Unitary property of scattering matrix

[promach]

From s-parameter lossless network definition , The sum of the incident powers at all ports is equal to the sum of the reflected powers at all ports.



But how do we derive the above unitary property expression involving S22 and S21 ?

[in3otd]

That same Wikipedia pages says that a lossless network has an unitary S-parameters matrix, that is \$\mathbf{S}^{H}\mathbf{S}=\mathbf{I}\$. If you do the matrix multiplication, you'll obtain three independent equations, one of which it the one you posted.

[promach]

Why S^HS=I ?

[ejeffrey]

Its just from the conservation law.  Conservation of energy states: \$\vec{a}^\dagger\cdot \vec{a} = \vec{b}^\dagger\cdot\vec{b}\$ (power in = power out).  But \$\vec{b} = \mathbf{S} \vec{a}\$.  That leaves you with \$\vec{a}^\dagger \mathbf{S}^\dagger \mathbf{S} \vec{a} = \vec{a}^\dagger\vec{a}\$.  That implies \$\mathbf{S}^\dagger\mathbf{S} = \mathbf{I}\$ which is the definition of unitary.