StudentsEducators

Nyquist Stability Margins

Nyquist Stability Margins are critical parameters used in control theory to assess the stability of a feedback system. They are derived from the Nyquist stability criterion, which employs the Nyquist plot—a graphical representation of a system's frequency response. The two main margins are the Gain Margin and the Phase Margin.

  • The Gain Margin is defined as the factor by which the gain of the system can be increased before it becomes unstable, typically measured in decibels (dB).
  • The Phase Margin indicates how much additional phase lag can be introduced before the system reaches the brink of instability, measured in degrees.

Mathematically, these margins can be expressed in terms of the open-loop transfer function G(jω)H(jω)G(j\omega)H(j\omega)G(jω)H(jω), where GGG is the plant transfer function and HHH is the controller transfer function. For stability, the Nyquist plot must encircle the critical point −1+0j-1 + 0j−1+0j in the complex plane; the distances from this point to the Nyquist curve give insights into the gain and phase margins, allowing engineers to design robust control systems.

Other related terms

contact us

Let's get started

Start your personalized study experience with acemate today. Sign up for free and find summaries and mock exams for your university.

logoTurn your courses into an interactive learning experience.
Antong Yin

Antong Yin

Co-Founder & CEO

Jan Tiegges

Jan Tiegges

Co-Founder & CTO

Paul Herman

Paul Herman

Co-Founder & CPO

© 2025 acemate UG (haftungsbeschränkt)  |   Terms and Conditions  |   Privacy Policy  |   Imprint  |   Careers   |  
iconlogo
Log in

Eigenvalues

Eigenvalues are a fundamental concept in linear algebra, particularly in the study of linear transformations and systems of linear equations. An eigenvalue is a scalar λ\lambdaλ associated with a square matrix AAA such that there exists a non-zero vector vvv (called an eigenvector) satisfying the equation:

Av=λvAv = \lambda vAv=λv

This means that when the matrix AAA acts on the eigenvector vvv, the output is simply the eigenvector scaled by the eigenvalue λ\lambdaλ. Eigenvalues provide significant insight into the properties of a matrix, such as its stability and the behavior of dynamical systems. They are crucial in various applications including principal component analysis, vibrations in mechanical systems, and quantum mechanics.

Biochemical Oscillators

Biochemical oscillators are dynamic systems that exhibit periodic fluctuations in the concentrations of biochemical substances over time. These oscillations are crucial for various biological processes, such as cell division, circadian rhythms, and metabolic cycles. One of the most famous models of biochemical oscillation is the Lotka-Volterra equations, which describe predator-prey interactions and can be adapted to biochemical reactions. The oscillatory behavior typically arises from feedback mechanisms where the output of a reaction influences its input, often involving nonlinear kinetics. The mathematical representation of such systems can be complex, often requiring differential equations to describe the rate of change of chemical concentrations, such as:

d[A]dt=k1[B]−k2[A]\frac{d[A]}{dt} = k_1[B] - k_2[A]dtd[A]​=k1​[B]−k2​[A]

where [A][A][A] and [B][B][B] represent the concentrations of two interacting species, and k1k_1k1​ and k2k_2k2​ are rate constants. Understanding these oscillators not only provides insight into fundamental biological processes but also has implications for synthetic biology and the development of new therapeutic strategies.

Recombinant Protein Expression

Recombinant protein expression is a biotechnological process used to produce proteins by inserting a gene of interest into a host organism, typically bacteria, yeast, or mammalian cells. This gene encodes the desired protein, which is then expressed using the host's cellular machinery. The process involves several key steps: cloning the gene into a vector, transforming the host cells with this vector, and finally inducing protein expression under specific conditions.

Once the protein is expressed, it can be purified from the host cells using various techniques such as affinity chromatography. This method is crucial for producing proteins for research, therapeutic use, and industrial applications. Recombinant proteins can include enzymes, hormones, antibodies, and more, making this technique a cornerstone of modern biotechnology.

Wkb Approximation

The WKB (Wentzel-Kramers-Brillouin) approximation is a semi-classical method used in quantum mechanics to find approximate solutions to the Schrödinger equation. This technique is particularly useful in scenarios where the potential varies slowly compared to the wavelength of the quantum particles involved. The method employs a classical trajectory approach, allowing us to express the wave function as an exponential function of a rapidly varying phase, typically represented as:

ψ(x)∼eiℏS(x)\psi(x) \sim e^{\frac{i}{\hbar} S(x)}ψ(x)∼eℏi​S(x)

where S(x)S(x)S(x) is the classical action. The WKB approximation is effective in regions where the potential is smooth, enabling one to apply classical mechanics principles while still accounting for quantum effects. This approach is widely utilized in various fields, including quantum mechanics, optics, and even in certain branches of classical physics, to analyze tunneling phenomena and bound states in potential wells.

Jacobi Theta Function

The Jacobi Theta Function is a special function that plays a crucial role in various areas of mathematics, particularly in complex analysis, number theory, and the theory of elliptic functions. It is typically denoted as θ(z,τ)\theta(z, \tau)θ(z,τ), where zzz is a complex variable and τ\tauτ is a complex parameter in the upper half-plane. The function is defined by the series:

θ(z,τ)=∑n=−∞∞eπin2τe2πinz\theta(z, \tau) = \sum_{n=-\infty}^{\infty} e^{\pi i n^2 \tau} e^{2 \pi i n z}θ(z,τ)=n=−∞∑∞​eπin2τe2πinz

This function exhibits several important properties, such as quasi-periodicity and modular transformations, making it essential in the study of modular forms and partition theory. Additionally, the Jacobi Theta Function has applications in statistical mechanics, particularly in the study of two-dimensional lattices and soliton solutions to integrable systems. Its versatility and rich structure make it a fundamental concept in both pure and applied mathematics.

Wiener Process

The Wiener Process, also known as Brownian motion, is a fundamental concept in stochastic processes and is used extensively in fields such as physics, finance, and mathematics. It describes the random movement of particles suspended in a fluid, but it also serves as a mathematical model for various random phenomena. Formally, a Wiener process W(t)W(t)W(t) is defined by the following properties:

  1. Continuous paths: The function W(t)W(t)W(t) is continuous in time, meaning the trajectory of the process does not have any jumps.
  2. Independent increments: The differences W(t+s)−W(t)W(t+s) - W(t)W(t+s)−W(t) are independent of the past values W(u)W(u)W(u) for all u≤tu \leq tu≤t.
  3. Normally distributed increments: For any time points ttt and sss, the increment W(t+s)−W(t)W(t+s) - W(t)W(t+s)−W(t) follows a normal distribution with mean 0 and variance sss.

Mathematically, this can be expressed as:

W(t+s)−W(t)∼N(0,s)W(t+s) - W(t) \sim \mathcal{N}(0, s)W(t+s)−W(t)∼N(0,s)

The Wiener process is crucial for the development of stochastic calculus and for modeling stock prices in the Black-Scholes framework, where it helps capture the inherent randomness in financial markets.