Low-noise amplifiers (LNA) set the general noise figure and sensitivity of receivers and other high-frequency systems. Although a variety of excellent low-noise transistors is currently available based on numerous process technologies, achieving good amplifier low-noise performance depends strongly on the required LNA device input impedances and the choice of impedance-matching topologies. By analyzing these choices, it should be possible to find an optimum topology for a given device impedance.

For the example amplifier in Fig. 1, R₀ is the signal source impedance (typically 50 Ω), Z_in is the amplifier input impedance, and M is a matching network. For simplicity, assume that the matching network is designed for an ideal conjugate (maximum power) match; i.e., it transforms the impedance Z_in to the impedance R₀. In most cases, the input of an LNA will be matched for minimum noise figure, which may result in a mismatch in terms of power performance. In the case of a lossless matching network, the perfect conjugate match condition implies the following property: if a plane is drawn anywhere within the matching network or at its terminals connected to R₀ and Z_in, impedances looking in opposite sides of the plane must be conjugate to each other.

For example, impedance looking into the left-hand side of the plane drawn in Fig. 1 must be Z_in*. Even if a matching network has loss, it is possible to rearrange its boundaries such that it is purely reactive and all losses are added to either R₀ or Z_in. So, the property of the conjugate impedances can still be used.

The losses in a matching network are due to the finite quality factors (Qs) of the components used in the matching network. The Qs of surface-mount or even on-chip metal-insulator-metal (MIM) capacitors are usually very high and their contributions to the losses can be neglected. For example, the Q of a typical 2-pF 0402-sized capacitor is about 600 at 1 GHz while the Q of a 10-nH 0402-sized ceramic inductor is only 32 at 1 GHz. From these values, it is apparent that inductors are responsible for most of the losses in a matching circuit. The inductor is also more expensive than a discrete capacitor.

For this analysis of LNA matching networks, all possible LNA input impedance, Z_in, will be considered but the number of matching network topologies will be limited to those that:

1. contain at most two reactive components, and
2. contain at most one inductor.

The assumption that capacitors are lossless implies that a matching network without inductors would also be lossless and need not be analyzed. A matching network with two inductors is assumed to have more loss than a matching network with a single inductor or no inductors.

Figure 2a shows how the Smith Chart can be divided into six regions for the purpose of categorizing different matching networks and LNA input impedance, Z_in. The input impedance Z_in can be presented by a simple equivalent circuit containing a resistor and either a capacitor or inductor. These circuit elements can be connected in either a series or parallel configuration. The analysis is much simpler with the parallel connection when the first matching element on the amplifier side is a shunt element; conversely, the series connection works better when the first matching element on the amplifier side is a series element (both connections can be easily interchanged to represent the same Z_in using the Thevenin theorem). Then, all the matching circuits and the analysis cases can be categorized as shown in Fig. 2b. Matching circuits for region E were not analyzed because a series-C/shunt-C matching network should be the obvious choice for this region. For every other region, there are two possible matching networks containing only one inductor. An analysis should show which of the two has lower loss or, equivalently, yields lower noise figure for the LNA.

The main derivations for noise figure require knowledge of the matching inductor value as a function of the matched impedances, which must be derived first. This derivation is simple. First, the impedance looking into the left terminal (from the source side) of a matching network is derived and, then, its imaginary part is forced to be zero and the real part is forced to be equal to R₀. The resulting two equations can be solved for the matching capacitor and inductor values. The derivations are omitted here for simplicity but can readily be completed using the MathCAD mathematical analysis software. The results of the derivation for the matching inductor are shown in Eqs. 1 through 6 (indexes 1 through 8 denote the matching inductors from the corresponding matching networks in Fig. 2b):

where:

Loss in the inductor is due to a series resistance that is frequency independent unless we take into account the skin effect. Figure 3 shows a simplified model of a surface-mount inductor (Fig. 3a) and its parallel Thevenin equivalent circuit (Fig. 3b). Resistor R models the ohmic losses in the winding conductor of the coil. Parameter V_n,R represents the noise voltage generated by:

It is easy to show that:

where: