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Here's a look at the theory of complex conjugate impedance matching when it comes to NFC.

To enable a device to communicate via near-field communication, you need a microcontroller, an NFC interface, and an antenna. The challenges are the filter and matching network to which the NFC antenna is connected.

Near-field communication (NFC) is a short-range, high-frequency, low-bandwidth wireless communication technology that enables standardized communication between two mobile devices such as smartphones, smartcards, stickers, or tags. NFC enjoys great popularity because it can be used with smartphones. In contrast to the comparable RFID standards on the same frequency of 13.56 MHz, NFC is better suited for sensitive applications such as data communication and secure financial transactions thanks to its short range of less than 10 cm.

It’s complex to calculate the design of the filtering and matching circuit for an NFC application, so several iterations may be necessary due to material tolerances of the components. The design of this circuit is the topic below.

**Theory of complex conjugate impedance matching**

Complex conjugate impedance matching is a very important technique in RF circuit design. It’s designed to ensure the maximum possible power transfer between a source and its load and to minimize signal reflections from the load. An example of the need for power transfer can be found in the front end of any sensitive receiver.

It’s clear that unnecessary losses cannot be tolerated in a circuit that already carries extremely small signal levels. Therefore, extreme care is taken in the initial design of such a front-end to ensure that each device in the chain is matched to its load. In high-frequency engineering, loads are often complex, i.e., they have an inductive or capacitive component in addition to their resistive component. For matching, the inductive or capacitive component must be compensated with its counterpart – the so-called complex conjugate component. For example, this means that an inductive component must be capacitively compensated.

Impedance matching is based on the maximum power transfer theorem. It states that to obtain the maximum external power from a source with a finite internal resistance, the resistance of the load must be equal to the resistance of the source as seen from its output terminals. In addition, it states that each reactive component of the source and the load should be equal in magnitude but opposite in sign. This means that the impedances of the load and source must be complex conjugate to each other.^{2}

In general, the complex conjugate of the impedance Z = R + jX is Z* = R – jX, where R is the real part and X is the imaginary part of the complex impedance Z. **Figure 1** shows the complex source impedance Z_{S} and the complex load impedance Z_{L}.

These impedances must meet the following condition to lead to an optimal match:

There are many possible circuit topologies that can be used to perform impedance matching. The simplest is the L topology, which consists of two reactances. This topology gets its name because of the component orientation, which resembles the shape of an L.^{1} **Figure 2** shows the two possible L topologies for impedance matching.

In **Figure 2**, X_{A} is the ideal reactance of the series branch and X_{B} is the ideal reactance of the parallel branch. Z_{S} and Z_{L} are the source and load impedances. Z_{IN} denotes the input impedance and includes the load and matching impedance, which must be the complex conjugate of Z_{S}. Before the matching components can be determined, the load and source impedances must be known. The source impedance is 50 Ω in most cases. In general, the source impedance can also be complex.

**Determining the complex load impedance**

The complex load impedance can be determined by measurement and calculation. The complex impedance measurement for the 13.56 MHz range can be performed with a vector network analyzer that measures the S-parameters of the DUT.

S-parameters describe the electrical behavior of linear electrical networks when exposed to different stationary electrical signals. For impedance matching, the parameter S11 is used, which is called the input reflection factor. The input reflection factor is a complex quantity whose absolute value is an indicator for the reflection. |S_{11}| = 0 means that the circuit is perfectly matched and that none of the incident “power waves” are reflected. |S_{11}| = 1 means that 100 % of the incident “power wave” is reflected back to the input. In NFC applications, the load is an antenna. For practical calculations and simulations, the electrical properties of the antenna are represented in an equivalent circuit diagram. The simplified series equivalent circuit of an NFC antenna is shown in **Figure 3**.

L_{a} is the inductance and R_{a} is the equivalent series resistance representing all ohmic losses of the antenna. C_{a} is the parallel equivalent capacitance of the antenna. The values L_{a} and R_{a} can be measured directly with a network analyzer or LCR meter.

The value C_{a} is a parasitic value and must be determined by measurement and calculation. The frequency dependencies of L_{a}, C_{a} and R_{a} are not considered in the calculations and simulations. If the inductance L_{a} is known, the parallel equivalent capacitance C_{a} at the self-resonance frequency f_{res} can be calculated with **formula (2)**.^{8}

The self-resonance frequency f_{res} is the first measuring point where the complex load impedance becomes real.

The impedance of the antenna Z_{L}, which is necessary for the impedance matching, can be calculated as follows:

The quality factor Q_{L} of the antenna is defined by the ratio of the imaginary part X_{L} and the real part R_{L} of the antenna impedance and can therefore be calculated with **equation (4)**.

**Determining the components of the matching circuit**

The matching procedure can be done by calculation and simulation. In general, both impedance networks shown in **Figure 2** can be used, but the network shown in **Figure 2a** is easier to calculate and was chosen to demonstrate the matching procedure. Since the load has an inductive behavior, the reactances X_{A} and X_{B} are capacitive. In general, X_{A} and X_{B} can also be inductive if the load has a capacitive behavior. X_{A }and X_{B} are considered ideal, i.e., they are assumed to have no resistive or parasitic component.

**Calculation of the components of the matching circuit**

For the network shown in **Figure 2a**, perfect matching is achieved for Z_{in} = Z_{S}*. Z_{in} can be calculated as follows^{5}:

Taking **equation (5)** into account, one obtains Z_{in} for the real and imaginary parts of the input impedance:

If **equation (6)** is solved for the reactance X_{B}, two different values X_{B1} and X_{B2} are obtained because **(6)** is a quadratic equation:

The reactance X_{A} also has two values, X_{A1} and X_{A2}, and based on **equation (7)**, taking **equation (8)** into account, gives the following expression:

The second part of this article will look at the application of impedance matching to an NFC output circuit.

**References:**

1; C. Bowick: RF Circuit Design, 2^{nd} Revised edition, Newnes, 2007.

2. K. Cartwright, Non-Calculus Derivation of the Maximum Power Transfer Theorem, Technology Interface, 8 (2), 2008.

3. M. Roland: Automatic Impedance Matching for 13.56MHz NFC Antennas, 2008.

4. Würth Elektronik eiSos: ANP057a, WE-MCA Multilayer Chip Antenna Placement & Matching**, **Application Note, 2018.

5.** **T. Baier: Automated Impedance Adjustment of 13.56MHz** **NFC Reader Antennas,** **Master Thesis, 2014.

6.** **A. Schober, M. Ciacci, and M. Gebhart: An NFC Air Interface coupling model for Contactless System Performance estimation, International Conference on Telecommunications (ConTEL), June 2013.

7. Rohde und Schwarz: Near Field Communication (NFC) Technology and Measurements, White Paper, 2011.

8.** **NXP Corporation: AN11564, Antenna Design and Matching Guide, Application Note, 2016.

*This article was originally published on **EE Times Europe**.*

**Christian Merz** is Product Manager for Wireless Power Transfer at Würth Elektronik eiSos GmbH & Co. KG.

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