Filters and Amplifiers
Learning Objectives
- Explain what a filter does in the frequency domain and identify low-pass, high-pass, band-pass, and band-reject types
- Calculate the cutoff frequency of a simple RC filter
- Describe the role of BJT common-emitter and op-amp based amplifier stages
- Explain how combining a filter with an amplifier (active filters) achieves better performance than passive filters alone
- Interpret a basic frequency response (Bode-style) description of gain vs. frequency
- Identify design tradeoffs: gain, bandwidth, impedance matching, and roll-off steepness
Quick Answer
Filters and amplifiers are the two most common building blocks in analog electronics: a filter reshapes a signal's frequency content (removing unwanted frequencies), while an amplifier increases a signal's amplitude without changing its shape. Together they let engineers extract a wanted signal from noise, boost it to a usable level, and prepare it for further processing or transmission. Nearly every real system — an audio system, a radio receiver, a sensor interface — is a chain of filter and amplifier stages, each doing one specific, well-understood job.
What a Filter Actually Does
A filter's job is described in the frequency domain: instead of asking "what does this signal look like over time," a filter description asks "how much does this circuit let through at each frequency?" The cutoff frequency (fc) marks the boundary where the circuit's output drops to about 70.7% of its input amplitude (down 3 dB) — beyond this point, attenuation increases with frequency.
Low-pass filter (RC):
Vin ──[ R ]──┬── Vout
|
[C]
|
GND
The resistor and capacitor form a voltage divider whose impedance ratio depends on frequency, since a capacitor's impedance (1/ωC) falls as frequency rises. At low frequencies, the capacitor's impedance is huge (it looks like an open circuit), so nearly all of Vin appears at Vout. At high frequencies, the capacitor's impedance shrinks toward zero, shorting Vout to ground. The cutoff frequency is:
fc = 1 / (2πRC)
Worked example: With R = 1.6 kΩ and C = 0.01 µF, fc = 1/(2π × 1600 × 0.00000001) ≈ 9,947 Hz ≈ 10 kHz. Frequencies below 10 kHz pass through with little attenuation; frequencies well above 10 kHz are progressively attenuated at a rate of 20 dB per decade (a tenfold increase in frequency reduces amplitude tenfold, for a simple single RC stage).
High-pass filter (RC): Simply swap the resistor and capacitor positions — the capacitor now sits in series with the signal path and the resistor sits to ground. Now DC and low frequencies are blocked (capacitor impedance is huge, dropping most of the voltage across the capacitor rather than reaching Vout), while high frequencies pass through easily. The same cutoff formula, fc = 1/(2πRC), applies.
Band-pass filter: Cascading a high-pass filter (removes very low frequencies) with a low-pass filter (removes very high frequencies) passes only a middle "band" of frequencies — used heavily in radio tuning, where you want only one station's frequency range.
Band-reject (notch) filter: The opposite of band-pass — it removes one specific frequency band while passing everything else. A classic use is removing 50/60 Hz mains hum from an audio signal.
What an Amplifier Actually Does
While a filter reshapes frequency content, an amplifier increases signal amplitude while (ideally) preserving its waveform shape exactly. Two families of amplifying devices dominate analog circuits:
BJT (bipolar junction transistor) amplifiers: The common-emitter configuration is the classic voltage amplifier stage. A small change in base current controls a much larger collector current, and that current flows through a collector resistor to produce a large output voltage swing. Common-emitter stages provide good voltage gain (typically tens to hundreds) but invert the signal (180° phase shift) and have moderate input impedance.
FET amplifiers: The common-source configuration does the same job using a field-effect transistor, offering very high input impedance (ideal for interfacing high-impedance sensors) at the cost of somewhat lower transconductance (gain per unit input voltage) than a comparable BJT stage.
Op-amp amplifiers: As covered in the previous chapter, op-amp based inverting and non-inverting amplifiers give the most predictable, resistor-set gain, at the cost of requiring an active IC rather than a single discrete transistor.
| Configuration | Voltage Gain | Input Impedance | Phase Shift | Common Use |
|---|---|---|---|---|
| Common-emitter (BJT) | High | Moderate | 180° (inverted) | General voltage amplification |
| Common-collector (BJT, emitter follower) | ≈1 | High | 0° | Buffering, impedance matching |
| Common-source (FET) | High | Very high | 180° (inverted) | High-impedance sensor amplification |
| Op-amp (inverting/non-inverting) | Set by Rf/Rin ratio | Very high (non-inv.) | 180° or 0° | Precision amplification, filtering |
Active Filters: Combining Filters and Amplifiers
A passive RC filter has two weaknesses: its output amplitude is always ≤ its input (it can only attenuate, never boost), and its cutoff behavior is "soft" — it takes many octaves to fully attenuate unwanted frequencies. An active filter places an op-amp in the circuit alongside the resistors and capacitors, which:
- Adds gain, so the filter can amplify the passband while attenuating the stopband
- Isolates the filter stage from whatever comes after it (high input impedance of the op-amp prevents loading)
- Achieves steeper roll-off (e.g., 40 dB/decade for a 2-pole active filter vs. 20 dB/decade for a single RC stage) by combining multiple RC sections with feedback
Real-world example: A graphic equalizer in an audio system uses several active band-pass filters, each centered on a different frequency band (bass, mid, treble), each with its own adjustable gain — this is only practical with active (op-amp based) filter stages, since passive RC filters alone can't provide gain.
Design Considerations
When combining filters and amplifiers into a signal chain, three practical factors dominate:
- Frequency response: Does the combined circuit pass the frequencies you care about and reject the ones you don't, with the right steepness (roll-off rate)?
- Gain: Enough gain to raise a weak signal to a useful level, without so much gain that noise or offset voltages get amplified into a problem, or that the amplifier saturates (clips) on large input signals.
- Impedance matching: Each stage's output impedance should be much lower than the next stage's input impedance, so voltage isn't lost across stage-to-stage interfaces (a "voltage divider" effect you don't want).
Why It Matters
Virtually every analog signal chain — a hearing aid, an AM/FM radio, a seismograph, an ECG machine — is built from a sequence of filter and amplifier stages: filter out unwanted frequencies (noise, hum, out-of-band signals), amplify the remaining wanted signal to a usable level, filter again if needed, amplify again if needed. Understanding these two building blocks individually, and how they combine into active filters, is what lets you read (and eventually design) almost any analog schematic.
Common Mistakes
Misconception 1: "A filter's cutoff frequency is a hard wall — everything above (or below) it is completely blocked." Why it's wrong: A single-stage RC filter's roll-off is gradual (20 dB/decade), meaning attenuation increases gradually well past the cutoff frequency rather than switching off instantly. At the cutoff frequency itself, the signal is only attenuated to about 70.7% of its original amplitude, not zero. Correct understanding: Cutoff frequency marks the −3 dB point (the "half power" point), and full attenuation requires being many multiples of fc away from that point, or using a higher-order (multi-stage) filter for a steeper roll-off.
Misconception 2: "An amplifier's gain is a single fixed number regardless of the input signal's frequency." Why it's wrong: Every real amplifier — transistor or op-amp — has a frequency-dependent gain (its frequency response), typically flat over some usable bandwidth and then rolling off at higher frequencies due to internal capacitances or gain-bandwidth limits. Correct understanding: A gain figure only applies within the amplifier's specified bandwidth; check the frequency response (or gain-bandwidth product for an op-amp) before assuming a stated gain holds at your operating frequency.
Misconception 3: "Passive filters and active filters do the same job, so the choice between them doesn't matter much." Why it's wrong: Passive RC/LC filters can only attenuate (gain ≤ 1) and interact with whatever load is connected after them (loading effects change the actual cutoff frequency). Active filters add gain and buffer against loading effects, but require a power supply and an active device (op-amp). Correct understanding: Choose passive filters for simple, low-power, non-critical filtering where signal loss is acceptable; choose active filters when you need gain, a sharp/steep response, or isolation from downstream loading.
Comparison and Connections
| Feature | Passive Filter (RC/LC) | Active Filter (Op-amp based) |
|---|---|---|
| Power source | None needed | Requires DC power supply |
| Gain | ≤ 1 (attenuation only) | Can exceed 1 (adds amplification) |
| Loading sensitivity | High — affected by what's connected next | Low — op-amp buffers the output |
| Roll-off steepness | Limited without cascading many stages | Easier to achieve steep roll-off |
| Feature | Common-Emitter (BJT) | Common-Source (FET) | Op-Amp Stage |
|---|---|---|---|
| Input impedance | Moderate (kΩ range) | Very high (MΩ+) | Very high |
| Gain predictability | Depends on transistor parameters (β) | Depends on transistor parameters | Set precisely by resistor ratio |
| Typical application | General discrete amplification | High-impedance sensor front-ends | Precision signal conditioning |
Practice Questions
Recall 1: State the formula for the cutoff frequency of a simple RC filter. Answer guidance: fc = 1/(2πRC).
Recall 2: Name the four basic filter types by their frequency-domain behavior. Answer guidance: Low-pass, high-pass, band-pass, band-reject (notch).
Understanding 1: Explain why a signal is attenuated to 70.7% of its original amplitude, rather than to zero, right at the cutoff frequency. Answer guidance: The cutoff frequency is defined as the point where output power drops to half of the passband power (the "−3 dB point"). Since power is proportional to voltage squared, a halving of power corresponds to a voltage ratio of 1/√2 ≈ 0.707, not zero.
Understanding 2: Why does adding an op-amp to a passive RC filter (making it "active") allow for a steeper roll-off? Answer guidance: An active filter can cascade multiple RC sections with feedback shaping, each contributing to the roll-off, while the op-amp's buffering prevents one stage from loading and distorting the next — allowing designers to combine stages predictably to sharpen the transition band.
Application 1: Design a low-pass RC filter with a cutoff frequency of 1 kHz using a 10 kΩ resistor. Find the required capacitor value. Answer guidance: C = 1/(2πfc R) = 1/(2π × 1000 × 10000) ≈ 15.9 nF.
Application 2: An audio system needs to remove 60 Hz mains hum without affecting the music signal (20 Hz–20 kHz). Which filter type should be used, and why? Answer guidance: A band-reject (notch) filter centered at 60 Hz, because it removes only the narrow hum frequency while passing the rest of the audio spectrum.
Analysis 1: Compare a common-emitter BJT amplifier stage to a non-inverting op-amp amplifier stage for amplifying a signal from a high-impedance sensor, and recommend which is better suited and why. Answer guidance: The op-amp non-inverting stage is generally better because its very high input impedance avoids loading the sensor and distorting the signal, and its gain is precisely set by resistor ratios rather than depending on transistor-to-transistor variation in β, as the BJT stage would.
Analysis 2: A student cascades two identical low-pass RC filters (each with the same fc) expecting the combined cutoff frequency to stay the same but with steeper roll-off. Evaluate this expectation. Answer guidance: Partially correct: the roll-off does get steeper (from 20 dB/decade to about 40 dB/decade with two stages), but if the stages are simply connected directly without a buffer between them, the second stage loads the first and shifts the effective cutoff frequency lower than expected — an active buffer (op-amp follower) between stages is needed to preserve the original fc of each stage.
FAQ
Q1: What does "3 dB down" mean when describing a filter's cutoff? A: It means the output power has dropped to half of its passband value at that frequency (equivalently, output voltage amplitude has dropped to about 70.7% of its passband value).
Q2: Why do some amplifiers invert the signal (180° phase shift) while others don't? A: It depends on which terminal of the active device the input is applied to and how the output is taken. Common-emitter/common-source stages and inverting op-amp configurations invert; common-collector/common-drain (follower) stages and non-inverting op-amp configurations do not.
Q3: Can a filter amplify a signal? A: A purely passive RC or LC filter cannot — it can only pass signals through with a gain of at most 1 (some loss is typical). An active filter, which includes an amplifying element like an op-amp, can provide gain greater than 1 in its passband.
Q4: Why is impedance matching important between amplifier and filter stages? A: If a stage's output impedance is comparable to or larger than the next stage's input impedance, a voltage-divider effect reduces the signal reaching the next stage, and can also shift the filter's actual cutoff frequency from its designed value.
Q5: What's the practical difference between a band-pass filter and cascading a high-pass and low-pass filter separately? A: They can achieve the same result — a band-pass filter is often literally built by cascading a high-pass stage (removes frequencies below the band) and a low-pass stage (removes frequencies above the band), so the two descriptions usually describe the same underlying circuit.
Quick Revision
- Filters reshape frequency content; amplifiers increase amplitude while preserving waveform shape.
- fc = 1/(2πRC) for a simple RC low-pass or high-pass filter.
- The −3 dB (cutoff) point means output has dropped to 70.7% amplitude (half power).
- Low-pass passes f < fc; high-pass passes f > fc; band-pass passes a middle range; band-reject blocks a middle range.
- Common-emitter (BJT) and common-source (FET) stages provide high voltage gain but invert the signal (180°).
- Common-collector/common-drain (follower) stages have gain ≈1 but very high input impedance — used for buffering.
- Op-amp amplifier gain is set precisely by resistor ratios, independent of transistor parameter variation.
- Passive filters can only attenuate (gain ≤1); active filters (with an op-amp) can also provide gain.
- Active filters buffer against loading and achieve steeper roll-off by cascading stages with feedback.
- Impedance matching (low output impedance driving high input impedance) prevents signal loss between stages.
Related Topics
Prerequisites: Analog Signal Fundamentals, Operational Amplifiers
Related: Signal Conditioning, Feedback Systems, Oscillators
Next: Feedback Systems, Oscillators