L-Network Matching: Low-Pass or High-Pass Version?
Matching 50 ohm to 200 ohm needs Q = sqrt(3) = 1.732. The low-pass version uses 13.78 nH and 1.378 pF, the high-pass version 1.838 pF and 18.38 nH, and the choice depends on the DC path and the filtering you want.
For practitioners
An L-network matches two real impedances with one series element and one shunt element. The shunt element always goes across the higher resistance. Two arrangements exist for the same pair of resistances: a low-pass version and a high-pass version. Both match perfectly at the design frequency. They behave differently everywhere else.
This page works through 50 ohm to 200 ohm at 1 GHz and shows how to choose between them.
How Q follows from the resistance ratio
An L-network has only two elements, so it has no free parameter. The ratio of the two resistances sets the loaded Q:
For 50 ohm and 200 ohm the ratio is 4, so Q = sqrt(3) = 1.732.
The series reactance and the shunt reactance follow from Q:
A larger ratio gives a higher Q. A higher Q gives a narrower bandwidth and larger reactances, so the network becomes more sensitive to component tolerance. A 50 ohm to 200 ohm match is a modest ratio, so the Q is low.
The worked example: 50 ohm to 200 ohm at 1 GHz
With Q = 1.732:
- Series reactance: Xs = 1.732 × 50 = 86.6 ohm.
- Shunt reactance: Xp = 200 / 1.732 = 115.5 ohm.
At 1 GHz, ω = 2π × 10⁹ rad/s. Convert the reactances to element values.
Low-pass version. The series element is an inductor and the shunt element is a capacitor.
- L = Xs / ω = 86.6 / (2π × 10⁹) ≈ 13.78 nH
- C = 1 / (ω Xp) = 1 / (2π × 10⁹ × 115.5) ≈ 1.378 pF
High-pass version. The series element is a capacitor and the shunt element is an inductor.
- C = 1 / (ω Xs) = 1 / (2π × 10⁹ × 86.6) ≈ 1.838 pF
- L = Xp / ω = 115.5 / (2π × 10⁹) ≈ 18.38 nH
You can check either result by hand. In the low-pass version, the 200 ohm load in parallel with the shunt capacitor gives 50 − j86.6 ohm. The series inductor adds +j86.6 ohm, which leaves 50 ohm. The high-pass version works the same way with the signs swapped: the shunt inductor makes the parallel combination 50 + j86.6 ohm, and the series capacitor cancels the reactance.
Where each element sits on the Smith chart
On the Smith chart, each element moves the impedance point along a fixed circle:
- The shunt element moves the point along a constant-conductance circle.
- The series element moves the point along a constant-resistance circle.
The path ends at the centre, which is 50 ohm. The shunt element is the first move from the 200 ohm side. It lands on the circle that passes through the centre. The series element then travels along that circle to the centre.
The low-pass and high-pass versions take the same two-step route. They use opposite sides of the chart. The low-pass version passes through the capacitive half, and the high-pass version passes through the inductive half. The figure in the plain-words section below shows this path for the 50 ohm to 200 ohm case.
How to choose between the two versions
The Q and the match at 1 GHz are identical. Choose by what else the network has to do.
Low-pass version
- It passes DC, so a bias or control voltage can pass through the network.
- It attenuates harmonics, which helps at an amplifier output.
- It adds series inductance, so a real inductor's self-resonance and loss matter at the top of the band.
High-pass version
- It blocks DC, so it can double as a DC block between stages.
- It attenuates low frequencies, which helps against low-frequency interference or noise.
- It does not reduce harmonics. They pass more easily than in the low-pass version.
Two more checks help before you decide:
- Is there a DC path requirement? If a stage needs DC through the network, the series capacitor of the high-pass version is a problem. If a stage must be DC-isolated, the low-pass version needs an extra blocking capacitor.
- What unwanted signals are nearby? Harmonics above the band point to the low-pass version. Interference below the band points to the high-pass version.
Element values like 1.378 pF and 13.78 nH are ideal. Real parts have tolerance, loss and parasitics, so check the final circuit with a simulation or a measurement before you trust it.
To try other resistance pairs, use the L-Network Impedance Matching Calculator.
In plain words · for students and beginners
L-network matching in plain words
An everyday analogy
Think of a gearbox between an engine and a wheel. The engine wants to turn fast with little force. The wheel wants to turn slowly with a lot of force. The gearbox changes one into the other so that no power is wasted.
An L-network does this for electrical signals. One side has a low resistance (50 ohm) and the other a high resistance (200 ohm). The network sits between them so the signal passes across without bouncing back.
The analogy breaks down in one place. A gearbox has many ratios to pick from. An L-network has only one Q for a given pair of resistances, and the pair decides it.
The same example in simple steps
- Divide the larger resistance by the smaller one: 200 / 50 = 4.
- Subtract 1 and take the square root: sqrt(3) = 1.732. This number is called Q. It tells you how "sharp" the network is.
- Multiply Q by 50 to get 86.6 ohm for the series part. Divide 200 by Q to get 115.5 ohm for the shunt part. A shunt part connects from the signal line to ground.
- Turn those ohm values into parts at 1 GHz. This gives either 13.78 nH in series with 1.378 pF in shunt, or 1.838 pF in series with 18.38 nH in shunt.
The figure below shows the same match as a path on a Smith chart. The Smith chart is a circular map of impedances. The match ends when the path reaches the centre.
The first version lets DC through and cuts high-frequency extras (harmonics). The second blocks DC and cuts low frequencies.
Mistakes beginners make
- Putting the shunt part on the wrong side. It goes across the higher resistance, the 200 ohm side.
- Mixing up the two versions. Check which part is in series and which is in shunt before you build.
- Forgetting DC. A series capacitor stops DC, so a stage that needs a bias voltage through that path will not work.
- Trusting the ideal values. Real parts differ from their labels, so the match will not be perfect without checking.
Remember
- Q = sqrt(Rhigh/Rlow − 1), so 50 ohm to 200 ohm gives Q = 1.732.
- Low-pass: 13.78 nH in series and 1.378 pF in shunt. High-pass: 1.838 pF in series and 18.38 nH in shunt.
- Choose low-pass to pass DC and reduce harmonics, and high-pass to block DC and reduce low frequencies.
Related pages on RF View
Frequently asked questions
How do I calculate the Q of an L-network?
Use Q = sqrt(Rhigh/Rlow - 1). For 50 ohm to 200 ohm the ratio is 4, so Q = sqrt(3) = 1.732. The Q is fixed by the two resistances, so you cannot choose it.
Should I use the low-pass or the high-pass L-network?
Use the low-pass version when you need a DC path or want harmonics reduced. Use the high-pass version when you need to block DC or attenuate low frequencies.
Which side does the shunt element go on in an L-network?
The shunt element goes across the higher resistance. The series element sits on the lower-resistance side.
Do the low-pass and high-pass L-networks match equally well at 1 GHz?
Both give an ideal match at the design frequency with the same Q. They differ in what they do away from 1 GHz and in whether they pass DC.