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A bipolar gradient is a magnetic field gradient waveform with lobes of opposite sign. In nuclear magnetic resonance, such waveforms can control sensitivity to position and motion. Opposite signs alone do not guarantee equal signed areas, and an applied waveform must be distinguished from its effective encoding waveform when refocusing pulses change phase evolution.

For an original counterexample, apply $+10mT/m$ for $1\mathrm{ms}$ and then $-10mT/m$ for $2\mathrm{ms}$. The gradient zeroth moment isThe waveform is bipolar but unbalanced.

If the two durations are instead equal to $\tau$, and the effective lobes are contiguous, $+G$ from time 0 to $\tau$ and $-G$ from $\tau$ to $2\tau$, then $m_0=0$ but the gradient first moment is $m_1=-G{\tau }^2$. Thus cancellation of stationary-position phase can coexist with sensitivity to constant velocity. The sign follows this stated ordering.

Bipolar pulses also appear outside flow imaging. A 1989 stimulated echo study addressed background gradients that distort diffusion measurements. Alternating pulses could cancel a cross term under specified conditions; tests used proton signals from homogeneous and granular samples.^[ref] This is a sequence-specific result, not a promise that every bipolar pair removes every imaging error.

The gradient first moment of an effective magnetic field gradient waveform $G(t)$ is its time-weighted integral,Here effective includes the sign changes caused by refocusing pulses. Its units are $\mathrm{T}{\mathrm{s}}^2/m$ when $G$ is in tesla per metre. The gradient zeroth moment is $m_0=\int G(t)dt$. For position $x(t)=x_0+vt$ with constant velocity, the accumulated nuclear magnetic resonance phase iswhere $\gamma$ is the angular gyromagnetic ratio. Thus $m_0=0$ removes the dependence on initial position while $m_1$ can retain sensitivity to velocity, the basis of an idealized phase-contrast magnetic resonance imaging encoding.^[ref]

For a teaching bipolar gradient, let the first lobe have signed area $A$ and time centroid $t_1$, and the second have signed area $-A$ and centroid $t_2$. ThenEqual and opposite lobe areas cancel stationary-spin phase, but their time separation prevents cancellation of the constant-velocity phase. For $A=10^{-5}\mathrm{T}s/m$ and $t_2-t_1=2\mathrm{ms}$, the first moment is $-2\times 10^{-8}\mathrm{T}{\mathrm{s}}^2/m$. These are illustrative waveform values, not measurements from the source.

The time origin is part of the definition: replacing $t$ by $t-t_0$ gives $m_1^{\prime }=m_1-t_0{m}_0$. Balanced $m_0=0$ makes the first moment independent of that choice. Two encodings with equal zeroth moments use their first-moment difference $\Delta m_1$, rather than either individual value, to infer velocity.

The gradient zeroth moment is the signed time integral of an effective magnetic field gradient waveform along a specified encoding axis:Its units are tesla seconds per metre. Effective means that the sign convention accounts for any refocusing pulses in the nuclear magnetic resonance sequence; it is not necessarily the unsigned physical pulse area.

For a stationary spin at position $x_0$, choose the phase convention $\varphi =\gamma \int G(t)x_{0}dt$. Then $\varphi =\gamma x_0{m}_0$, where $\gamma$ is the angular gyromagnetic ratio. A balanced waveform with $m_0=0$ cancels this position-dependent contribution within the model. It does not assert that every possible background contribution to phase vanishes.

As an original rectangular-pulse example, $5mT/m$ for $2\mathrm{ms}$ has $m_0=10^{-5}\mathrm{T}s/m$. Adding $-10mT/m$ for $1\mathrm{ms}$ cancels it. Equal areas, rather than equal pulse heights, produce the balance.

For a moving spin with $x(t)=x_0+vt$, the remaining contribution can be $\gamma v{m}_1$, involving the gradient first moment. That separation explains how magnetic resonance velocity encoding can suppress sensitivity to stationary position while preserving motion information. Early phase-sensitive flow imaging supplied the experimental motivation for extracting such information from resonance phase.^[ref]

Magnetic resonance velocity encoding makes a component of motion measurable through the phase of a nuclear magnetic resonance signal. Controlled magnetic field gradients give different phase shifts to spins with different trajectories. Phase-contrast magnetic resonance imaging compares encodings to extract a velocity component, rather than interpreting ordinary image brightness as speed.^[ref]

In an original idealized model, assume constant velocity $v$ along one encoding axis and effective gradients with equal gradient zeroth moments. Then the encoding difference obeyswhere $\gamma$ is the angular gyromagnetic ratio and $\Delta m_1$ is the gradient first moment difference. For $\gamma =2\pi \cdot 42.58\times 10^6\mathrm{rad}{\mathrm{s}}^{-1}{\mathrm{T}}^{-1}$, $\Delta m_1=2\times 10^{-8}\mathrm{T}{\mathrm{s}}^2/m$ and $v=0.10m/s$, this gives $\Delta \varphi \simeq 0.535\mathrm{rad}$. These stipulated values illustrate the conversion, rather than a particular scanner experiment.

Reversing the moment difference reverses the phase sign under this convention. Reducing its magnitude decreases sensitivity to a given velocity while increasing the velocity encoding limit. A measured voxel can contain a range of velocities, and background phase or phase aliasing can complicate reconstruction.

The encoded component also differs from volumetric flow rate: estimating transported volume requires combining the appropriate normal velocity with area and time, not merely recording one peak speed.

The velocity encoding limit, commonly written $v_{\mathrm{enc}}$, specifies the velocity scale represented by a phase difference of magnitude $\pi$ in a chosen phase-contrast magnetic resonance imaging encoding. With equal gradient zeroth moments and first-moment difference $\Delta m_1$, an ideal constant-velocity model givesThe gyromagnetic ratio $\gamma$ is angular, in radians per second per tesla, and $\Delta m_1$ is the difference of gradient first moments. A sign convention must be specified separately. A 1982 original paper described obtaining motion information through phase-sensitive nuclear magnetic resonance encoding.^[ref]

Choose the convention $\Delta \varphi =\pi v/v_{\mathrm{enc}}$. If $v_{\mathrm{enc}}=100cm/s$, then $v=50cm/s$ produces $\Delta \varphi =\pi /2$. But $v=120cm/s$ produces $1.2\pi$, which wraps to $-0.8\pi$ when phase is represented in $[-\pi ,\pi )$. A reconstruction that ignores this aliasing would report $-80cm/s$. The limit describes an encoding ambiguity, not a physical maximum speed of the fluid.

For a fixed phase uncertainty ${\sigma }_{\varphi }$ in this simple model, propagation gives ${\sigma }_v=(v_{\mathrm{enc}}/\pi ){\sigma }_{\varphi }$. Reducing $v_{\mathrm{enc}}$ improves velocity sensitivity under that assumption while narrowing the unambiguous range. For example, ${\sigma }_{\varphi }=0.02$ radians and $v_{\mathrm{enc}}=100cm/s$ give ${\sigma }_v\simeq 0.64cm/s$. Real measurements must also account for background phase, intravoxel variation and timing. Gated measurements of cerebrospinal fluid show why timing within the cardiac cycle matters even when net displacement is small.^[ref]