What S21 and S11 of a Low-Pass Filter Look Like
At the -3 dB cutoff a Butterworth low-pass filter transmits half the power and reflects half, so S21 and S11 are both -3.01 dB. Above the cutoff S21 falls and S11 climbs towards 0 dB, because a lossless filter reflects what it cannot pass.
For practitioners
A low-pass filter passes low frequencies and blocks high ones. Its S-parameters show this directly. S21 tells you how much signal gets through. S11 tells you how much comes back to the source. Once you can read these two curves together, you can read most filter plots.
The two curves on one plot
Plot S21 and S11 in dB against frequency. The result has a recognizable shape.
The figure shows S21 and S11 of a low-pass sample filter, computed from the circuit model. It is not measured data. Three regions stand out:
- Pass band. S21 is flat and close to 0 dB. S11 is low, which means a good match.
- Cutoff. S21 has dropped by 3 dB. S11 has risen to meet it.
- Stop band. S21 keeps falling. S11 sits close to 0 dB.
The two curves move in opposite directions. When one goes down, the other goes up. The rest of this article explains why.
Pass band, cutoff and roll-off
In the pass band, an ideal lossless filter has S21 near 0 dB and a very small S11. Nearly all the power reaches the load.
The cutoff frequency is usually defined where S21 has fallen by 3 dB. That is half of the power, or 0.707 of the voltage:
The two forms agree because power goes as voltage squared.
Above the cutoff, S21 falls at a rate set by the filter order. For a Butterworth response of order n, the transmitted power is:
Far above fc, the 1 in the denominator becomes small compared to the second term. Then S21 in dB drops by about 20·n dB per decade. A 3rd-order filter falls by about 60 dB per decade, or about 18 dB per octave. Each extra order steepens the roll-off.
Why S11 rises when S21 falls
For a lossless two-port, no power can disappear inside the filter. Every watt that goes in either leaves through port 2 or comes back out of port 1:
So when S21 falls, S11 has to rise. The filter does not turn the unwanted signal into heat. It reflects it. In the stop band, the filter presents a mismatch to the source, and that mismatch sends the power back.
This also explains why S11 is not a flaw above the cutoff. A low S11 there would mean the filter was absorbing the signal. A lossless filter that blocks a signal does so by reflecting it. That is also why a filter placed after a power amplifier needs care: the reflected out-of-band energy goes back towards the amplifier.
S11 near 0 dB in the stop band is normal for a lossless filter. Do not read it as a bad match in the sense of a design error. It is the mechanism of the filtering.
Worked example: 3rd-order Butterworth
Take a lossless 3rd-order Butterworth low-pass filter with n = 3. Use the formula above.
At the cutoff, f = fc. |S21|² = 1 / (1 + 1) = 0.5, so S21 = -3.01 dB. Then |S11|² = 1 - 0.5 = 0.5, so S11 = -3.01 dB as well. The power is split equally: half is transmitted and half is reflected.
At twice the cutoff, f = 2·fc. |S21|² = 1 / (1 + 2⁶) = 1/65, so S21 = -18.1 dB. |S11|² = 64/65, so S11 = 10·log10(64/65), which is about -0.07 dB. Almost everything is reflected.
At half the cutoff, f = 0.5·fc. |S21|² = 1 / (1 + (0.5)⁶) = 64/65, so S21 is about -0.07 dB. |S11|² = 1/65, so S11 is about -18.1 dB.
The numbers at 0.5·fc and 2·fc mirror each other. S21 at one is S11 at the other. This follows from the lossless relation, and it makes a handy check when you look at a plot.
What a real filter changes
A real filter also has dissipation loss. Inductors and capacitors are not perfect, and the traces and the board add loss too. Two effects follow:
- The pass-band S21 sits a little below 0 dB, not on it.
- The sum |S11|² + |S21|² is now less than 1. The missing share of the power is turned into heat inside the filter.
The picture stays the same. S21 still has a flat region, a cutoff and a roll-off, and S11 still rises in the stop band. But the two curves no longer add up exactly to 1, and the cutoff is easier to read if you measure the 3 dB drop from the actual pass-band level rather than from 0 dB.
You can load an S2P file of a filter and look at both curves yourself. See How to Open an S2P (Touchstone) File for ways to do it.
In plain words · for students and beginners
Low-pass filter curves in plain words
An everyday analogy
Think of a sieve that lets small grains through and stops big stones. Fine sand is the low frequency. Stones are the high frequency. Sand that falls through is S21. Stones that bounce off the sieve and fly back at you are S11.
The analogy breaks down in one place. A sieve changes smoothly from "lets everything through" to "blocks everything" only if you choose a sieve with a gradual mesh. A filter has no hard edge either. Its S21 slides downwards over a range of frequencies, and the cutoff is just the point where we agree to say "it has dropped enough".
The same example in simple steps
Our filter is a 3rd-order Butterworth low-pass filter. It has no loss.
- At the cutoff, the filter lets half of the power through and reflects the other half. Half of the power is -3.01 dB. So S21 and S11 are both -3.01 dB.
- At twice the cutoff, only 1/65 of the power gets through. That is -18.1 dB for S21. The other 64/65 comes back, which is about -0.07 dB for S11.
- At half the cutoff, the numbers swap. S21 is about -0.07 dB and S11 is about -18.1 dB.
The power that goes through plus the power that comes back always makes one whole. Nothing gets lost.
The drawing shows a two-port device and its four S-parameters. S21 is the signal from port 1 to port 2. S11 is the signal that comes back out of port 1.
Mistakes beginners make
- Thinking S11 high means the filter is broken. Above the cutoff a high S11 is the filter doing its job. It pushes the signal back.
- Mixing up dB and plain numbers. -3 dB does not mean "3 percent lost". It means about half of the power is left.
- Expecting exactly 0 dB in the pass band. Real parts have a little loss, so S21 sits slightly below 0 dB.
- Forgetting that the cutoff is only a convention. The -3 dB point is a label we agreed on, not a wall.
Remember
- S21 is what gets through. S11 is what comes back.
- At the -3 dB cutoff of a Butterworth filter, half the power goes through and half is reflected.
- For a lossless filter, power in S11 plus power in S21 always equals 1, so when one falls the other must rise.
Related pages on RF View
Frequently asked questions
Why does S11 go up when S21 goes down in a low-pass filter?
A lossless filter cannot absorb power. Power that is not transmitted (S21) must be reflected (S11), because the power in S11 plus the power in S21 is 1 for a lossless two-port.
How is the cutoff frequency of a low-pass filter defined?
It is usually the frequency where S21 has fallen by 3 dB from its pass-band level. That is half of the power, or 0.707 of the voltage.
What are S21 and S11 of a 3rd-order Butterworth filter at the cutoff?
Both are -3.01 dB. At the cutoff the power is split equally: half is transmitted and half is reflected.
Why is the pass-band S21 of a real filter below 0 dB?
A real filter also dissipates some power as heat in its inductors, capacitors and lines. That dissipation loss puts the pass-band S21 a little below 0 dB.