8.1 Introduction to Network Interconnections

• Definition:
• Combining multiple two-port networks to form more complex systems while preserving their individual characteristics
• Key Applications:
• Cascaded amplifier stages
• Filter design (low-pass, high-pass, bandpass)
• Impedance matching networks

8.2 Fundamental Interconnection Methods

8.2.1 Series Connection (Z-Parameters Add)

Network A Network B
Z1 Z2
V1─┬─□□□──┬──□□□─┬─V2
│ │ │
I1 I1' I2

• Combined Z-Matrix:
  \[Z_{total} = Z_A + Z_B\]
• Conditions:
• Input currents must be equal (I₁ = I₁')
• Output currents must be equal (I₂ = I₂')

8.2.2 Parallel Connection (Y-Parameters Add)

+──Network A──+
│ │
V1─┤ ├─V2
│ │
+──Network B──+

• Combined Y-Matrix:
  \[Y_{total} = Y_A + Y_B\]
• Conditions:
• Input voltages must be identical
• Output voltages must be identical

8.2.3 Cascade Connection (ABCD Matrices Multiply)

V1─Network A─┬─Network B─V2
│
I1'=I2'

• Combined ABCD Matrix:
  \[ABCD_{total} = ABCD_A × ABCD_B\]
• Most common for amplifier stages and filter design

8.3 Advanced Interconnection Topologies

8.3.1 Series-Parallel (h-Parameters Add)

Network A
│
├─Network B
│

• Combined h-Matrix:
  \[h_{total} = h_A + h_B\]

8.3.2 Parallel-Series (g-Parameters Add)

Network A
│
┼─Network B
│

• Combined g-Matrix:
  \[g_{total} = g_A + g_B\]

8.4 Practical Considerations

8.4.1 Loading Effects

• Impedance Mismatch:
  \[
  ext{Actual gain} = rac{A_V}{1 + Z_{out}/Z_{in}}
  \]
• Solution: Use buffer stages (emitter/source followers)

8.4.2 Stability Analysis

• Rollett Stability Factor (K):
  \[
  K = rac{1 - |S_{11}|^2 - |S_{22}|^2 + | ext{Δ}|^2}{2|S_{12}S_{21}|}
  \]
  where \(Δ = S_{11}S_{22} - S_{12}S_{21}\)

8.5 Design Examples

8.5.1 Cascaded Amplifier Design

Stage 1 (CE) Stage 2 (CC)
Z1 Z2
V1─□□□─┬─────────□□□─V2
│ ▲
└─Coupling Cap

• Total Gain:
  \[A_{V(total)} = A_{V1} × A_{V2}\]

8.5.2 Ladder Filter Design

Series L Shunt C Series L
V1─□□□─┬───||───┬───□□□─V2
│ │
GND GND

• ABCD Matrix Multiplication:
  \[ABCD_{filter} = ABCD_L × ABCD_C × ABCD_L\]

8.6 Verification Methods

8.6.1 Experimental Verification

1. Impedance Measurements:
2. Network Analyzer for S-parameters
3. LCR meter for Z/Y parameters
4. Signal Tracing:
5. Input test signal, measure output at each stage

8.6.2 Simulation Techniques

* Cascaded Amplifier Example
X1 1 2 CE_Amplifier
X2 2 3 CC_Buffer
.subckt CE_Amplifier ...
.subckt CC_Buffer ...
.ac dec 10 1Hz 100MHz
.probe V(3)/V(1)
.end

8.7 Summary Table: Interconnection Methods

8.8 Key Equations

1. Cascaded Networks:
   \[
   \begin{bmatrix}
   V_1 \
   I_1
   \end{bmatrix}
   =
   \begin{bmatrix}
   A_1 & B_1 \
   C_1 & D_1
   \end{bmatrix}
   \begin{bmatrix}
   A_2 & B_2 \
   C_2 & D_2
   \end{bmatrix}
   \begin{bmatrix}
   V_2 \
   -I_2
   \end{bmatrix}
   \]
2. Stability Criterion:
   \[
   K > 1 ext{ and } | ext{Δ}| < 1
   \]

8.9 Laboratory Experiment

8.9.1 Cascaded RC Networks

1. Setup:
2. Stage 1: R=1kΩ, C=100nF (low-pass)
3. Stage 2: R=1kΩ, C=10nF (high-pass)
4. Measurements:
5. Individual and combined frequency response
6. Compare measured vs. calculated ABCD parameters
7. Expected Results:
8. Bandpass characteristic (f_L ≈ 160Hz, f_H ≈ 16kHz)
9. -6dB/octave roll-off on both sides

8.10 Summary

1. Interconnection method determines which parameter set to use
2. Cascade connections are most common in signal processing
3. Loading effects must be accounted for in practical designs
4. Stability is critical for active networks