Two-Port Networks and Network Graphs
Z, Y, h and ABCD parameters, conditions for reciprocity and symmetry, interconnection of two-ports, network graph theory, trees, cut-sets and incidence matrices.
A two-port network is a box with an input pair and an output pair of terminals. Four parameters describe it completely, whatever is inside.
Parameter sets
| Parameters | Equations | Found by |
|---|---|---|
| Z (impedance) | V₁ = z₁₁I₁ + z₁₂I₂; V₂ = z₂₁I₁ + z₂₂I₂ | Opening a port (I = 0) |
| Y (admittance) | I₁ = y₁₁V₁ + y₁₂V₂; I₂ = y₂₁V₁ + y₂₂V₂ | Shorting a port (V = 0) |
| h (hybrid) | V₁ = h₁₁I₁ + h₁₂V₂; I₂ = h₂₁I₁ + h₂₂V₂ | Transistor models |
| ABCD (transmission) | V₁ = AV₂ − BI₂; I₁ = CV₂ − DI₂ | Cascaded networks, power lines |
[Y] = [Z]⁻¹.
Reciprocity and symmetry
| Parameters | Reciprocal | Symmetric |
|---|---|---|
| Z | z₁₂ = z₂₁ | z₁₁ = z₂₂ |
| Y | y₁₂ = y₂₁ | y₁₁ = y₂₂ |
| h | h₁₂ = −h₂₁ | Δh = 1 |
| ABCD | AD − BC = 1 | A = D |
Networks of R, L, C (and transformers) are reciprocal.
Interconnections
- Series: Z matrices add.
- Parallel: Y matrices add.
- Cascade: ABCD matrices multiply (in order).
Network graph theory
- A graph has nodes (n) and branches (b). A tree connects all nodes without loops: it has n − 1 branches (twigs); the rest, b − n + 1, are links.
- Fundamental loops (tie-sets): one per link → b − n + 1 loop equations. Fundamental cut-sets: one per twig → n − 1 node equations.
- Incidence matrix A (reduced: (n − 1) × b): KCL is A I = 0.
- Number of trees of a complete graph with n nodes: nⁿ⁻² (Cayley).
Worked example
A T-network has series arms Z₁ = 2 Ω and Z₂ = 3 Ω and shunt arm Z₃ = 4 Ω.
- z₁₁ = Z₁ + Z₃ = 6 Ω; z₂₂ = Z₂ + Z₃ = 7 Ω; z₁₂ = z₂₁ = Z₃ = 4 Ω (reciprocal, not symmetric).
Practice
A two-port has A = 2, B = 3 Ω and D = 2. If it is reciprocal, what is C (in S)?
Show answer and explanation
Answer: 1AD − BC = 1 gives 4 − 3C = 1, so C = 1 S.
A connected graph has 6 nodes and 10 branches. How many links does any tree leave?
Show answer and explanation
Answer: 5Twigs = n − 1 = 5, links = b − n + 1 = 5.
Two two-port networks are connected in cascade. The overall parameters are obtained by:
Show answer and explanation
Answer: CIn a cascade, the output of one is the input of the next, so transmission matrices multiply.
How many spanning trees does a complete graph on 4 nodes have?
Show answer and explanation
Answer: 16Cayley's formula nⁿ⁻² = 4² = 16.