State variable filters use three (or more) cascading operational amplifier circuits (active elements) to produce individual filter outputs. However, an additional aggregation amplifier can also be added to produce a fourth notch filter output response if necessary.
State variable filters are second-degree RC active filters consisting of two identical op-amp integrators, each of which functions as a primary, unipolar low pass filter, aggregation amplifier from which we can adjust the filter gain and damping feedback network. Output signals from all three op-amp stages feed back into the input, which allows us to define the state of the circuit.
One of the most important advantages of state variable filter design is that the three main parameters of filters, gain (A), corner frequency, εC and filters Q, can be adjusted or adjusted independently without affecting the performance of filters.
In fact, if designed correctly, the corner frequency (εc) point of -3db for both low pass amplitude response and high transition amplitude response should be the same as the Central frequency point of the band transition phase. This means that εLP(-3db) is equal to εhp (- 3dB), which is equal to εBP(Center). In addition, the damping factor (ζ) for tape-passing filter response must be equal to 1/Q because it will be set in Q-3db (0.7071).
Although the filter provides low pass (LP), high transition (HP), and band pass (BP) outputs, the main application of this type of filter circuit is the filter design that passes a state variable band with a Central frequency set by two RC integers.
Although we have previously seen that the properties of filters that pass a band can be obtained by combining a low-pass filter with a high-pass filter, state variable band passing filters have the advantage that they can be adjusted to be high selectors (high Q) that offer high gains at the Central frequency point.
There are various state variable filter designs based on the standard filter design with both inverted and non-inverted variations. However, the basic filter design will be the same for both variations, as shown in the block diagram demonstration below.
Status Variable Filter Block Diagram
Next, from the basic block diagram above we can see that the state variable filter has three possible outputs, VHP, VBP and VLP, each of which is one of three op-amp. A notch filter response can also be performed by adding a fourth op-amp.
The VIN output from the collection amplifier with a constant input voltage also produces a high transition response, which becomes the input of the first RC Integrator. The output of this integrator produces a tape-passing response that becomes the input of the second RC Integrator, which produces a low transition response at the output. As a result, separate transfer functions can be available for each output according to the input voltage.
Status Variable Filter Circuit
Normalized Response to Status Variable Filter
One of the main design elements of a state variable filter is the use of two op-amp integrators. As we see in the integrator tutorial, op-amp integrators use a frequency-dependent impedance in the form of a capacitor within the feedback loop. When using a capacitor, the output voltage is proportional to the integral of the input voltage as shown.
Op-amp Integrator Circuit
The Vout output voltage is fixed at 1/RC times the integral of the input voltage of the V according to the time. Integrators produce a phase delay with a Minus sign ( – ) indicating 180o phase shift, since the input signal is connected directly to the reverse input terminal of the op-amp.
Op-amp A2 Transfer Function
To find the transfer function for the other OP-amp integrator A3, exactly the same assumption can be made as above.
Op-amp A3 Transfer Function
Thus, the two op-amp integrators, A2 and A3, are connected in stages. Thus, the output from the first (VBP) becomes the input of the second. In this way, we can see that the band migration response is generated by integrating the high migration response, and the low migration response is generated by integrating the band migration response. Therefore, the transfer function between VHP and VLP is given as follows:
Note that each stage of the integrator provides inverted output, but the total outputs will be positive because they invert integrators. If exactly the same values are used for R and C for the two circuits to have the same integrator time constant, the two amplifier circuits can be accepted with a single integrator circuit with a corner frequency of εC.
In addition to the two integrator circuits, the filter also has a differential collection amplifier, which provides a weighted sum of its inputs. The advantage here is that inputs to the A1 collection amplifier combine oscillation feedback damping and input signals into the filter, as all three outputs are fed back into the aggregation inputs.
Amplifier Collection Circuit
The operational amplifier, A1, is connected as a collector-extractor circuit. That is, it collects the input signal, the vn with the VBB output of the op-amp A2 and extracts the VLP output of the op-amp A3 from it.
As differential inputs, the +V and-V of an operational amplifier are the same, that is: +V – -V, A1 output, we can rearrange the above two statements to find the transfer function for high transition output.
Status Variable Filter Transfer Function
We have previously said that a state variable filter produces three filter responses, low transition, high transition, and band transition response is the response of a very narrow high Q filter.
Normalized 2. Degree Transfer Function
State Variable Filter Corner Frequency
If we make the R3 and R4 feedback resistors the same values, the corner frequency of each filter that comes out of the state variable filter is simply:
The state variable corner frequency is performed only if that setting changes or adjustment resistance, R or capacitor, C.
State variable filters are characterized not only by individual output responses, but also by "Q" filters, which are the quality factor. Q is related to the "sharpness" of the amplitude response curve of tape-passing filters. The higher Q, the higher or sharper the output response, which leads to a highly selective filter.
Q Factor of a Status Variable Filter
Status Variable Filter Design
Now, we can draw individual output response curves for the state variable filter circuit in a frequency range from 1Hz to 1MHz to a Bode chart as shown.
Next, from the filter response curves above, we can see that the DC gain of the filter circuit is at 5.57 db, which is equal to open loop voltage gain, Ao or 1.9, as calculated above. It also indicates that the output curves reached the peak in the maximum voltage gain of 25.6 dB at the corner frequency depending on the Q value. Since Q also associates the Central frequency of tape-passing filters with bandwidth, the bandwidth of the filter will be as follows: εo/10 = 100Hz.
In this case variable filter tutorial, we found that instead of an active filter that produces some kind of frequency response, we can use multiple feedback techniques to produce three filter responses, low pass, high pass, and band pass at the same time.
Notch Filter Design
A notch filter filter is basically the opposite of a tape-passing filter, because it rejects or stops a specific frequency band. Then the notch filter is also known as "band stop filter". To get the response of a notch filter from the basic state variable filter design, we must combine high-pass and low-pass output responses using another op-amp aggregation amplifier.
Here we assumed that the two input resistances, R5 and R6, as well as the feedback resistance, the r7 all have the same 10kΩ value as R3 and R4 to keep everything simple. Therefore, 1 gives the notch filter a gain of unity.
The output response of the notch filter and the tape-passing filter is related to the fact that the Central frequency of the tape-passing response is equal to the zero response point of the notch filter, and in this example it will be 1kHz.