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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.