Chapter 1.4 First-Order Physical Systems (Impedance Analogy)

Vocabulary flashcards covering key terms and concepts from Chapter 1.4 on First-Order Physical Systems and the impedance analogy.

Impedance analogy

A framework that models systems across disciplines by relating a driving (potential) variable to a resisting (flow) variable, using impedance to link energy transfer (Maxwell’s approach).

Power conjugate variables

A pair whose product has units of power (W); examples include voltage and current, force and velocity, pressure and volumetric flow, or temperature and heat flow.

Potential variable

The driving quantity that causes flow in a system (e.g., voltage, force, pressure, temperature).

Flow variable

The responding quantity that carries energy or flux (e.g., current, velocity, volumetric flow, heat flow).

Transfer function

The ratio of output to input in the frequency domain, expressed as H(s) = Y(s)/X(s); depends only on system properties, not the signals.

Frequency domain

A representation in terms of s (complex frequency) after transformation, where differential equations become algebraic.

Laplace transform

A transform that converts time-domain signals to the s-domain, enabling algebraic solutions for linear differential equations.

First-order differential equation

An equation involving only the first derivative with respect to time, common in Chapter 1.4 and leading to a first-order denominator in H(s).

Ohm’s law

V = IR; relationship between voltage, current, and resistance in electrical systems.

Poiseuille’s law

Q = (Δp π r^4)/(8 μ L); laminar flow relation; defines hydraulic resistance b = 8 μ L/(π r^4).

Resistance

Impedance in the electrical domain; R = V/I; units are Ohms (Ω).

Stokes’ drag

Viscous drag F_drag = b v; drag proportional to velocity, with b the friction coefficient; leads to terminal velocity when balanced by gravity.

Inductance (mass analogy)

In the impedance analogy, electrical inductance L corresponds to mechanical mass m; V = L dI/dt models F = m dv/dt.

Capacitance (spring analogy)

In the impedance analogy, electrical capacitance C corresponds to a mechanical spring; I = C dV/dt models energy storage and rate relations.

d’Alembert’s principle

F − ma = 0; used to form equations by including inertial terms, bridging dynamics with force/acceleration in the analysis.

Kirchhoff’s voltage law

The sum of voltages around any closed loop equals zero.

Inductor/capacitor relationships

In the impedance analogy: V across an inductor = L dI/dt and current through a capacitor = C dV/dt.

Free body diagram vs circuit diagram

Two graphical representations of the same system; mechanics uses free-body diagrams, circuits use circuit diagrams, yielding equivalent equations.

Step input

An input that turns on at t = 0, often modeled as a unit step to study system response.

Unit step function (u(t))

A function that is 0 for t < 0 and 1 for t ≥ 0, used to model turning on inputs in time-domain analysis.

Hybrid transfer function

A transfer function that maps a signal of one type to a signal of another type (e.g., force to velocity).

LTI system (SciPy lti)

Linear Time-Invariant system; has constant coefficients and obeys superposition; represented and solved using tools like SciPy’s lti object.

Step response

The output of an LTI system in response to a unit step input.

Terminal velocity

Steady velocity where drag force equals driving force (e.g., gravity), resulting in a constant velocity.