Experiment #6 EEL 4314L Active Filters Page 5

Experiment #6 EEL 4314L Active Filters Page 5

Objective: To gain experience with active filters.

General Information:

1.  Introduction

In circuit theory, a filter is an electrical network that alters the amplitude and/or phase characteristics of a signal with respect to frequency. Ideally, a filter will not add new frequencies to the input signal, nor will it change the component frequencies of that signal, but it will change the relative amplitudes of the various frequency components and/or their phase relationships.

Filters are often used in electronic systems to emphasize signals in certain frequency ranges and reject signals in other frequency ranges. Such a filter has a gain which is dependent on signal frequency.

As an example, consider a situation where a useful signal at frequency f1 has been contaminated with an unwanted signal at f2. If the contaminated signal is passed through a circuit (Figure 1.1.) that has very low gain at f2 compared to f1, the undesired signal can be removed, and the useful signal will remain. Note that in the case of this simple example, we are not concerned with the gain of the filter at any frequency other than f1 and f2. As long as f2 is sufficiently attenuated relative to f1, the performance of this filter will be satisfactory.

In general, however, a filter gain may be specified at several different frequencies, or over a band of frequencies. Since filters are defined by their frequency-domain effects on signals, it makes sense that the most useful analytical and graphical descriptions of filters also fall into the frequency domain. Thus, curves of gain vs frequency and phase vs frequency are commonly used to illustrate filter characteristics,and the most widely-used mathematical tools are based in the frequency domain.

The frequency-domain behavior of a filter is described mathematically in terms of its transfer function or network function. This is the ratio of the Laplace transforms of its output and input signals. The voltage transfer function H(s) of a filter can therefore be written as:

where VIN(s) and VOUT(s) are the input and output signal voltages and s is the complex frequency variable. The transfer function defines the filter response to any arbitrary input signal, but we are most often concerned with its effect on continuous sine waves. Especially important is the magnitude of the transfer function as a function of frequency, which indicates the effect of the filter on the amplitudes of sinusoidal signals at various frequencies. Knowing the transfer function magnitude (or gain) at each frequency allows us to determine how well the filter can distinguish between signals at different frequencies. The transfer function magnitude versus frequency is called the amplitude response or sometimes, especially in audio applications, the frequency response.

Similarly, the phase response of the filter gives the amount of phase shift introduced in sinusoidal signals as a function of frequency. Since a change in phase of a signal also represents a change in time, the phase characteristics of a filter become especially important when dealing with complex signals where the time relationships between signal components at different frequencies are critical.

By replacing the variable s in (1) with jω, where j is equal to , and ω is the radian frequency (2πf), we can find the filter's effect on the magnitude and phase of the input signal. The magnitude is found by taking the absolute value of the transfer function as:

The magnitude is and the phase is .

Figure 1.1. Using a Filter to Reduce the Effect of an Undesired Signal at Frequency f2, while Retaining Desired Signal at Frequency f1

2.  Active Filters

Active filters use amplifying elements, especially op amps, with resistors and capacitors in their feedback loops, to synthesize the desired filter characteristics. Active filters can have high input impedance, low output impedance, and virtually any arbitrary gain.

They are also usually easier to design than passive filters. Possibly their most important attribute is that they lack inductors, thereby reducing the problems associated with those components. Still, the problems of accuracy and value spacing also affect capacitors, although to a lesser degree. Performance at high frequencies is limited by the gain-bandwidth product of the amplifying elements, but within the amplifier's operating frequency range, the op amp-based active filter can achieve very good accuracy, provided that low-tolerance resistors and capacitors are used. Active filters will generate noise due to the amplifying circuitry, but this can be minimized by the use of low-noise amplifiers and careful circuit design. In Figure 2.1., low – pass, high – pass and band – pass filters are shown.

Figure 2.1. Filter response vs frequency for low, high and band pass filters.

2.1 Low – Pass Filter

A low – pass filter passes all frequencies from DC (zero frequency) up to its cutoff frequency . At frequencies above the cutoff frequency, the output is greatly attenuated. There are a number of active low-pass filter circuits, and one of the more commonly used is shown in Figure 2.1.1. This circuit is a second-order low – pass active filter, and is a type of voltage – controlled – voltage – source (VCVS). It is also known as a Sellen – Key filter. Because there are two R – C pairs that control the frequency response, it is second order. These are R1, C1 and R2, C2.

Figure 2.1.1. Active Low – Pass Filter

If we set R1 = R2 and C1 = 2C2, the cutoff frequency is given by:

This is a unity – gain VCVS filter. It provides a maximally flat pass- band response (Butterworth) below the cutoff frequency, and a response that rolls off (attenuates) at 40dB/decade above the cutoff frequency. At R1 = R2, place R3 ≈ 2R1 to minimize the DC offset.

Figure 2.1.2. Frequency response of second order VCVS low-pass filter

2.2 High – Pass Filter

The complement of the low – pass filter is the high – pass filter. A high – pass version of the second order VCVS filter is shown in Figure 2.2.1.

Figure 2.2.1. High – pass filter

For C1 = C2 and R2 = 2R1, the cutoff frequency is given by:

To minimize DC offset, R3 ≈ R2. This circuit has unity gain above ,and rolls off at 40 dB per decade below . In Figure 2.2.2, the frequency response of the high – pass filter vs frequency is given.

Figure 2.2.2. Frequency response of second order VCVS high-pass filter

2.3  Band – Pass Filter

A band – pass filter is a circuit designed to pass signals only in a certain band of frequencies while rejecting all signals outside this band. Looking at Figure 2.1C, the bandwidth (BW) of the filter is defined as the difference between the upper cutoff frequency and the lower cutoff frequency . The center frequency is . Actually is a geometric mean, because the frequency scale of Figure 2.1C is logarithmic.

The ratio of the center frequency to the bandwidth is called the Q or quality factor. It is a measure of the selectivity of the circuit.

A band – pass filter may readily be formed by cascading a low – pass filter and a high – pass filter so that their pass – bands overlap. This is often done to form filters with a wide bandwidth.

A very narrow band – pass filter may be constructed by using a notch filter as a feedback element .The notch filter blocks the negative feedback signal over a very narrow band of frequencies and creates a very high gain over the narrow range of frequencies. The notch filter supplies heavy feedback – low amplifier gain – outside its pass band and attenuates frequencies in these ranges.

Pre-Lab

Compute the following tranfer function and show them in details in your report:

1)  Compute the transfer function and cut – off frequency of the circuit given in the first step of the Procedure.

2)  Compute the transfer function and cut – off frequency of the circuit given in the second step of the Procedure.

3)  Compute the transfer function and cut – off frequency of the circuit given in the third step of the Procedure.

Procedure :

1. i) Wire the following low–pass amplifier circuit. The 0.02 μF capacitor consists of two 0.01 μF in parallel.

ii)  Apply a sinusoidal input voltage with Vpp = 2 V and position the input signal above the output signal on the oscilloscope screen.

iii)  Sweep the input frequency over the range of 200Hz to 10 KHz as defined in the below table and observe the effect on the output amplitude.

iv)  Observe and record the filter gain at the indicated frequencies.

Frequency / Measured Gain (Vo/Vin)
200 Hz
500 Hz
1000 Hz
1100 Hz
1200 Hz
2000 Hz
10 kHz

v)  Find and record the cutoff frequency. This is the frequency where Vo/Vin = 0.707. Record the oscilloscope display at cutoff frequency. Show the simulation result at cutoff frequency in your report as well.

2.  Wire the following high–pass filter and repeat the above steps as in the low-pass filter. Fill in the following table.

Frequency / Measured Gain (Vo/Vin)
200 Hz
500 Hz
1000 Hz
1100 Hz
1200 Hz
2000 Hz
10 kHz

3.  i) Wire the following pass- band circuit.

ii) Apply power and position the input signal above the output signal on the oscilloscope screen. Sweep the frequency over the range of 200Hz to 2 kHz as the below table and observe the effect on the output amplitude.

iii) Observe and record the filter gain at the indicated frequencies.

Frequency / Measured Gain (Vo/Vin)
200 Hz
400 Hz
600 Hz
800 Hz
1200 Hz
1400 Hz
2000 Hz

iv) Measure the center frequency. This is the frequency at which the output reaches maximum amplitude. Record the oscilloscope display. Show the simulation result at center frequency in your report as well.