L-Network Matching: Calculate Inductor and Capacitor Values

Open almost any antenna tuner and, somewhere inside, you will find a coil and a capacitor arranged in the shape of the letter L. That tiny circuit does a remarkable job: it takes an impedance that does not suit your transmitter and transforms it into one that does. This guide to L-network matching explains how the circuit works, how to choose between its two basic forms, how to calculate the inductor and the capacitor from three simple formulas, and what limits the result. We work through two examples with real numbers, and we look at where the circuit is used in a ham station, from a remote tuner at the base of a vertical to a feedpoint network for an end-fed wire. If you have ever wondered what your tuner is actually doing, this is the answer.

What an L-Network Is

An L-network is the simplest matching circuit that can transform one resistance into another over a narrow band. It has two parts, one in series with the signal path and one in parallel (shunt) across it. One part is an inductor, and the other is a capacitor. The impedance matching article describes it as the L-section: a network with a series and a shunt element in which one must be an inductor and the other a capacitor, able to match a source and a load of different resistances.

The name comes from the drawing. The series element is the vertical stroke of the letter and the shunt element is the horizontal one, though the diagram is often turned around. There are only two ways to connect the parts for a given pair of resistances, and the choice decides whether the network passes low or high frequencies, a point that we return to below.

Why match at all? A transmitter such as a modern transceiver wants to see about 50 ohms, and an antenna system that presents something else causes a high SWR, as explained in our article on what an SWR meter tells you and what it hides. A matching network does not change the SWR on the antenna side of the network, and it cannot fix a poor antenna, but it makes the radio side see the right load.

The Three Formulas

The L-network calculation is short enough to do with a pocket calculator. Let R_high be the larger of the two resistances and R_low the smaller one. The network is designed by choosing its Q, which is set by the ratio of the resistances.

1. Q = square root of (R_high divided by R_low, minus 1).
2. Series reactance X_s = Q times R_low. This is the element in series with the low-resistance side.
3. Shunt reactance X_p = R_high divided by Q. This is the element across the high-resistance side.

The same relation appears in the impedance matching article, which gives the Q of the section as the square root of (R1 minus R2) divided by R2, with R1 the larger resistance. Once you have the reactances, convert them to component values at your operating frequency f:

• Inductor: L = X divided by (2 times pi times f).
• Capacitor: C = 1 divided by (2 times pi times f times X).

One more rule decides where each part goes: the shunt element is placed at the high-resistance side of the network, and the series element is placed at the low-resistance side, so that the shunt part sits across the higher resistance and the series part sits in line with the lower one.

Low-Pass or High-Pass

You can use an inductor in series and a capacitor in shunt, which makes a low-pass network, or a capacitor in series and an inductor in shunt, which makes a high-pass network. Both give the same match at the design frequency. The difference is how they treat other frequencies. A low-pass network lets the wanted signal through and attenuates harmonics, and the impedance matching article points out that this gives the network an additional feature of harmonic suppression, which is why most amateur tuners use it. A high-pass network passes harmonics more easily and blocks low frequencies. For a transmitter, the low-pass form is usually the better choice, as our guide to the low-pass filter for harmonics explains.

Worked Example 1: 50 Ohms to 200 Ohms on 40 Metres

Suppose you want to match a 200 ohm load to a 50 ohm transmitter on 7.1 MHz. R_high is 200 and R_low is 50.

1. Q is the square root of (200 divided by 50, minus 1), which is the square root of 3, or about 1.73.
2. The series reactance is 1.73 times 50, which is about 86.6 ohms.
3. The shunt reactance is 200 divided by 1.73, which is about 115.5 ohms.

For the low-pass version, the series element is an inductor of 86.6 ohms. At 7.1 MHz, 2 pi f is about 44.6 million, so L is 86.6 divided by 44.6 million, which is about 1.94 microhenries. The shunt element is a capacitor of 115.5 ohms, so C is 1 divided by (44.6 million times 115.5), which is about 194 picofarads. The capacitor goes across the 200 ohm side and the inductor sits in line with the 50 ohm side.

Worked Example 2: 50 Ohms to 450 Ohms on 80 Metres

The next case has a higher ratio. Match 450 ohms to 50 ohms on 3.6 MHz.

1. Q is the square root of (450 divided by 50, minus 1), the square root of 8, or about 2.83.
2. The series reactance is 2.83 times 50, about 141.4 ohms.
3. The shunt reactance is 450 divided by 2.83, about 159.1 ohms.

At 3.6 MHz, 2 pi f is about 22.6 million. The inductor is 141.4 divided by 22.6 million, about 6.25 microhenries. The capacitor is 1 divided by (22.6 million times 159.1), about 278 picofarads. Compare the two examples: the higher the ratio, the larger the Q, and the bigger the reactances become.

Check the Voltage and the Current

Matching networks carry real power, and the components must survive it. With 100 W flowing into 450 ohms, the voltage is the square root of 100 times 450, which is about 212 volts RMS and about 300 volts at the peak. A capacitor and a coil rated for far more than that are needed, particularly on SSB peaks and in any case where the SWR is high. The ratings of variable capacitors, the wire gauge of the coil and the core material of toroids all matter.

Which Side Gets the Shunt Part

The rule that the shunt element goes on the high-resistance side means the network can face either way. If the load is higher than 50 ohms, the shunt capacitor sits at the load end and the series inductor sits at the transmitter end. If the load is lower than 50 ohms, the shunt capacitor goes at the transmitter end, across the 50 ohm side, and the series inductor goes toward the load. The table shows the arrangement.

Many antenna tuners contain an L-network with a bank of switched inductors and capacitors and a relay that swaps the shunt capacitor between the input and the output. This is how a single tuner covers loads both above and below 50 ohms. The antenna tuner article describes the common types, and our guide to choosing an external antenna tuner explains when you need one.

Limits of the L-Network

The L-network is simple, and simplicity has a cost. The first limit is that you cannot choose the Q. For a given pair of resistances there is exactly one L-network, and its Q is fixed by the ratio. A high ratio means a high Q, which means a narrower bandwidth and more circulating current in the components. The Q of the network also sets its losses. The impedance matching article notes that the magnitude of the ratio is limited by reactance losses such as the Q of the inductor, because real coils have resistance, and some of the power is lost as heat in them.

The second limit is bandwidth. The network is exact at one frequency and degrades as you move away from it. On a narrow amateur band the effect is small, and on 80 metres a high ratio can mean a re-tune within the band. If you need a bigger ratio or a wider bandwidth, you can cascade two sections, which the same article lists as a way to achieve a larger ratio, or you can use a transformer of the kind described in our guide to baluns and ununs.

The third limit is that the formulas assume purely resistive source and load. A real antenna usually has some reactance in addition to its resistance. The practical solution is to absorb the reactance into the network: if the load has a series inductive reactance, subtract it from the series element, and if it has a capacitive reactance, add it. The calculation is slightly longer, and a computer program or a vector network analyser makes it simpler. Our article on the NanoVNA shows how to measure the impedance you need to match before you pick the components.

Where You Will Use It

• Remote tuner at the antenna. A box at the base of a vertical or a wire matches the antenna impedance there, so that the coax carries a low SWR back to the shack.
• Feedpoint network for a voltage-fed wire. An end-fed antenna presents a high resistance at the end, and an L-network or a transformer brings it to 50 ohms.
• Short verticals and mobile antennas. A short whip has a low resistance and a large capacitive reactance, and an inductor at the base cancels the reactance while a matching network lifts the low resistance to 50 ohms.
• Amateur-built tuners. A coil with taps and a variable capacitor make a very capable tuner for a modest cost.
• Test and experiments. An L-network on the bench is an easy way to see matching at work with an analyser.

How the L-Network Differs From a Pi Network

A pi network has three elements and lets you choose the Q, which is why transmitter output stages use it. The extra element gives more harmonic suppression and flexibility, at the price of more parts. An L-network has fewer parts and lower loss for a given ratio, because it uses the lowest possible Q. If you only need to match two resistances and have no reason to pick a specific Q, the L-network is the best choice.

Building and Testing a Simple L-Network

1. Measure the impedance you need to match with an antenna analyser, and note the resistance and the reactance at your operating frequency.
2. Calculate the reactances from the formulas, and decide on the low-pass form unless you have a reason to use the high-pass one.
3. Choose the components. Use a coil wound on an air core or a suitable toroid with the right power rating, and a capacitor rated for the voltage with a safe margin.
4. Connect the network on the bench with the analyser, using a resistor of the load value, and check the match. Adjust the inductance by a turn or the capacitance by a small amount.
5. Install the network, check the SWR with a low power, and then increase the power while watching for heating in the parts.
6. Record the final values for the band, and keep a note of the date and conditions.

Test the network at low power first, and never run high power through a new matching circuit until you have verified it with a dummy load or a low-power measurement. A mistake in the ratings can cost you a capacitor or a coil, and it can damage the transmitter if the SWR becomes high.

The Bottom Line

L-network matching uses one inductor and one capacitor to transform one resistance into another at a single frequency. You calculate the Q from the ratio of the resistances, find the series and shunt reactances from the Q and the lower and higher resistance, and convert them to inductance and capacitance for your band. Use the low-pass form for transmitters to help with harmonics, place the shunt element on the high-resistance side and check the voltage and current ratings. Remember that the network has a fixed Q and a narrow bandwidth, and that real antennas have reactance that needs to be absorbed. With a calculator, an analyser and a few components, you can build the same circuit that sits inside your tuner and see exactly how your antenna is being matched.

Frequently Asked Questions

What is an L-network in ham radio?

It is a matching circuit with one series and one shunt element, one an inductor and the other a capacitor. It transforms one resistance into another at a single frequency and is the basis of many antenna tuners.

How do I calculate an L-network?

Find Q as the square root of (R_high divided by R_low, minus 1). The series reactance is Q times R_low, and the shunt reactance is R_high divided by Q. Convert to inductance and capacitance for your frequency.

Should I use a low-pass or a high-pass L-network?

Use low-pass for most transmitting work, because it attenuates harmonics. High-pass gives the same match at the design frequency, but it passes harmonics more easily and blocks low frequencies.

Which side does the shunt capacitor go on?

On the side with the higher resistance. For a load above 50 ohms it is at the load, and for a load below 50 ohms it is at the transmitter side.

Why does a high matching ratio cause losses?

A larger ratio needs a higher Q, which increases the circulating current and the loss in the coil resistance. The match gets narrower in bandwidth as well.

Can an L-network match any antenna?

It can match a wide range of impedances, but the reactance of the load must be absorbed into the calculation. Extreme impedances or very wide bandwidths may need two sections or a transformer, and the components must handle the voltage and current.