An RLC network is shown in Figure P7.30. The nominal values...

An RLC network is shown in Figure P7.30. The nominal values (normalized) of the network elements are $L-C=1$ and $R=2.5$. Show that the root sensitivity of the two roots of the input impedance $Z(s)$ to a change in $R$ is different by a factor of 4 .

Key Concepts

Normalization of Component Values

Normalization is a technique used to simplify circuit analysis by scaling component values to a common reference, which reduces the number of parameters and clarifies underlying relationships. In the context of the RLC network discussed, normalizing the inductance and capacitance values (L-C=1) allows for a focus on the relative effect of the resistor’s value on the circuit’s dynamic behavior.

Root Sensitivity Analysis

Root sensitivity analysis examines how the poles (roots) of a circuit’s transfer function or impedance equation change in response to variations in circuit parameters. This analysis is crucial for understanding the robustness of the circuit’s performance. In this context, determining that the sensitivities of the two roots vary by a factor indicates that changes in a component such as the resistor can affect different modes of the circuit in significantly different ways.

RLC Circuit Analysis

RLC circuit analysis involves understanding circuits that contain resistors (R), inductors (L), and capacitors (C). These components interact to create frequency-dependent behaviors such as resonance and filtering, and their dynamics are often modeled using differential equations. Analyzing the circuit’s response, including stability and transient response, is essential to determine how the circuit reacts under various conditions.

Input Impedance

Input impedance is a fundamental concept in circuit theory that describes the opposition a circuit presents to an incoming signal at its input terminals. In RLC networks, the input impedance is a complex function of frequency and component values, and its poles (or roots) determine critical aspects of the circuit’s response such as stability and resonance.

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