Why insertion loss; double-section shunt-arm LPF

Why the insertion loss technique is used

The insertion loss method designs a filter from a chosen power loss ratio:

PLR = 1 / (1 − |Γ(ω)|²) = 1 + M(ω²)/N(ω²)
Butterworth: PLR = 1 + k²(ω/ωc)^(2N)
Chebyshev  : PLR = 1 + k² T_N²(ω/ωc)

It is preferred over the older image-parameter method because:

• the complete response (passband ripple, cut-off sharpness, phase) is specified and achieved exactly,
• the design is systematic: g-value tables, then scaling, then transformation, with no trial and error,
• the order N gives a clear trade-off between performance and size or loss,
• one LP prototype can be turned into HP, BP or BS filters and then into microstrip (Richards, Kuroda, stepped-impedance).

Double-section shunt-arm microwave LPF in microstrip

In a shunt-arm LPF, the shunt capacitors are made as open-circuited stubs (or wide pads) connected across the line, and the series inductors are short high-impedance line sections. A double-section filter has two shunt arms.

 Lumped:  o--UUU--+--UUU--+--UUU--o
            L1    |   L2  |   L3
                  C1      C2
                  |       |
          o-------+-------+-------o

 Microstrip (top view):
              stub C1      stub C2
               | |          | |    open end
               | |  l_s     | |    (λ/8 at fc)
 50 Ω ====+----+-+----+-----+-+----+==== 50 Ω
     feed  L1 (hi-Z)  L2 (hi-Z)  L3   feed
           -------- ground plane below ------

Labels

• 50 Ω feed lines at input and output.
• L₁, L₂, L₃: narrow high-impedance (about 100–120 Ω) line sections, βl = L·R₀/Z_h (normalised L), acting as series inductors. With Richards and Kuroda design, these are λ/8 unit elements.
• C₁, C₂: open-circuited shunt stubs. From Richards' transformation each is λ/8 long at fc with Z₀ = R₀/g_k. (Wide low-impedance pads can be used instead.)
• Substrate with εr and height h, and a full ground plane underneath.