Low Pass Filters
A low pass filter is one of the most useful circuits in electronics. As the name suggests, it passes low-frequency signals while attenuating (reducing) high-frequency signals. Below a certain frequency — the cutoff frequency — signals pass through with little loss. Above the cutoff frequency, signals are increasingly attenuated the higher their frequency goes.
In amateur radio, the most critical application of the low pass filter is harmonic suppression. Every transmitter generates harmonics — signals at 2, 3, 4, and more times the operating frequency. A well-designed low pass filter placed between your transmitter and antenna passes your signal on 40 meters (7 MHz) while suppressing the second harmonic on 20 meters (14 MHz), the third harmonic near 21 MHz, and higher harmonics beyond. The FCC requires this suppression; the low pass filter provides it. Understanding how these filters work — and how to calculate them — makes you a better operator and builder.
Low pass filter frequency response on a logarithmic scale. The passband is flat up to the cutoff frequency fc. At fc, the output is −3 dB (0.707× the input voltage). Above fc, the single-pole filter rolls off at 20 dB per decade — each factor of 10 increase in frequency reduces the output by another 20 dB.
View LargerHow a Low Pass Filter Works
To understand a low pass filter, start with the concept of a voltage divider. A basic voltage divider uses two resistors in series — one connected to the input, one to ground — and takes the output from the junction between them. The output voltage is a fixed fraction of the input, regardless of frequency.
Now replace the bottom resistor with a capacitor. A capacitor's impedance (XC) is high at low frequencies and low at high frequencies. At low frequencies, XC is large compared to the series resistor R — most of the voltage appears across the capacitor (the output), and little is dropped across R. At high frequencies, XC is small compared to R — most of the voltage is dropped across R, and little appears at the output.
This is the essence of an RC low pass filter: the capacitor divides the signal voltage in a way that is frequency-dependent, passing low frequencies and attenuating high frequencies. The same principle applies to an RL low pass filter, where the inductor acts as the top element (high impedance at high frequency, low impedance at low frequency) with a resistor to ground at the output.
The RC Low Pass Filter
The RC low pass filter consists of a resistor in series with the input and a capacitor from the junction to ground. The output is taken across the capacitor.
The transfer function (ratio of output to input voltage) as a function of frequency is:
Vout / Vin = XC / √(R² + XC²) = 1 / √(1 + (f/fc)²)
where the cutoff frequency fc is:
fc = 1 / (2πRC)
At f = fc: Vout/Vin = 1/√2 = 0.707 = −3 dB
The cutoff frequency fc is the half-power frequency — the frequency at which output power has fallen to half its maximum value, or equivalently, voltage has fallen to 0.707 of maximum (same −3 dB definition as bandwidth). Below fc, the filter is in the passband. Above fc, the filter is in the stopband (rolloff region).
At the cutoff frequency, R = XC, meaning the resistor's impedance equals the capacitor's impedance. This gives a phase shift of exactly 45° between input and output at fc.
A receiver audio circuit picks up 60 Hz hum from the power supply. You want a low pass filter with a cutoff frequency of 3,400 Hz (top of voice bandwidth) using a 10 kΩ resistor. Find the required capacitance.
Rearranging fc = 1/(2πRC) for C:
C = 1 / (2π × fc × R) = 1 / (2π × 3,400 × 10,000)
= 1 / (213,628,300) = 4.68 nF ≈ 4.7 nF (standard value)
Verification: fc = 1/(2π × 10,000 × 4.7×10⁻⁹) = 1/295,310 = 3,387 Hz ✓
Attenuation at 60 Hz: Vout/Vin = 1/√(1 + (60/3387)²) = 1/√(1.000314) = 0.9998 — essentially zero attenuation at 60 Hz!
Correction: A filter with fc = 3,400 Hz passes 60 Hz freely. To attenuate 60 Hz, you would need fc far below 60 Hz — like 10 Hz — using a much larger capacitor: C = 1/(2π × 10 × 10,000) = 1.59 µF. At 3,400 Hz, attenuation would then be: Vout/Vin = 1/√(1 + (3400/10)²) = 1/√115,601 = 0.00294 = −50.6 dB. This effectively removes audio above 10 Hz, which is not useful for voice.
The lesson: filter cutoff frequency selection requires understanding what you want to pass and what you want to reject — the cutoff lies between them.
RC Low Pass Cutoff Frequency Calculator
RC Low Pass Filter Cutoff Frequency Calculator
Enter R and C to find the cutoff frequency, or enter any two known values to solve for the third. Attenuation at any frequency is also calculated.
The RL Low Pass Filter
The RL low pass filter replaces the capacitor with an inductor. The inductor is placed in series with the signal path (not the resistor as in RC filters), and the resistor is connected from the output to ground. The inductor has low impedance at low frequencies (allowing signal to pass) and high impedance at high frequencies (blocking the signal), while the resistor provides the output load.
Wait — that sounds backwards from the RC filter. Let us clarify both configurations:
| Filter Type | Series Element | Shunt Element | Why It Low-Passes |
|---|---|---|---|
| RC Low Pass | Resistor (R) | Capacitor (C) to ground | At high f, C shorts the output to ground; at low f, C is open and output ≈ input |
| RL Low Pass | Inductor (L) | Resistor (R) to ground | At high f, L blocks the signal; at low f, L passes signal freely to the R output |
The cutoff frequency formula for an RL low pass filter is:
fc = R / (2πL)
where R is in ohms and L is in henrys
At f = fc: XL = R (inductor and resistor impedances are equal), output = −3 dB
The RL filter is less common than the RC filter in general electronics because inductors are bulkier and more expensive than capacitors for the same cutoff frequency. However, RL filters are used in power supply output stages (where the inductor also stores energy) and in RF circuits where inductors are more natural elements than large capacitors.
A regulated power supply includes an output RL filter. The load resistance is 10 Ω. You want the cutoff frequency to be 100 Hz so that the 120 Hz ripple component from the rectifier is attenuated by at least 20 dB. Find the required inductance and verify the attenuation at 120 Hz.
Step 1 — Find L for fc = 100 Hz:
L = R / (2π × fc) = 10 / (2π × 100) = 10 / 628.3 = 15.9 mH
Step 2 — Attenuation at 120 Hz:
Vout/Vin = 1/√(1 + (f/fc)²) = 1/√(1 + (120/100)²) = 1/√(1 + 1.44) = 1/√2.44 = 1/1.562 = 0.640
Attenuation = 20 log10(0.640) = −3.87 dB
Problem: Only 3.87 dB of ripple attenuation — far less than the required 20 dB. A single-pole RL filter with fc = 100 Hz gives only 3.87 dB at 120 Hz because 120 Hz is barely above cutoff.
Solution: To get 20 dB at 120 Hz from a single pole, you need fc much lower, OR use a two-pole LC filter (which rolls off at 40 dB/decade). With a two-pole filter, 20 dB at 120 Hz requires fc ≈ 38 Hz. This illustrates why power supply filters often use multiple L-C stages.
RL Low Pass Cutoff Frequency Calculator
RL Low Pass Filter Cutoff Frequency Calculator
Enter R and L to find the cutoff frequency. Optionally enter a test frequency to find attenuation.
Rolloff Rate and Filter Order
A single RC or RL section is called a first-order or single-pole filter. Above the cutoff frequency, its attenuation increases at a rate of 20 dB per decade — for every ten-fold increase in frequency above fc, attenuation increases by 20 dB. This slope corresponds to −6 dB per octave (per doubling of frequency).
−20 dB/decade = −6 dB/octave
At 10× fc: −20 dB attenuation
At 100× fc: −40 dB attenuation
At 1000× fc: −60 dB attenuation
For many applications, −20 dB/decade is not steep enough. Adding more filter sections — each adding one pole — increases the rolloff rate:
| Filter Order (Poles) | Rolloff Rate | Attenuation at 10× fc | Components Needed |
|---|---|---|---|
| 1st order (1 pole) | −20 dB/decade | −20 dB | 1 R + 1 C, or 1 L + 1 R |
| 2nd order (2 poles) | −40 dB/decade | −40 dB | 1 L + 1 C (LC filter) |
| 3rd order (3 poles) | −60 dB/decade | −60 dB | 2 L + 1 C, or 1 L + 2 C |
| 5th order (5 poles) | −100 dB/decade | −100 dB | 3 L + 2 C (typical HF low pass) |
| 7th order (7 poles) | −140 dB/decade | −140 dB | 4 L + 3 C (high-performance HF) |
Transmitter output low pass filters typically use 5 or 7 poles to achieve the required 40 dB of harmonic suppression with a cutoff frequency close to the operating frequency. Placing the cutoff close to the operating frequency provides more attenuation at the second harmonic while still passing the fundamental with minimal loss.
LC Low Pass Filters for Transmitters
The RC and RL low pass filters described above work well for audio and low-frequency applications, but RF transmitter output filters must use LC (inductor-capacitor) circuits. The reason is efficiency: resistors dissipate power as heat, and even a 1 Ω resistor in the output circuit of a 100 W transmitter carrying 1.4 A of RF current would dissipate I²R = 1.4² × 1 = 2 W of transmitter power unnecessarily. Inductors and capacitors (ideal) dissipate no power — they only store and return it.
The standard transmitter output low pass filter is the Chebyshev or Butterworth LC ladder network. These networks alternate shunt capacitors (to ground) and series inductors, creating a cascade of poles. The component values are calculated from filter design tables based on the cutoff frequency and desired impedance.
For a 7-element (7-pole) Chebyshev low pass filter designed for 50 Ω systems, the cutoff frequency formula for each element follows:
For a low pass filter with cutoff fc and system impedance Z0:
Cn = cn / (2π × fc × Z0)
Ln = ln × Z0 / (2π × fc)
where cn and ln are normalized values from filter design tables
For practical ham radio use, published filter designs from the ARRL Handbook provide component values directly for common band low pass filters. These have been optimized and verified — no need to derive from tables for typical applications.
A 5-element Chebyshev low pass filter has fc = 10 MHz in a 50 Ω system. The transmitter operates at 7.1 MHz. Estimate attenuation at the second harmonic (14.2 MHz) and third harmonic (21.3 MHz).
At the second harmonic (14.2 MHz):
Frequency ratio: 14.2/10 = 1.42 — only 42% above cutoff
For a 5-pole Chebyshev filter, attenuation ≈ 5 × 20 × log10(1.42) ≈ 5 × 20 × 0.152 = 15.2 dB
(Actual Chebyshev attenuation is higher due to equiripple design, typically 25–35 dB at 1.42× cutoff for 5 poles with 0.5 dB ripple.)
At the third harmonic (21.3 MHz):
Frequency ratio: 21.3/10 = 2.13
Approximate attenuation: 5 × 20 × log10(2.13) ≈ 5 × 20 × 0.328 = 32.8 dB
(Actual Chebyshev: typically 45–55 dB at 2.13× cutoff.)
Conclusion: A single 5-pole LC low pass filter gives excellent harmonic suppression for a 40-meter transmitter. Combined with the tank circuit's own selectivity, total harmonic suppression typically exceeds 50 dB — well above the FCC minimum of 40 dB.
Low Pass Filters in Your Shack
HF Transmitter Output Low Pass Filter
Every HF transmitter that complies with FCC Part 97 has a low pass filter at its output. In commercial transceivers this is internal. For home-built transmitters and QRP kits, the low pass filter is often a separate module soldered during assembly. The ARRL Handbook publishes complete designs for each amateur HF band. If you homebrew a transmitter, the low pass filter is not optional — it is a legal and technical requirement.
VHF/UHF Low Pass Filters
VHF and UHF transmitters also require low pass filters, though the component values are much smaller (picofarads and nanohenrys at 144 MHz and above). These filters are often implemented as microstrip transmission line sections on circuit boards rather than discrete L and C components at GHz frequencies.
Receiver Audio Low Pass Filters
Many receivers include a switchable audio low pass filter that reduces the bandwidth to about 2,400 Hz for SSB reception, improving intelligibility by cutting high-frequency hiss. The RC filter for this application typically uses R = 10–47 kΩ and C chosen to set fc at 2.4 kHz.
CW Narrow Filter
For CW (Morse code) reception, a low pass filter with fc around 700–900 Hz removes much of the adjacent-channel interference. Combined with a high pass filter (covered in the next lesson) to form a bandpass filter centered around the CW note frequency, this dramatically improves weak-signal CW reception.
Experiment: Build and Test an RC Audio Low Pass Filter
You need: 10 kΩ resistor, 4.7 nF capacitor, breadboard, audio signal generator or PC sound card with signal generator software, oscilloscope or audio voltmeter.
Theory: fc = 1/(2π × 10,000 × 4.7×10⁻⁹) = 3,387 Hz.
Build: Connect 10 kΩ from input to junction. Connect 4.7 nF from junction to ground. Output is taken at the junction.
Test:
- Apply 1 V at 100 Hz. Measure output — should be nearly 1 V (well below cutoff).
- Apply 1 V at 3,387 Hz. Measure output — should be 0.707 V (−3 dB at cutoff).
- Apply 1 V at 10 kHz. Measure output — should be 0.323 V (−9.8 dB, close to single-pole prediction of about −9.5 dB).
- Apply 1 V at 33.87 kHz (10× fc). Measure output — should be 0.0995 V (about −20 dB).
Expected result: Measured values should be within 5–10% of theoretical at each frequency. Any large discrepancy suggests a wiring error or component out of specification. This experiment demonstrates the −20 dB/decade rolloff of a single-pole filter directly.
Frequently Asked Questions
Why is the cutoff frequency defined at −3 dB rather than where the signal is completely blocked?
A simple RC or RL filter never completely blocks any frequency — attenuation increases gradually and approaches but never reaches infinite attenuation. The −3 dB point (0.707 × voltage, half power) was chosen as the standard cutoff because it represents the half-power frequency and sits at the mathematically significant point where the reactive impedance equals the resistance. It is a precise, measurable, and universally agreed-upon definition. In practice, "useful" signal passes below this frequency and is "significantly attenuated" above it, making it a practical engineering boundary.
Can I use a low pass filter as an antenna tuner?
Not directly — an antenna tuner must match impedances (transform between the transmitter's 50 Ω and the antenna's impedance), while a simple low pass filter does not change impedance. However, LC low pass filter topologies (like the L-network) can simultaneously filter harmonics and transform impedances when designed for that purpose. The L, Pi, and T matching networks covered in a later lesson are effectively impedance-transforming filters that serve both functions at once.
Does a low pass filter affect my transmitted signal quality?
A well-designed low pass filter with its cutoff above the operating frequency passes the fundamental with negligible loss and does not affect signal quality. A filter with a cutoff too close to the operating frequency can attenuate the signal and introduce distortion. For a 40-meter transmitter at 7.1 MHz, a filter with fc = 10 MHz provides good harmonic suppression with minimal effect on the fundamental. Filters designed for each amateur band are available from the ARRL Handbook.
Test Your Knowledge
Answer the questions below to check your understanding. Every answer can be found in the lesson above.