Two-Port, Ladder and Terminated Network Synthesis

Introduction

Network synthesis begins with a prescribed immittance or transfer characteristic and seeks a physically realizable circuit that produces it. Two-port synthesis extends this task from a single driving-point function to networks that transmit energy or signals between a source and a load. It underlies passive filters, matching networks, equalizers, attenuators, microwave structures and power-distribution interfaces.

Three viewpoints are central:

1. Two-port synthesis, in which a matrix or transfer function specifies terminal behavior.
2. Ladder synthesis, in which poles and zeros are successively extracted as series and shunt elements.
3. Terminated synthesis, in which source and load impedances form part of the design specification.

A mathematically admissible transfer function does not automatically correspond to a passive network. Positive-realness, reciprocity, stability, causality and permissible element types must therefore be checked before realization.

Learning Objectives

After studying this chapter, the reader should be able to:

• apply impedance, admittance and transmission descriptions of two-ports;
• derive terminated voltage and power-transfer functions;
• test essential passive-network realizability conditions;
• synthesize LC, RC or RL ladders by successive extraction;
• calculate image and iterative impedances without confusing their conditions;
• use insertion-loss specifications to obtain Butterworth or Chebyshev ladders;
• perform frequency and impedance scaling;
• evaluate practical limitations caused by losses, tolerances and parasitics.

1. Two-Port Representation

A two-port has input variables V₁,I₁ and output variables V₂,I₂. Unless otherwise stated, both currents are defined as entering the network. Voltages are measured from the upper terminal to the common return.

1.1 Impedance and admittance parameters

The impedance description is

Each z_ij has units of ohms. The admittance description is

where each y_ij is measured in siemens. When the relevant inverse exists, \mathbf{Y}=\mathbf{Z}^-1.

For a reciprocal network, z₁₂=z₂₁ and y₁₂=y₂₁. Reciprocity does not imply symmetry: symmetry additionally requires z₁₁=z₂₂.

1.2 Transmission parameters

For cascaded synthesis it is convenient to define the output current I'₂=-I₂ as leaving port 2:

Here A and D are dimensionless, B is in ohms, and C is in siemens. Cascade matrices multiply in physical order from input to output.

For a reciprocal two-port,

For a symmetric two-port, A=D. These conditions are independent: a reciprocal network need not be symmetric.

Elementary cascade sections have the matrices

Their determinants equal one, as expected for reciprocal elements.

2. A Properly Terminated Two-Port

Figure: A source impedance and a load connected to a two-port; the load is connected across the output port and all return connections are explicit.

Let the load satisfy V₂=Z_L I'₂. The transmission equations become

With a Thévenin source V_s and series impedance Z_s, V_s=V₁+Z_sI₁. Hence

This expression applies to linear two-ports with the stated current convention. All terms in its denominator have units of ohms.

Worked Example 1: Terminated voltage transfer

A measured two-port has A=2, B=50\ \Omega, C=0.01\ \mathrm{S} and D=1. It is driven by Z_s=50\ \Omega and terminated by Z_L=100\ \Omega. No reciprocity assumption is made.

Thus the voltage magnitude ratio is approximately 0.286. The matrix determinant is AD-BC=1.5, so this particular measured two-port is not reciprocal.

2.1 Transducer power gain

If Z_s=R_s and Z_L=R_L are positive real resistances and the network is sinusoidally operated, the transducer power gain is

It is the load power divided by the available power of the source. For passive networks, G_T\leq1. Insertion loss is

when the reference is the source’s available power and the direct reference connection is matched.

3. Realizability Requirements

3.1 Positive-real functions

A real-rational driving-point immittance F(s) is positive real (PR) if:

1. it is real for real s wherever finite;
2. it has no poles in \operatorname{Re}(s)>0;
3. \operatorname{Re}F(s)\geq0 for \operatorname{Re}(s)>0;
4. any poles on the imaginary axis, including infinity where applicable, are simple and have nonnegative real residues.

These conditions characterize finite passive RLC driving-point immittances. For a lossless LC network, an impedance is an odd rational reactance function: Z(-s)=-Z(s). Its imaginary-axis poles and zeros are simple and interlace, apart from permitted endpoints at zero or infinity.

A stable voltage-transfer function alone need not be a PR driving-point function. Transfer synthesis therefore requires both the desired response and a compatible passive embedding.

Worked Example 2: Valid LC realization

Consider the normalized impedance

Polynomial division gives

The first term is a series inductor L₁=1\ \mathrm{H}. The second term is the impedance of a parallel LC branch because

Equating Z_p(s)=2s/(s²+1) gives L₂=2\ \mathrm{H} and L₂C₂=1, hence C₂=0.5\ \mathrm{F}. All extracted values are positive. The residues at s=\pm j are positive real numbers, so the reactance satisfies the relevant lossless realizability condition.

A decomposition such as s-s/(s²+2) would contain a negative resonant coefficient and cannot be accepted as a passive Foster LC realization.

4. Ladder Synthesis by Successive Extraction

A ladder alternates series impedances and shunt admittances. It is especially valuable because extraction is algorithmic and its topology is compatible with filter prototypes.

4.1 Cauer extraction

Suppose a PR impedance has a pole at infinity. A series inductance may be extracted as

provided the limit is finite and positive. Define Z_r(s)=Z(s)-sL₁. Invert the remainder to obtain an admittance, from which a shunt element is extracted. Repeated subtraction and inversion generate a continued fraction.

If a finite series resistance exists at high frequency, it is extracted using the appropriate limiting constant only when the resulting remainder remains PR. Extraction about s=0 instead gives the alternative Cauer form.

4.2 Foster versus Cauer forms

Foster forms expose modal residues and resonances. Cauer forms generally produce compact ladders and are better suited to terminated-filter implementation.

4.3 Brune and Darlington perspectives

Not every PR function can be synthesized using a simple canonical LC ladder. Brune synthesis realizes general PR impedances and may require ideal transformers or tightly coupled inductors when a minimum real part is extracted at a finite frequency.

Darlington synthesis embeds a prescribed PR impedance into a lossless reciprocal two-port terminated by a resistor. Dissipation is thereby concentrated in the termination. This is conceptually fundamental to insertion-loss filter synthesis, although the resulting realization is not always the minimum-element circuit.

5. Image and Iterative Impedances

5.1 Image impedances

For a reciprocal two-port with AD-BC=1, the image impedances are the mutually matched terminating impedances. Under nondegenerate conditions,

The square-root branches are selected to give physically admissible impedances, normally with nonnegative real parts. These formulas require the relevant products to be nonzero and must not be applied blindly to degenerate cases such as an ideal transformer with B=C=0. They satisfy

For a reciprocal symmetric network, A=D, and therefore

5.2 Iterative impedance

The iterative impedance is the impedance that reproduces itself when an identical section is terminated in that impedance. For a general two-port it obeys

or

Thus

when C\neq0. The physically relevant root is chosen according to passivity and the direction of propagation. Only for a symmetric section does this reduce to Z_it=\sqrt{B/C}. The general quadratic must not be replaced by the symmetric special case.

6. Terminated Filter Synthesis by Insertion Loss

The insertion-loss method designs the complete source–two-port–load system. For equal resistive terminations and a lossless reciprocal network, the power transmission has the form

where \Omega is normalized frequency, N is filter order, \varepsilon controls passband ripple, and F_N is a characteristic polynomial.

6.1 Butterworth response

For a normalized low-pass Butterworth filter,

It is maximally flat at the origin. If passband and stopband attenuation limits are A_p and A_s at \Omega_p and \Omega_s, respectively, then

The smallest integer satisfying the inequality is selected.

6.2 Chebyshev response

A type-I Chebyshev low-pass filter has equal ripple in the passband:

where T_N(\Omega)=\cos(N\arccos\Omega) for |\Omega|\leq1. Its minimum order is

Chebyshev filters provide faster transition than Butterworth filters of the same order but introduce passband ripple and greater sensitivity.

6.3 Correct Chebyshev prototype recurrence

For an equal-termination type-I Chebyshev low-pass prototype with g₀=1, define

The first element requires a separate formula:

Subsequent values are

The load coefficient is parity dependent:

Therefore, g_N+1=1 is not generally valid for even-order Chebyshev prototypes under this convention. Other tabulation conventions may absorb the unequal terminal resistance into an impedance transformation; conventions must not be mixed.

7. Prototype Scaling and Transformations

For a normalized low-pass ladder with cutoff angular frequency 1\ \mathrm{rad/s} and reference resistance 1\ \Omega, frequency scaling to \omega_c and impedance scaling to R₀ give

Series prototype elements become inductors and shunt prototype elements become capacitors. Units are henries and farads, respectively.

Common low-pass substitutions include:

Here B=\omega₂-\omega₁ is bandwidth and \omega₀=\sqrt{\omega₁\omega₂} is the geometric center frequency. These transformations replace individual prototype elements by resonant branches where required.

8. Applied Case: Butterworth Anti-Aliasing Ladder

This is an illustrative engineering case.

Background and problem

A sensor interface has a 50\ \Omega source and a 50\ \Omega acquisition input. A passive third-order low-pass filter with cutoff f_c=10\ \mathrm{kHz} is required to reduce high-frequency interference without passband ripple.

Analysis and approach

A third-order Butterworth prototype is selected. Its normalized element coefficients are

With \omega_c=2\pi(10⁴)=62,831.9\ \mathrm{rad/s} and R₀=50\ \Omega, the series inductors are

The shunt capacitor is

At f=20\ \mathrm{kHz}, the normalized frequency is \Omega=2. The ideal attenuation is

Outcome

The synthesized ladder is a series inductor, shunt capacitor and series inductor between equal terminations. Practical values might be selected from standard component series and then re-optimized using measured component models.

Learning points

• Terminations are part of the transfer response, not external details.
• Normalized prototypes separate response approximation from physical scaling.
• Ideal attenuation must be corrected for inductor resistance, capacitor loss, source variation and load tolerance.
• If the acquisition input is not approximately 50\ \Omega, a matching or buffering strategy is necessary.

9. Practical Limitations and Best Practice

9.1 Nonideal components

Real inductors exhibit winding resistance, finite quality factor, parasitic capacitance and self-resonance. Capacitors exhibit equivalent series resistance, lead inductance, dielectric absorption and voltage dependence. Distributed behavior becomes significant when physical dimensions are no longer small relative to wavelength.

9.2 Sensitivity and tolerance

High-order and narrow-band filters can be highly sensitive to element variation. Monte Carlo tolerance analysis, worst-case analysis and sensitivity derivatives should supplement nominal synthesis. Chebyshev and elliptic designs often require tighter tolerances than Butterworth designs.

9.3 Passivity and numerical verification

Rational fitting of measured data can create unstable poles or violate passivity even when the measurements came from a passive device. Vector fitting should therefore be followed by causality, stability and passivity enforcement. For high-frequency models, checking the positive-semidefinite condition associated with the Hermitian part of the impedance or admittance matrix is essential.

9.4 Technology and contemporary practice

Modern synthesis combines symbolic extraction with numerical optimization. Electromagnetic simulation is used for distributed filters, while convex optimization and passivity-preserving model reduction assist broadband macromodeling. Machine-learning tools may accelerate topology selection or surrogate modeling, but they do not replace realizability checks, traceable specifications or physical validation.

9.5 Governance and professional responsibility

Filters used in medical, transport, energy or communication systems must satisfy applicable electromagnetic-compatibility, insulation, thermal and safety requirements. Designers should document assumptions, component derating, tolerance coverage and validation data. Automated synthesis results must remain auditable, especially where failure may affect safety or regulatory compliance.

Recommended practice is to:

1. define source and load models before approximation;
2. verify PR, stability and reciprocity conditions separately;
3. state the prototype and scaling convention explicitly;
4. retain adequate numerical precision until final component selection;
5. simulate with lossy component models;
6. validate the fabricated network using impedance or scattering measurements.

Chapter Summary

Two-port synthesis links prescribed transfer behavior to realizable networks. Transmission matrices are particularly effective for cascaded ladders, while terminated voltage and power formulas explicitly incorporate the source and load. Passive driving-point immittances must be positive real; lossless LC reactances additionally exhibit odd symmetry and interlacing imaginary-axis poles and zeros.

Foster synthesis uses partial fractions, whereas Cauer synthesis uses continued fractions and successive series–shunt extraction. Brune synthesis addresses general positive-real impedances, and Darlington synthesis embeds a PR impedance in a lossless terminated two-port. Image impedances apply to mutually matched terminations, while iterative impedance is obtained from a general quadratic fixed-point equation.

Insertion-loss synthesis transforms Butterworth or Chebyshev approximations into normalized ladders and then applies impedance and frequency scaling. Accurate implementation requires attention to termination conventions, element losses, tolerance sensitivity, passivity and regulatory constraints.

Key Terms

Two-port network: A four-terminal network represented as an input port and an output port.

Transmission parameters: The A,B,C,D parameters used conveniently for cascaded networks.

Reciprocity: The property that transmission between two ports is unchanged when source and response locations are interchanged under compatible conditions.

Symmetry: Port interchange invariance; in transmission form it implies A=D.

Positive-real function: A real-rational immittance with nonnegative real part in the right half-plane.

Reactance function: A lossless LC driving-point immittance having odd rational form.

Ladder network: A network formed by alternating series and shunt branches.

Foster synthesis: Realization based on the partial-fraction expansion of an immittance.

Cauer synthesis: Continued-fraction realization through successive element extraction.

Brune synthesis: A general procedure for realizing positive-real impedances.

Darlington synthesis: Embedding a prescribed impedance in a lossless two-port terminated by a resistor.

Image impedance: One of a pair of mutually matching port terminations.

Iterative impedance: A fixed-point impedance reproduced by repeated identical sections.

Insertion loss: Reduction of delivered power caused by insertion of a network.

Butterworth response: A maximally flat magnitude response.

Chebyshev response: An equal-ripple response with a sharper transition than Butterworth for equal order.

Prototype filter: A normalized network from which practical filters are obtained by scaling and transformation.

Impedance scaling: Multiplication of prototype impedance level to obtain a specified reference resistance.

Frequency transformation: A substitution that converts a normalized low-pass response into another frequency response.

Passivity enforcement: Modification of a fitted model so that it cannot generate net energy.

Review Questions

Short Answer Questions

1. Why are transmission parameters convenient for cascaded networks?
2. State the positive-real conditions for a rational driving-point immittance.
3. Distinguish reciprocity from symmetry in a two-port.
4. What is the physical meaning of iterative impedance?
5. Why must source and load impedances be included in terminated-filter synthesis?

Long Answer Questions

1. Derive the terminated voltage-transfer function of an ABCD two-port using the current convention adopted in this chapter.
2. Compare Foster, Cauer, Brune and Darlington synthesis methods in terms of mathematical basis and realization scope.
3. Derive the general iterative-impedance quadratic and explain its reduction for a symmetric network.
4. Explain insertion-loss synthesis for Butterworth and type-I Chebyshev low-pass filters, including order determination.
5. Discuss how component losses, tolerances, parasitics and termination mismatch affect an ideal ladder-filter design.

Analytical / Application Questions

1. A reciprocal two-port has A=2, B=3\ \Omega, C=1\ \mathrm{S} and D=2. Calculate both image impedances and verify reciprocity.
2. Synthesize Z(s)=s+2s/(s²+1) as a passive LC network and verify every element value.
3. Design and scale a third-order Butterworth low-pass ladder for R₀=75\ \Omega and f_c=5\ \mathrm{MHz}, then discuss likely high-frequency implementation limitations.

Multiple Choice Questions with Answers

Question 1

For the transmission convention [V₁\ I₁]^T=[A\ B;C\ D][V₂\ I'₂]^T, what condition characterizes a reciprocal two-port?

A. A=D

B. AD-BC=1

C. B=C

D. AD+BC=1

Correct Answer: B. AD-BC=1

Question 2

A two-port has A=2, B=50\ \Omega, C=0.01\ \mathrm{S}, D=1, Z_s=50\ \Omega and Z_L=100\ \Omega. What is V₂/V_s?

A. 0.286

B. 0.400

C. 0.500

D. 0.714

Correct Answer: A. 0.286

Question 3

Which expression is the correct general equation for iterative impedance?

A. AZ_it²+(C-D)Z_it-B=0

B. CZ_it²+(D-A)Z_it-B=0

C. BZ_it²+(A-D)Z_it+C=0

D. CZ_it²+(A+D)Z_it+B=0

Correct Answer: B. CZ_it²+(D-A)Z_it-B=0

Question 4

For a reciprocal symmetric two-port with nonzero B and C, the image impedance is

A. B/C

B. \sqrt{C/B}

C. \sqrt{B/C}

D. 1/(BC)

Correct Answer: C. \sqrt{B/C}

Question 5

The impedance Z(s)=s+2s/(s²+1) is realized by which passive arrangement?

A. A 1 H series inductor followed by a parallel branch with 2 H and 0.5 F

B. A 2 H series inductor followed by a parallel branch with 1 H and 2 F

C. A 1 F series capacitor followed by a 2 H series inductor

D. A 1 H series inductor followed by a negative capacitor

Correct Answer: A. A 1 H series inductor followed by a parallel branch with 2 H and 0.5 F

Question 6

Which synthesis method is based primarily on continued-fraction expansion?

A. Foster synthesis

B. Cauer synthesis

C. Fourier synthesis

D. Nodal synthesis

Correct Answer: B. Cauer synthesis

Question 7

For a normalized low-pass ladder scaled to resistance R₀ and cutoff \omega_c, a series prototype inductor g_k becomes

A. L_k=g_k/(R₀\omega_c)

B. L_k=R₀\omega_cg_k

C. L_k=R₀g_k/\omega_c

D. L_k=\omega_cg_k/R₀

Correct Answer: C. L_k=R₀g_k/\omega_c

Question 8

A third-order Butterworth response has what ideal attenuation at normalized frequency \Omega=2?

A. Approximately 6.02 dB

B. Approximately 12.04 dB

C. Approximately 18.13 dB

D. Exactly 30 dB

Correct Answer: C. Approximately 18.13 dB

Question 9

Under the Chebyshev prototype convention described in the chapter, when is g_N+1=1?

A. For every filter order

B. Only when N is odd

C. Only when N is even

D. Only when the ripple is zero

Correct Answer: B. Only when N is odd

Question 10

Which statement about a positive-real rational function is correct?

A. It may have arbitrary poles in the open right half-plane

B. Every imaginary-axis pole may have a negative residue

C. Its real part is nonnegative in the open right half-plane

D. It must always be an even function of s

Correct Answer: C. Its real part is nonnegative in the open right half-plane