{c})^{2}}}$

The phase shift of the filter is given by

$\varphi = tan^{-1}\left(\frac{\omega}{\omega_{c}}\right)$

The cut off frequency and phase-shift of the filter can be calculated as follows

$f_{c}= \frac{1}{2\pi RC}$

$\phi = tan^{-1}(2\pi fRC)$

Similar filter configuration can be obtained for nth order, when RC stages are cascaded together.

First Order Active Low Pass Filter Design And Example

Design a non-inverting active low pass filter circuit that has a gain of ten at low frequencies, a high-frequency cut-off or corner frequency of 175Hz and an input impedance of 20KΩ.

The voltage gain of the non-inverting amplifier is given as

$A_{f}=1+\frac{R_{2}}{R_{1}} =10$

Now assume the value of R1 to be 1KΩ and calculate the value R2 from the above equation.

$R_{2}=(10-1)x R_{1}= 9x\times1k\Omega = 9k\Omega$

Hence for a voltage gain of 10, values of R1 and R2 are 1KΩ and 9KΩ respectively. Gain in dB is given as 20LogA = 20Log10 = 20dB

Now we are given with the cut-off frequency value as 175Hz and input impedance value as 20KΩ. By substituting these values in the equation and value of C can be calculated as follows.

$f_{c}= \frac{1}{2\pi RC}$

$C= \frac{1}{2\pi Rf_{c}}=\frac{1}{2\pi\times 175\times 20k\Omega}$ = 45.47nF

Thus the final design of filter and its equivalent frequency response curve is shown below.

Active Low Pass Filter Circuit

A typical circuit for an active low pass filter is given below:

Active Low Pass Filter Frequency Response Curve

The frequency response curve for an active low pass filter is given below:

Frequency- Response- of -the- problem — Frequency Response of the problem

Non-Inverting Amplifier Filter

A simple non-inverting amplifier filter is given below:

Non-inverting -circuit- for-the- problem — Non-inverting circuit for the problem

Inverting Amplifier Filter

An equivalent inverting amplifier filter is given below:

Equivalent -inverting- circuit- for -the -problem — Equivalent inverting circuit for the problem

Second Order Active Low Pass Filter

Second-Order Filters are also attributed to as VCVS filters since Op-Amp used here is Voltage Controlled Voltage Source Amplifier. This is another important type of active filter used in applications.

The frequency response of the second-order low pass filter is indistinguishable to that of the first-order type besides that the stopband roll-off will be twice the first-order filters at 40dB/decade. Consequently, the design steps wanted of the second-order active low pass filter are identical. A simple method to get a second-order filter is to cascade two first-order filters.

Second -Order- Active -Low -Pass -Filter — Second-Order Active Low Pass Filter

When filter circuits are cascaded into higher-order filters, the filter’s overall gain is equal to the product of each stage. Active second-order (two-pole) filters are essential because they can be used to design higher-order filters. Filters with an order value can be built by cascading first and second-order filters.

Third - Order - Active - Low - Pass - Filter - Configuration — Third Order Active Low Pass Filter Configuration

Fourth -Order -Active -Low- Pass- Filter- Configuration — Fourth Order Active Low Pass Filter Configuration