# Wilkinson power divider

Power divider in microstrip technology

In the field of microwave engineering and circuit design, the Wilkinson Power Divider is a specific class of power divider circuit that can achieve isolation between the output ports while maintaining a matched condition on all ports. The Wilkinson design can also be used as a power combiner because it is made up of passive components and hence reciprocal. First published by Ernest J. Wilkinson in 1960,[1] this circuit finds wide use in radio frequency communication systems utilizing multiple channels since the high degree of isolation between the output ports prevents crosstalk between the individual channels.

It uses quarter wave transformers, which can be easily fabricated as quarter wave lines on printed circuit boards. It is also possible to use other forms of transmission line (e.g. coaxial cable) or lumped circuit elements (inductors and capacitors).[2]

## Theory

Picture shows a typical output expected from a Wilkinson power divider. The ${\displaystyle S_{21},S_{31}}$ are almost -3 dB, and the ${\displaystyle S_{11}}$ is low near the design frequency.
Picture demonstrates a very high isolation between output ports (port 2 & 3) of a Wilkinson power divider

The scattering parameters for the common case of a 2-way equal-split Wilkinson power divider at the design frequency is given by[3]

${\displaystyle [S]={\frac {-j}{\sqrt {2}}}{\begin{bmatrix}0&1&1\\1&0&0\\1&0&0\\\end{bmatrix}}}$

Inspection of the S matrix reveals that the network is reciprocal (${\displaystyle S_{ij}=S_{ji}}$), that the terminals are matched (${\displaystyle S_{11},S_{22},S_{33}=0}$), that the output terminals are isolated (${\displaystyle S_{23},S_{32}}$=0), and that equal power division is achieved (${\displaystyle S_{21}=S_{31}}$). The non-unitary matrix results from the fact that the network is lossy. An ideal Wilkinson divider would yield ${\displaystyle S_{21}=S_{31}=-3\,{\text{dB}}=10\log _{10}({\frac {1}{2}})}$.

Network theorem governs that a divider cannot satisfy all three conditions (being matched, reciprocal and loss-less) at the same time. Wilkinson divider satisfies the first two (matched and reciprocal), and cannot satisfy the last one (being loss-less). Hence, there is some loss occurring in the network.

No loss occurs when the signals at ports 2 and 3 are in phase and have equal magnitude.

However, some modification can be done to achieve unequal power division at the output ports. By cascading, the input power might be divided to any ${\displaystyle n}$-number of outputs.