## 2 SSFP/CWFP Theory
Under this condition, due to the short time pulse intervals T\({}_{\text{P}}\) < T\({}_{2}\)*, the FID and echo signals are fully overlapped. Depending on the precession angle \(\Phi\), the FID/echo interference can be constructive for \(\Phi=(2n+1)\pi\) with maximum amplitude (Figure 1D), or destructive for \(\Phi=n2\pi\) with minimum of amplitude (Figure 1E). Thus, the SSFP signal does not decay, forming the Continuous Wave Free Precession (CWFP) signal. Figure 1: Nuclear magnetic resonance (NMR) signals simulated numerically using \(\theta\) = 90\({}^{\circ}\), T\({}_{1}\) = 200 ms, T\({}_{2}\) = 150 ms, T\({}_{2}\)* = 0.5 ms and several T\({}_{\text{P}}\) values. **(A)** T\({}_{\text{P}}\) = 5T\({}_{1}\), **(B)** T\({}_{\text{P}}\) = T\({}_{2}\), **(C)** T\({}_{\text{P}}\) = 2T\({}_{2}\)*, in **(D)** and **(E)** T\({}_{\text{P}}\) = 0.3 ms \(<\) T\({}_{2}\)*. The frequency offset is **(A–D)** 5.0 kHz and **(E)** 3.3333 kHz. Grey bars indicate the position of the applied pulses. According to Equation (2), the magnitude of the CWFP regimes, using \(\theta\) = 90\({}^{\circ}\) and \(\Phi=(2n+1)\pi\), is dependent on the T\({}_{1}\)/T\({}_{2}\) ratio:  
\[|M_{ss}|=\frac{M_{0}}{1+T_{1}/T_{2}} \tag{3}\]  
This equation shows that the magnitude of the CWFP regimes does not depend on the pulse interval T\({}_{\rm p}\), i.e., does not depend on T\({}_{1}\)[12; 13]. Therefore, it is possible to acquire thousands of CWFP signals during one T\({}_{1}\) period, enhancing the SNR by more than one order of magnitude during the same average time used for FID or echo signals in a conventional pulse sequences [12]. Given these proprieties, CWFP methods have been used for quantitative analysis in conventional benchtop spectrometers and in line quality control systems, as is discussed in the next section. Figure 2 shows the two main schemes of CWFP sequences, where CP-CWFP is the Carr-Purcell pulse sequence using 90\({}^{\circ}\) pulse that also reaches a CWFP regime (CP-CWFP). Depending on the pulse sequence parameters and phase alternation, different properties and names are assigned to the CWFP sequences [9; 23; 24; 27]. In Figure 2A, CWFP is the regime when \(\theta_{1}\) = 90\({}^{\circ}\) and \(\phi_{1}\) = \(\phi_{2}\), CWFPx-x is when \(\theta_{1}\) = 90\({}^{\circ}\) and \(\phi_{1}\) = \(-\phi_{2}\). In Figure 2B, CP-CWFP is the sequence with \(\theta\) = \(\theta_{1}\) = 90\({}^{\circ}\) and \(\phi\) = \(\phi_{1}\) = \(\phi_{2}\), and CP-CWFPx-x is \(\theta\) = \(\theta_{1}\) = 90\({}^{\circ}\) and \(\phi\) = \(\phi_{1}\) = \(-\phi_{2}\). For these sequences, the CWFP regime of the magnetization is obtained after a transition time T\({}^{*}\)[30], given by:  
\[T^{*}=\frac{2T_{1}T_{2}}{T_{1}(1-cos\theta)+T_{2}(1+cos\theta)} \tag{4}\]  
These sequences have been used to calculate T\({}_{1}\) and T\({}_{2}\) relaxation times in a single experiment using 90\({}^{\circ}\) pulses. To calculate T\({}_{1}\) using T\({}^{*}\), it is necessary to use small flip angle (\(\theta_{1}\) \(\sim\) 5\({}^{\circ}\) to 10\({}^{\circ}\)) in CP-CWFPx-x. The sequence was named CWFP-T\({}_{1}\)[23]. One of the major differences between CWFP sequences with and without phase alternation is in the excitation profile (Figure 3). Figure 3A shows the CWFP and CP-CWFP excitation profiles for pulses with same phase (\(\phi_{1}\) = \(\phi_{2}\)). The maximum and minimum amplitudes are obtained when \(\Phi=(2n+1)\pi\) and \(\Phi=2n\pi\), respectively. Note that the amplitude for resonance experiments is the minimum, and the distance between minimums of signal amplitudes are spaced in frequency by \(\Delta f=1/T_{p}\). Figure 3A shows the importance of setting the resonance frequency according to T\({}_{\rm p}\) to form signals with maximum amplitude. For example, for T\({}_{\rm p}\) = 300 \(\upmu\)s (solid line), the maximum signal occurs at 2 KHz and shifts to 2.5 KHz for T\({}_{\rm p}\) = 200 \(\upmu\)s (dashed line). [PERSON] et al.