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The Impedance Transforming Property of the L-Match Network

Key Takeaways

  • Impedance transforming networks convert the load impedance into smaller values to make the circuit power-efficient.

  • There are various types of impedance transforming networks, including the L-match network.

  • The L-match circuit cancels the imaginary part of the source and load impedance and converts the real part to a lower value so that maximum power is transferred to the load. 

RF mixer

The characteristic impedance of most RF circuits is 50 Ω

Characteristic impedances in RF and microwave systems most often take the fixed value of 50 Ω. Power amplifiers and antennas in RF systems have both an input and output impedance equal to 50 Ω. To achieve higher output power in RF systems with various load impedances, the insertion of an extra network made of inductors and capacitors is required. The intention behind this insertion is for the extra network to possess impedance transforming properties and convert the 50Ω into a smaller value at specific frequencies. If the impedance transforming network is lossless, the power delivered to it reaches the load impedance directly, and higher power is achieved. 

Networks with impedance transforming properties often get confused with impedance matching networks. Impedance transforming networks convert the load impedance into smaller values to make the circuit power-efficient. There are various types of impedance transforming networks—in this article, we will focus on the L-match network, its impedance transforming properties, and its use in electronics circuitry. 

The L-Match Network

an image of the L-match network

The L-match network reduces load impedance

L-match circuits usually consist of an inductor and/or capacitor placed in the shape of the letter ‘L’. However, this placement can be changed according to the given input. There are several configurations of L-match circuits (shown above). The insertion of the L-match network between the source and load impedance converts the 50 Ω load resistance to a lower value. The L-match network then cancels the reactive part of the load impedance and reduces the value of the real part. This adjustment helps to deliver full power to the real part of the load impedance. 

In summary, the L-match circuit cancels the imaginary part of the source and load impedance and converts the real part to a lower value so that maximum power is transferred to the load. 

Equations Governing the L-Match Network

L-match network in a PA circuit

The L-match network in a power amplifier (PA) circuit

The L-match circuit inserted between the input and output in the figure above converts the load resistance RL of 50 Ω to a lower value, Rin

Let’s analyze how RL transforms into Rin with the L-match network. The equivalent circuit reduction principles are the basic equations governing the L-match circuit impedance transformation. 

The steps involved in transforming RL into Rin are:

  1. Convert Cm and RL into series equivalent CS and Rm, where QC is the quality factor of Cm and RL.

RmRLQC2(1)  CSCm(2)

  1. Let  the operating frequency of the power amplifier (PA) circuit. To make the circuit resistive (real part only), the inductor Lm is made to resonate with the capacitor Cs. The equation can be given by:

o=1LmCs(3)

  1. At resonance, the resistive input impedance Rin of the L-match network equals Rm.

  2. The impedance transformation ratio of the L-match circuit is given by: 

  1. The quality factor Qm of the transformation network is given by:

 Qm=r-1(5)

  1. The Lm and Cm values can be obtained from the quality factor Qm as follows:

Cm=QmRL(6)  Lm=QmRm(7) 

E=Output power with L-match networkOutput power without L-match network=RLRin=r(8)

  1. The output power is enhanced with the incorporation of the L-match network. The power enhancement ratio is as follows:

The steps above can be pictorially represented below:

L-match network impedance transformation

L-match network transformation

In power amplifiers, antennas, and antenna filters used in RF and microwave circuits, L-match circuits with impedance transforming properties can be inserted to ensure maximum power transfer to the load. The L-match network is one example of an impedance transforming network, however, there are a variety of such networks available to transform load impedance to smaller values. When working on RF and microwave systems, make sure that an impedance transformation network is incorporated to achieve power-efficient circuit operation. 

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