--- title: "Realizability conditions and lossless two-port properties" subject: "Filter Design" chapter: 4 key: "semester_5/filter_design:realizability-lossless-two-port" source: https://www.bctengineer.com/q/semester_5/filter_design/realizability-lossless-two-port site: BCT Engineering --- # Realizability conditions and lossless two-port properties Filter Design · Chapter 4: Properties and Synthesis of Passive Networks - Asked in: 2071 Shrawan, 3+3 marks - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/filter_design/realizability-lossless-two-port - Whole chapter: https://www.bctengineer.com/md/semester_5/filter_design/chapter-4 ## Question What are the required properties of a function to be realizable? Explain the properties of lossless two port function. ## Short answer A driving-point immittance must be positive real: ratio of polynomials with real positive coefficients, poles/zeros in LHP or jω axis with simple jω-axis poles and positive residues, ReZ(jω)≥0. For a lossless two-port: z11, z22 are LC driving-point functions; z21 is odd rational with poles on jω axis, shared by z11 and z22, with residue condition k11k22−k21²≥0; transmission zeros symmetric about axes. ## Answer #### Properties for a function to be realizable A driving-point immittance $F(s)$ is realizable with passive $R$, $L$, $C$ (and ideal transformers) if and only if it is a **positive real (PR) function**: $F(s)$ is real for real $s$, and $\text{Re}\,F(s) \ge 0$ whenever $\text{Re}\,s \ge 0$. In practice this is tested through these properties: 1. $F(s) = N(s)/D(s)$ is a ratio of polynomials with real, positive coefficients. 2. Poles and zeros lie in the left half plane or on the $j\omega$ axis; $N(s)$ and $D(s)$ are Hurwitz. 3. Poles and zeros on the $j\omega$ axis are simple, with real and positive residues. 4. The degrees of $N$ and $D$ differ by at most 1, and so do their lowest powers (no multiple pole or zero at $s=\infty$ or $s=0$). 5. $\text{Re}\,F(j\omega) \ge 0$ for all $\omega$. For a **transfer function** to be realizable, it must be stable (poles strictly in the left half plane, denominator strictly Hurwitz), and the numerator degree must not exceed the denominator degree. #### Properties of a lossless two-port For a lossless (LC) reciprocal two-port described by $z_{11}, z_{12}=z_{21}, z_{22}$ (or the $y$-parameters): 1. $z_{11}$ and $z_{22}$ are LC driving-point functions (odd rational functions with alternating simple poles and zeros on the $j\omega$ axis, positive residues). 2. $z_{21}$ is also an odd rational function; its poles lie on the $j\omega$ axis and are simple. 3. Every pole of $z_{21}$ is also a pole of $z_{11}$ and $z_{22}$. $z_{11}$ or $z_{22}$ may have extra **private poles** not present in $z_{21}$. 4. **Residue condition** at every pole: $k_{11} \ge 0$, $k_{22} \ge 0$, $k_{21}$ real (may be negative), and $$ k_{11}k_{22} - k_{21}^2 \ge 0 $$ When equality holds at every pole the network is called **compact**. 5. The zeros of transmission (zeros of $z_{21}$) are the roots of an even or odd polynomial, so they are symmetric about both axes. In ladder realizations they lie on the $j\omega$ axis, including $0$ and $\infty$. They need not alternate with the poles. 6. Since no power is lost, $|S_{11}(j\omega)|^2 + |S_{21}(j\omega)|^2 = 1$ when the two-port is resistively terminated. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. A driving-point function F(s) is passively realizable if and only if: A. It has all poles in the right half plane B. Re F(jw) <= 0 for all w C. It is real for real s and Re F(s) >= 0 whenever Re s >= 0 D. It has only real poles and zeros Answer: C. Such a function is called positive real (PR); this is the basic realizability test. (Key point: Positive real condition) 2. What constraint on N(s) and D(s) of a realizable immittance holds? A. Their highest powers differ by at most 1, and so do their lowest powers B. Their degrees must be equal C. Numerator degree must be at least 2 more than the denominator D. Both must have repeated roots on the jw axis Answer: A. Otherwise there would be a multiple pole or zero at s=0 or infinity, which is not PR. (Key point: Degree condition) 3. In a lossless two-port, a pole of z11 that does not appear in z21 is called: A. A transmission zero B. A compact pole C. A private pole D. A residue pole Answer: C. Every pole of z21 must be a pole of z11 and z22, but z11 or z22 may have extra private poles. (Key point: Private poles) 4. The residue condition at each pole of a lossless two-port is: A. k11*k22 + k21^2 = 0 B. k11*k22 - k21^2 >= 0, with equality everywhere for a compact network C. k11 + k22 = k21 D. k21 must be positive and k11*k22 <= k21^2 Answer: B. k11, k22 >= 0, k21 is real, and k11k22 - k21^2 >= 0; equality at every pole makes the network compact. 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