Impedance matching
Matching 5 Ω to 50 Ω with an L network forces a Q of 3, and the match lasts about 23 MHz around 100. The transformation ratio decides the bandwidth.
01Why match
An RF power transistor presents a few ohms at its output; an antenna, 50 Ω; a low-noise amplifier delivers its best noise figure with a particular source impedance. Connecting them directly wastes power and, on lines, produces the standing waves of transmission lines. A lossless matching network transforms one impedance into another at one frequency:
02The L network
The simplest network has two reactances: one in series on the side of the lower resistance and another in parallel on the side of the higher one. The ratio between the resistances sets its quality factor, and the reactances follow from it:
The Q is not free: it is set by the transformation ratio, and with it the bandwidth of the match, which is approximately inversely proportional to Q. A large transformation in a single step is narrowband.
A low-pass L network that matches a resistive load to a 50 Ω source at 100 MHz. The curve is the return loss seen from the source; the dotted line, 10 dB, the usual threshold of an acceptable match.
03π and T networks, multiple sections and transformers
With three elements, the π and T networks make it possible to choose the Q independently of the transformation ratio: the path goes through a virtual intermediate resistance. They serve to narrow the band, for example to filter harmonics at the output of a transmitter. To widen the band the opposite is done: several L networks in cascade with intermediate resistances in geometric progression, each with a lower Q. At higher frequencies, the networks are built with line sections and stubs, and designed on the Smith chart. Broadband transformers, with a ferrite core or a transmission-line type, give fixed ratios over several octaves.
04In the lab
Design an L network that matches a 12 Ω resistance to 50 Ω at 50 MHz, build it with SMD components and measure the return loss with a network analyzer.
Redesign the same match with two L networks and an intermediate resistance of √(12·50) ≈ 24.5 Ω, and compare the bandwidth.
Plot on the Smith chart the path of the impedance as the components are added.
05Common mistakes
- Forgetting the parasitics of the components. At hundreds of megahertz a capacitor has inductance and an inductor, capacitance.
- Expecting broadband from a large transformation in one step. The high Q prevents it; several sections are needed.
- Matching a power amplifier to the conjugate. It is matched to the load that gives the required power and efficiency.
- Ignoring inductor losses. With a high network Q, the circulating current dissipates in the winding resistance.
06Self-assessment
What Q does an L network that matches 10 Ω to 50 Ω have?
\( \sqrt{5 - 1} = 2 \).
What reactances does it need in series and in parallel?
20 Ω in series on the 10 Ω side and 25 Ω in parallel on the 50 Ω side.
What advantage does a low-pass L network have at the output of a transmitter?
Besides matching, it attenuates harmonics.
How is the bandwidth of a large-ratio match widened?
With several sections in cascade and intermediate resistances in geometric progression.
07Further reading
- Chris Bowick. RF Circuit Design. 2nd ed., Newnes, 2008. L, π and T networks with numerical design examples.
- David M. Pozar. Microwave Engineering. 4th ed., Wiley, 2012. Matching with lumped elements, stubs and quarter-wave transformers.
- Les Besser and Rowan Gilmore. Practical RF Circuit Design for Modern Wireless Systems. Artech House, 2003. Impedance matching in amplifiers and the use of the Smith chart.