Search research articles

ABOUT JoVE

Overview Leadership Blog JoVE Help Center

AUTHORS

Publishing Process Editorial Board Scope & Policies Peer Review FAQ Submit

LIBRARIANS

Testimonials Subscriptions Access Resources Library Advisory Board FAQ

RESEARCH

JoVE Journal Methods Collections JoVE Encyclopedia of Experiments Archive

EDUCATION

JoVE Core JoVE Business JoVE Science Education JoVE Lab Manual Faculty Resource Center Faculty Site

Terms & Conditions of Use

Related Concept Videos

Hyperbolic and Inverse Hyperbolic Functions: Problem Solving

Hyperbolic and Inverse Hyperbolic Functions: Problem Solving

An arched gate can be effectively modeled using a hyperbolic cosine profile because this type of function is smooth and symmetric about the vertical axis. When the arch is centered at the origin, its maximum height occurs at the center point. This symmetry ensures that any height below the crown of the arch is reached at two horizontal positions that are equal in distance from the centerline but lie on opposite sides.To determine where the gate reaches a height of five meters, the height of the...

Geometry of Hyperbolas

Geometry of Hyperbolas

A hyperbola consists of all points where the absolute difference of distances to two fixed points, called foci, remains constant. The standard equation isEach branch extends infinitely and approaches two asymptotes, which guide the curve’s behavior. The parameters a and b define key features: a measures the distance from the center to each vertex along the transverse axis, while b influences the slopes of the asymptotes. The asymptotes have equationsA rectangle centered at the origin with...

Hyperbolas

Hyperbolas

A hyperbola is a conic section produced when a double-napped cone is intersected by a plane at an angle steeper than the slope of the cone, such that it cuts through both nappes. This intersection yields two separate, mirror-image curves known as branches, which open away from each other along the transverse axis. The nearest points on each branch to the hyperbola’s center are termed vertices, and the distance from the center to a vertex is denoted by a. Perpendicular to the transverse...

Relation between Mathematical Equations and Block Diagrams

Relation between Mathematical Equations and Block Diagrams

In a spring-mass-damper system, the second-order differential equation describes the dynamic behavior of the system. When transformed into the Laplace domain under zero initial conditions, this equation can be effectively analyzed and manipulated. The transformation into the Laplace domain converts differential equations into algebraic equations, simplifying the process of isolating the output.

Tangent Line

Tangent Line

In differential calculus, understanding how a quantity changes at an exact point is central to interpreting dynamic systems. This can be illustrated by analyzing a car traveling along a winding road. The car’s trajectory is represented as a continuous curve, and the direction in which it moves at any instant is given by the tangent to that curve. In contrast, the secant line, intersecting the curve at two points, captures how the car’s position changes over an interval — an...

Linear Differential Equations

Linear Differential Equations

The integrating factor method provides a systematic way to solve first-order linear differential equations, especially those that cannot be handled by separation of variables. This method is particularly useful in modeling time-dependent physical systems influenced by both constant inputs and resistive forces. A common example is the motion of a car subjected to a constant engine force while experiencing air resistance proportional to its velocity.In such scenarios, Newton’s second law...

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by

Same author

Suppression of Oscillation and Ghosting in RF-Spoiled Gradient-Echo-Based Dynamic Imaging.

Magnetic resonance in medicine·2026

Same author

Modulating IL-11-dependent matrix stiffness to delay ovarian aging.

Nature aging·2026

Same author

CAV1-dependent mitochondrial transfer from hucMSCs reprograms epithelial lipid metabolism to relieve pulmonary fibrosis.

Stem cell research & therapy·2026

Same author

Baozhu Granule as a Novel Treatment Strategy for Diminished Ovarian Reserve: A Prospective Clinical Trial.

Rejuvenation research·2026

Same author

Prokaryotically expressed IBDV VP2 virus-like particles: Stability and protective efficacy.

Microbial pathogenesis·2026

Same author

A ferritin-based mosaic-like nanovaccine elicits effective cross-protection against H1N1 and H3N2 swine influenza viruses.

Journal of nanobiotechnology·2026

Search research articles

Related Experiment Video

Updated: Mar 21, 2026

Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180° Curved Artery Test Section

Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180° Curved Artery Test Section

Published on: July 19, 2016

Full analytical solution of the bloch equation when using a hyperbolic-secant driving function.

Jinjin Zhang¹, Michael Garwood¹, Jang-Yeon Park^2,3

¹Center for Magnetic Resonance Research and Department of Radiology, University of Minnesota, Minneapolis, MN, USA.

Magnetic Resonance in Medicine

|May 13, 2016

Summary

This summary is machine-generated.

A new analytical solution for the Bloch equation driven by hyperbolic-secant (HS) pulses was derived. This provides insights into HS pulse parameter effects on magnetization, crucial for NMR techniques like SWIFT imaging.

Keywords:

Bloch equation HS pulse adiabatic pulse analytical solution hyperbolic secant

More Related Videos

Lens-free Video Microscopy for the Dynamic and Quantitative Analysis of Adherent Cell Culture

Lens-free Video Microscopy for the Dynamic and Quantitative Analysis of Adherent Cell Culture

Published on: February 23, 2018

Visualizing Hyporheic Flow Through Bedforms Using Dye Experiments and Simulation

Visualizing Hyporheic Flow Through Bedforms Using Dye Experiments and Simulation

Published on: November 18, 2015

Related Experiment Videos

Last Updated: Mar 21, 2026

Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180° Curved Artery Test Section

Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180° Curved Artery Test Section

Published on: July 19, 2016

Lens-free Video Microscopy for the Dynamic and Quantitative Analysis of Adherent Cell Culture

Lens-free Video Microscopy for the Dynamic and Quantitative Analysis of Adherent Cell Culture

Published on: February 23, 2018

Visualizing Hyporheic Flow Through Bedforms Using Dye Experiments and Simulation

Visualizing Hyporheic Flow Through Bedforms Using Dye Experiments and Simulation

Published on: November 18, 2015

Area of Science:

Magnetic Resonance Imaging (MRI)
Nuclear Magnetic Resonance (NMR) Spectroscopy
Applied Physics

Background:

Hyperbolic-secant (HS) pulses are widely used in NMR for adiabatic spin inversion.
HS pulses are also valuable for excitation and refocusing in gradient-echo and spin-echo sequences.
Applications include ultrashort echo-time imaging (e.g., SWIFT) and B1 mapping.

Purpose of the Study:

To derive the complete theoretical solution of the Bloch equation driven by HS pulses for arbitrary initial magnetization.
To facilitate the analysis of NMR techniques employing HS pulses.

Main Methods:

Analytical derivation of the Bloch-Riccati equation solution for transverse and longitudinal magnetization.
Comparison of the analytical solution with Runge-Kutta numerical methods and the small-tip approximation.

Main Results:

The analytical solution was validated across various initial states and frequency offsets.
Transverse magnetization evolution is significantly influenced by HS pulse parameters.
The small-tip approximation deviates by <5% for flip angles <30° but >10% for angles >40°.

Conclusions:

The derived analytical solution offers critical insights into HS pulse parameter effects on magnetization dynamics.
This solution aids in understanding and optimizing HS pulse applications in NMR and MRI.