Solved Numerical 4

$ S_{12}=0 \quad\text{and}\quad S_{21}\neq0 $

Part (a): Proof for a Reciprocal Lossless Two-Port Network

Step 1: Write the Scattering Matrix

For a reciprocal two-port network:

$ [S] = \begin{bmatrix} S_{11} & S_{12} \\ S_{21} & S_{22} \end{bmatrix} $

Reciprocity requires:

$ S_{12}=S_{21} $

Therefore:

$ [S] = \begin{bmatrix} S_{11} & S_{12} \\ S_{12} & S_{22} \end{bmatrix} $

Step 2: Apply the Lossless Condition

For a lossless network, the scattering matrix must be unitary:

$ [S]^\dagger[S]=[I] $

From the first row (or first column) of the unitary matrix condition:

$ |S_{11}|^2+|S_{12}|^2=1 $

Step 3: Rearrange the Equation

$ |S_{12}|^2 = 1-|S_{11}|^2 $

Since the network is reciprocal:

$ S_{21}=S_{12} $

Hence:

$ |S_{21}|^2 = 1-|S_{11}|^2 $

Result

$ \boxed{ |S_{21}|^2 = 1-|S_{11}|^2 } $

This proves that in a reciprocal lossless network, any reduction in reflected power directly increases transmitted power. The sum of reflected and transmitted power must always equal the incident power.

Part (b): Proof that Unidirectional Transmission is Impossible

Step 1: Assume a Lossless Non-Reciprocal Network

Consider a lossless network where:

$ S_{12}=0 $

but:

$ S_{21}\neq0 $

The scattering matrix becomes:

$ [S] = \begin{bmatrix} S_{11} & 0 \\ S_{21} & S_{22} \end{bmatrix} $

Step 2: Apply the Lossless Condition to the First Row

Since the network is lossless, the scattering matrix must satisfy the unitary condition:

$ [S]^\dagger[S]=[I] $

One of the resulting equations is obtained from the first row:

$ |S_{11}|^2+|S_{12}|^2=1 $

Given that:

$ S_{12}=0 $

Substituting into the equation:

$ |S_{11}|^2+0=1 $ $ |S_{11}|^2=1 $

Therefore:

$ |S_{11}|=1 $

This means all incident power at Port 1 is reflected back, with no power absorbed by the network.

Step 3: Apply the Lossless Condition to the First Column

Another requirement for a lossless network is:

$ |S_{11}|^2+|S_{21}|^2=1 $

Substituting:

$ |S_{11}|^2=1 $

gives:

$ 1+|S_{21}|^2=1 $

Subtracting 1 from both sides:

$ |S_{21}|^2=0 $

Therefore:

$ |S_{21}|=0 $

which implies:

$ S_{21}=0 $

Step 4: Identify the Contradiction

The original assumption was:

$ S_{12}=0 \quad \text{and} \quad S_{21}\neq0 $

However, the lossless condition forces:

$ S_{21}=0 $

This directly contradicts the assumption that forward transmission exists while reverse transmission is zero.

Final Result

For a lossless non-reciprocal two-port network:

$ S_{12}=0 \quad \Longrightarrow \quad S_{21}=0 $

Therefore, a lossless network cannot support true one-way transmission where signal travels in one direction but not the other.

$ \boxed{ S_{12}=0 \Rightarrow S_{21}=0 } $

Hence, unidirectional transmission is impossible in a lossless two-port network.

Many students assume that a non-reciprocal device automatically allows perfect one-way signal flow. The mathematics shows otherwise. A lossless two-port network must obey power conservation. If one transmission coefficient is forced to zero, the unitary condition forces the other transmission coefficient to become zero as well.

This is why practical one-way RF components such as isolators and circulators rely on special non-reciprocal structures and termination mechanisms rather than ideal lossless unidirectional transmission.

Key Exam Takeaway

This result is frequently asked in Microwave Engineering, RF Engineering, Communication Systems, and Network Theory examinations. Many proof-based questions can be solved directly using the lossless network conditions without performing lengthy matrix calculations.

• For a reciprocal and lossless two-port network: $ |S_{21}|^2 = 1 - |S_{11}|^2 $
• For every lossless network, the scattering matrix must satisfy the unitary condition: $ [S]^\dagger [S] = [I] $
• If a lossless network has: $ S_{12}=0 $ then the lossless condition forces: $ S_{21}=0 $
• Therefore, a lossless two-port network cannot provide perfect one-way transmission. If signal transmission is completely blocked in one direction, it must also be blocked in the opposite direction.
• To achieve true unidirectional transmission, the network must introduce loss or use non-reciprocal devices such as ferrite-based isolators and circulators.

Exam Shortcut

For IOE, IIT, B.Tech Electronics, BE Electronics & Communication, CSIT, BCA, and Microwave Engineering examinations, memorize the following two equations. They appear repeatedly in derivations, numerical problems, and theory questions involving lossless two-port networks.

For a lossless two-port network:

$ |S_{11}|^2 + |S_{21}|^2 = 1 $

$ |S_{22}|^2 + |S_{12}|^2 = 1 $

These equations represent power conservation. The total outgoing power must equal the incoming power because a lossless network cannot dissipate energy internally.

A quick exam trick is to start from these identities whenever you're asked to:

• Prove a network is not lossless.
• Verify whether an S-matrix is physically realizable.
• Show reciprocity or non-reciprocity conditions.
• Find an unknown S-parameter magnitude.
• Prove that unidirectional transmission is impossible in a lossless two-port network.

In many university examinations, these two equations alone are sufficient to derive the final answer within a few steps.

Memory Tip for Exams

Think of a lossless two-port network as a perfectly sealed pipe. If some power is reflected at a port, the remaining power must be transmitted somewhere else. Nothing can disappear inside the network.

That simple idea leads directly to:

$ |S_{11}|^2 + |S_{21}|^2 = 1 $

and

$ |S_{22}|^2 + |S_{12}|^2 = 1 $

Remember these identities, and many S-parameter proofs become straightforward.