An inverting active low-pass filter has passband gain = -Rf/Rin and cutoff frequency fc = 1/(2πRfC). An engineer wants to double the passband gain without changing the cutoff frequency. What should she do?
AHalve Rin while keeping Rf and C unchanged — gain doubles, cutoff is unaffected since Rf and C don't change
BDouble Rf while keeping all other components unchanged — gain doubles and cutoff shifts proportionally
CDouble C while keeping Rf and Rin unchanged — the capacitor controls both gain and cutoff equally
DDouble both Rf and Rin proportionally — this maintains the gain ratio while adjusting for higher frequency
Passband gain = Rf/Rin; cutoff fc = 1/(2πRfC). To double gain, double the ratio Rf/Rin. To keep cutoff fixed, keep Rf and C unchanged. Therefore halve Rin: new gain = Rf/(Rin/2) = 2Rf/Rin. Option 1 (doubling Rf only) changes both gain and cutoff simultaneously. This demonstrates the key advantage of active filters: gain (set by resistor ratio) and cutoff (set by Rf × C) are independently adjustable through separate components.
Question 2 Multiple Choice
A first-order active low-pass filter and a first-order passive RC low-pass filter are both designed with a cutoff frequency of 1 kHz. How do their roll-off rates above 1 kHz compare?
ABoth roll off at -20 dB/decade — the active version adds passband gain and buffering but does not change the filter order or roll-off rate
BThe active filter rolls off at -40 dB/decade, because the op-amp adds an additional pole to the transfer function
CThe active filter rolls off at -20 dB/decade; the passive rolls off at -10 dB/decade due to loading effects
DThe active filter maintains flat gain at all frequencies; only the passive version has a roll-off region
Roll-off rate is determined by filter ORDER — a first-order filter always rolls off at -20 dB/decade, regardless of whether it is active or passive. The op-amp does NOT add a pole in a first-order active filter; it provides gain and isolation. Option 1 (op-amp adds a pole, -40 dB/decade) is the most tempting misconception — active does not mean sharper. To get steeper roll-off, you need a higher-order filter design.
Question 3 True / False
A first-order active high-pass filter has a steeper roll-off below the cutoff frequency than an equivalent first-order passive RC high-pass filter.
TTrue
FFalse
Answer: False
Both are first-order filters and both roll off at exactly -20 dB/decade below the cutoff frequency. Adding an op-amp adds passband gain and prevents loading effects when cascaded, but it does not change the filter order. Roll-off slope is governed by the number of poles in the transfer function, not by whether the circuit is active or passive.
Question 4 True / False
The gain-bandwidth product (GBW) of an op-amp limits the practical performance of active filters at high frequencies.
TTrue
FFalse
Answer: True
An op-amp's open-loop gain drops at high frequencies. For a filter with passband gain A and cutoff frequency fc, the op-amp must supply gain × bandwidth = A × fc. If this product exceeds the op-amp's GBW, the op-amp's own rolloff will interfere with the designed filter response. For example, a 40 dB gain (100×) filter with a 10 kHz cutoff requires GBW ≥ 1 MHz. At very high frequencies, passive filters are often preferred precisely because they have no power supply or bandwidth limitations.
Question 5 Short Answer
What is the fundamental capability that active filters have over passive RC filters, and what circuit property of the op-amp makes this possible?
Think about your answer, then reveal below.
Model answer: Active filters can provide passband gain greater than unity — passive filters can only attenuate (maximum passband gain is 0 dB). The op-amp's near-zero output impedance also prevents loading effects when stages are cascaded, enabling each stage to be designed independently. These two properties — gain and isolation — are unavailable in passive designs and make active filters essential when signal amplification and multistage filtering are needed simultaneously.
The near-infinite input impedance and near-zero output impedance of the op-amp solve both problems passive filters face: inability to amplify and sensitivity to loading. When cascading two passive RC stages, the second stage loads the first, shifting the cutoff frequency. Active stages are perfectly isolated: the op-amp output drives the next stage from zero impedance, so the second stage has no effect on the first.