Electronics · Ch 5 — Operational Amplifiers

Active filters

5.8

Active filters

Active filters use passive components (resistors and capacitors) together with active components (op-amps) to select a band of frequencies. The op-amp provides amplification and also isolates or buffers the filter stage from the following circuit.

Active low-pass filter

An ideal low-pass filter passes signals from zero frequency up to a cut-off frequency $f_c$ with constant gain, and blocks everything above $f_c$ . A first-order low-pass active filter is a single RC low-pass stage followed by an op-amp connected as a non-inverting amplifier (figure 5.8.1); its frequency response is shown in figure 5.8.2. Using two RC sections gives a second-order low-pass active filter, whose gain rolls off faster and is therefore closer to the ideal response.

The theoretical cut-off frequency and the voltage gain are

f_c = \dfrac{1}{2\pi R_1 C_1}, \qquad A_V = 1 + \dfrac{R_f}{R_i}

Active high-pass filter …

Figure 1Circuit of a first-order low-pass active filter: a passive RC low-pass network feeding an op-amp non-inverting amplifier.

Fig. 1 — Circuit of a first-order low-pass active filter: a passive RC low-pass network feeding an op-amp non-inverting amplifier.

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

Figure 5.8.1 shows the first-order low-pass active filter. A passive RC low-pass network (source Vi in series with resistor R1, and capacitor C1 from that node to ground, enclosed in a dashed box) drives the non-inverting (+) input of the op-amp. The op-amp is wired as a non-inverting amplifier with input resistor Ri (to a grounded node) and feedback resistor Rf, giving output Vo. The RC netw …

Figure 2Frequency response curve of a first-order low-pass active filter, showing a flat gain up to the cut-off frequency fc and a falling response beyond it.

Fig. 2 — Frequency response curve of a first-order low-pass active filter, showing a flat gain up to the cut-off frequency fc and a falling response beyond it.

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

Figure 5.8.2 shows the low-pass frequency response. The vertical axis is voltage gain Av and the horizontal axis is frequency in Hz. The gain is flat at Av(max) from zero up to the cut-off, then slopes down. A dashed line at Av(max)/sqrt(2) meets the falling slope, and a vertical dotted line drops from that p …

Formula 3Cut-off frequency and voltage gain of a first-order active filter

Cut-off frequency $f_c = \dfrac{1}{2\pi R_1 C_1}$ ; voltage gain (op-amp as non-inverting amplifier) $A_V = 1 + \dfrac{R_f}{R_i}$ . The same two expressions apply to both the low-pass and the h …

Figure 4Circuit of a first-order high-pass active filter: a passive RC high-pass network feeding an op-amp non-inverting amplifier.

Fig. 4 — Circuit of a first-order high-pass active filter: a passive RC high-pass network feeding an op-amp non-inverting amplifier.

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

Figure 5.8.3 shows the first-order high-pass active filter. A passive RC high-pass network (source Vi in series with capacitor C1, and resistor R1 from the C1-R1 junction to ground, in a dashed box) drives the non-inverting (+) input of the op-amp. The op-amp is a non-inverting amplifier with input resistor Ri and feedback resistor Rf, supply pins +Vcc and -Vee, output Vo. Placing the capacit …

Figure 5Frequency response curve of a first-order high-pass active filter, showing gain rising to the cut-off frequency fc and then flat beyond it.

Fig. 5 — Frequency response curve of a first-order high-pass active filter, showing gain rising to the cut-off frequency fc and then flat beyond it.

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

Figure 5.8.4 shows the high-pass frequency response. The vertical axis is gain and the horizontal axis is frequency in Hz. The curve rises from the origin, crosses Av(max)/sqrt(2) at the cut-off frequency fc (marked by dashed lines), then flattens to a plate …

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