For a passive driving-point immittance function of an RLC network, what must be true about the locations of its poles and zeros in the complex plane?
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AThe real parts of all poles must be negative
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BThe real parts of all poles and zeros must be negative
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CThe real parts of all poles and zeros must be negative or zero
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DThe real parts of all zeros must be negative
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EPoles may lie anywhere as long as zeros are in the left half-plane
Answer
Correct Answer: The real parts of all poles and zeros must be negative or zero
Explanation
Introduction / Context:Driving-point immittance functions (impedance or admittance seen at a single pair of terminals) of passive, linear, time-invariant RLC networks are positive-real (PR) functions. PR functions impose strict constraints on pole–zero locations and are foundational in network synthesis (Foster, Cauer, Brune).
Given Data / Assumptions:
- Passive RLC elements only; no active components or dependent sources.
- Linear, time-invariant behavior; standard Laplace-domain representation.
- Driving-point function is positive-real.
Concept / Approach:A function is positive-real if Re{Z(jω)} ≥ 0 for all real ω, with poles and zeros only in the closed left half-plane (CLHP). Simple poles/zeros may lie on the jω-axis, but none can be in the right half-plane. Consequently, the real parts of all poles and zeros are ≤ 0. This ensures stability and passivity and allows synthesis via canonical ladder forms.
Step-by-Step Solution:
State PR constraint: Poles and zeros reside in the CLHP; jω-axis singularities must be simple and non-repeated.Translate to coordinates: For every pole/zero s_k = σ_k + jω_k, we must have σ_k ≤ 0.Therefore, the correct statement is that the real parts are negative or zero.Verification / Alternative check:
Example: An inductor's impedance sL has a zero at s = 0 (σ = 0); a capacitor's impedance 1/(sC) has a pole at s = 0. Both are on the jω-axis and satisfy σ ≤ 0.Why Other Options Are Wrong:
'Poles must be negative' excludes allowable jω-axis poles (e.g., capacitors).'Poles and zeros must be negative' forbids jω-axis elements, contradicting passive components.Statements focusing only on zeros or allowing RHP poles violate PR conditions and passivity.Common Pitfalls:
Confusing stability (poles in LHP) with passivity (PR), and forgetting that simple axis singularities are allowed in driving-point functions.Final Answer:
The real parts of all poles and zeros must be negative or zero