*3.1. Assumption and Parameter Definition*

The machine structure chosen to be studied in this paper was a 24 slot 11 pole-pair SCP-MGM and DCP-MGM, as shown in Figure 2. A few assumptions must be made to simplify the mathematical modeling:

Since there exists a z-direction current within the windings of the studied machines and the machine topology has a circular shape, a vector magnetic potential (VMP) *A*z in a polar coordinate is adopted to calculate the magnetic flux density distribution within the machines. The machine structures are then divided into several ring-like regions based on the different material interfaces, as can be seen in Figure 3, where α represents the angle of the inner PM arc, β represents the angle of the slot opening in the modulator, δ is the slot opening angle, and γ is the stator slot angle. The whole machine is divided into ten subdomains: the innermost one (region I) represents the shaft part; region II is the rotor yoke; region III is the inner consequent-pole PM; region IV is the inner air gap; region V is the modulator pieces (it should be noted that for SCP-MGMs, the gap between each two modulator pieces is air, while for DCP-MGM, bipolar PMs are inserted in that gap). Region VI is the outer air gap; region VII is the stator teeth; region VIII is the stator slots together with windings; region IX is the stator yoke; region X is the outside of the studied machines.

The angular position of the *j*-th PM part of the inner rotor θ*PM*, the position of the *k*-th modulator piece θ*Mod*, the position of the *t*-th stator tooth part θ*tooth*, and the position of the *s*-th stator slot part θ*slot* can be defined, respectively, as:

$$
\theta\_{\rm PM} = q\_{\rm in} + \frac{j \cdot 2\pi}{P\_i} \tag{3}
$$

$$
\theta\_{M\text{od}} = \frac{k \cdot 2\pi}{Q} + \theta\_0 - \frac{\beta}{2} \tag{4}
$$

$$
\theta\_{toch} = \mathbf{t} \cdot 2\pi/\mathbf{P} \tag{5}
$$

$$
\theta\_{\text{slot}} = \text{s} \cdot 2\pi / P \tag{6}
$$

where ϕ*in*, and θ0 the initial angular positions of the inner rotor and outer rotor, respectively. Due to the symmetrical structure of the inner rotor and outer rotor, ϕ*in,* θ0 has a range of [0, 2π/*Pi*], [0, 2π/*Q*], respectively. Specifically, ϕ*in* is defined as zero when the lower edge of the PM in the inner rotor coincides with the positive direction of the angular axis; θ0 is defined as zero when the center of the slot of the outer rotor (air in SCP-MGM and PM in DCP-MGM) coincides with the positive direction of the angular axis, as shown in Figure 2.

**Figure 3.** Subdomain divisions of SCP-MGM and DCP-MGM.

The VMP → *A*, the magnetic flux density → *B*, the magnetic field strength → *H*, and the current density distribution → *J* in stator windings can be written in vector form as:

$$
\overrightarrow{A} = A\_z(r, \theta) \cdot \overrightarrow{u}\_z \tag{7}
$$

$$
\overrightarrow{B} = B\_r(r, \theta) \cdot \overrightarrow{\boldsymbol{\mu}}\_r + B\_\theta(r, \theta) \cdot \overrightarrow{\boldsymbol{\mu}}\_\theta \tag{8}
$$

$$
\overrightarrow{H} = H\_{\rm r}(r, \theta) \cdot \overrightarrow{\boldsymbol{u}}\_{\rm r} + H\_{\rm 0}(r, \theta) \cdot \overrightarrow{\boldsymbol{u}}\_{\rm 0} \tag{9}
$$

$$
\overrightarrow{J} = J\_z(r, \theta) \cdot \overrightarrow{u}\_z \tag{10}
$$

To simplify the solving process, all the parameters related to magnetic field are expressed in terms of complex Fourier series. Thus, the vector amplitude above can be further expressed as:

$$A\_z(r, \theta) = \sum\_{n=-\infty}^{n=\infty} A\_{z,n}(r) \cdot e^{-in\theta} \tag{11}$$

$$B\_r(r, \theta) = \sum\_{n = -\infty}^{n = \infty} \mathcal{B}\_{r, n}(r) \cdot e^{-in\theta} \text{ and } B\_0(r, \theta) = \sum\_{n = -\infty}^{n = \infty} \mathcal{B}\_{0, n}(r) \cdot e^{-in\theta} \tag{12}$$

$$H\_{\mathbf{r}}(\mathbf{r},\theta) = \sum\_{n=-\infty}^{n=\infty} \hat{H}\_{\mathbf{r},\theta}(\mathbf{r}) \cdot \mathbf{e}^{-in\theta} \text{ and } H\boldsymbol{\varrho}(\mathbf{r},\theta) = \sum\_{n=-\infty}^{n=\infty} \hat{H}\boldsymbol{\varrho}\_{\mathbf{r}}(\mathbf{r}) \cdot \mathbf{e}^{-in\theta} \tag{13}$$

$$f\_z(r, \theta) = \sum\_{n = -\infty}^{n = \infty} f\_{z,n}(r) \cdot e^{-in\theta} \tag{14}$$

where *n* represents the *n*-th order coefficient of the corresponding Fourier series. It should be noticed that, in numerical calculation, a reasonable harmonic order *N* is used to truncate the infinite Fourier series. If *N* is too small, the Fourier series will have a large error, if *N* is too large, the calculation time will be rather long.

#### *3.2. Partical Di*ff*erential Equation Solution*

The magnetic field within the machine follows quasistatic Maxwell equations:

$$
\nabla \times \overrightarrow{H} = \overrightarrow{J} \tag{15}
$$

$$
\nabla \cdot \overrightarrow{B} = 0\tag{16}
$$

The relationship between → *B* and → *A* can be further expressed as:

$$
\overrightarrow{B} = \nabla \times \overrightarrow{A}\tag{17}
$$

The radial component and tangential component matrix of magnetic flux density → *B* are then obtained in matrix form as [31]:

$$\mathbf{B}\_r = \frac{1}{r} \frac{\partial \mathbf{A}\_z}{\partial \theta} = -i \frac{1}{r} \mathbf{K} \mathbf{A}\_z \tag{18}$$

$$\mathbf{B}\_0 = -\frac{\partial \mathbf{A}\_z}{\partial r} \tag{19}$$

where **K** represents the harmonic order coefficient diagonal matrix that is related to *N*, given by:

$$\mathbf{K} = \begin{bmatrix} -N & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & N \end{bmatrix} \tag{20}$$

Similar to Equations (11)–(14), the relative permeability of each region can also be expressed in a complex Fourier series form:

$$\mu(\theta) = \sum\_{n=-\infty}^{n=\infty} \hat{\mu}\_n \cdot e^{-in\theta} \tag{21}$$

Next, based on the relation between → *B* and → *H***,** as expressed below:

$$
\overrightarrow{B} = \mu \overrightarrow{H} + \mu\_0 \overrightarrow{M} \tag{22}
$$

where → *M* is the magnetization vector. The first item on the right is a product of two Fourier series, which can be rewritten in matrix form by using the Cauchy product theorem:

$$\mathbf{B}\_{\mathbf{r}} = \mu\_{\mathbf{r}, \text{cov}} \mathbf{H}\_{\mathbf{r}} + \mu\_0 \mathbf{M}\_{\mathbf{r}} \tag{23}$$

$$\mathbf{B}\_{\partial} = \mu\_{\partial, \text{cov}} \mathbf{H}\_{\partial} + \mu\_{\partial} \mathbf{M}\_{\partial} \tag{24}$$

where μ*r*,cov and μθ,cov are convolution matrices of the radial and tangential components of permeability, respectively. **M***r* and **M**θ are the radial and tangential components of magnetization intensity, respectively. **M***r* and **M**θ can all written in complex Fourier series. The convolution matrix μ*r*,cov can be defined as:

$$
\boldsymbol{\mu}\_{r, \text{cov}} = \begin{bmatrix}
\boldsymbol{\mu}\_0 & \cdots & \boldsymbol{\mu}\_{-2N} \\
\vdots & \ddots & \vdots \\
\boldsymbol{\mu}\_{2N} & \cdots & \boldsymbol{\mu}\_0 \\
\end{bmatrix} \tag{25}
$$

where μˆ *n* is the *n*-th order coefficient of the Fourier series of corresponding μ.

In Equation (24), **H**θ is continuous at the interface between two regions, but Bθ is discontinuous at the interface. Hence, the matrix μθ,cov cannot be settled using Equation (25). Instead, a fast Fourier factorization is applied to calculate μθ,cov, for the sake of keeping the rate of convergence the same for the left and right side of Equation (24) [34]. μθ,cov can be given by [31]:

$$\boldsymbol{\mu}\_{0,\text{cov}} = \begin{bmatrix} \hat{\mu}\_0^{\text{rec}} & \cdots & \hat{\mu}\_{-2N}^{\text{rec}} \\ \vdots & \ddots & \vdots \\ \hat{\mu}\_{2N}^{\text{rec}} & \cdots & \hat{\mu}\_0^{\text{rec}} \end{bmatrix}^{-1} \tag{26}$$

From Equation (26), it can be seen that μθ,cov is acquired by replacing μˆ *n* with the corresponding *n*-th order Fourier coefficient of 1/μθ for each element, and there is a matrix inversion outside.

The region V in SCP-MGM is used to illustrated the convolution matrix with respect to relative permeability, as shown in Figure 4.

**Figure 4.** The calculation instance of the convolution matrix with respect to relative permeability.

The relative permeability distribution on the circumferential direction can be expressed as:

$$\mu(\theta) = \begin{cases} \mu\_0 & \theta \in [\varrho\_{out} + \frac{2(k-1)\pi}{Q}, \varrho\_{out} + \frac{2(k-1)\pi}{Q} + \beta) \\\ \mu\_{iron,k} & \theta \in [\varrho\_{out} + \frac{2(k-1)\pi}{Q} + \beta, \varrho\_{out} + \frac{2k\pi}{Q}) \end{cases} \tag{27}$$

where ϕ*out* = θ0 − β 2 .

When a Fourier expansion on [0, 2π] is applied to Equation (27), the expressions of μˆ *k* and μˆ*rec k* are given by:

$$\mu\_n = \begin{cases} \sum\_{k=1}^{\mathcal{Q}} \frac{\mu\_0}{2\pi i n} e^{in(\frac{2\pi}{\mathcal{Q}} + \mathcal{O}\_0)} (e^{\frac{in\mathcal{C}}{2}} - e^{-\frac{in\mathcal{C}}{2}}) + \sum\_{k=1}^{\mathcal{Q}} \frac{\mu\_{\text{inv}k}}{2\pi i n} e^{in(\frac{2\pi}{\mathcal{Q}} + \mathcal{O}\_0)} (e^{in(\frac{2\pi}{\mathcal{Q}} - \frac{\mathcal{C}}{2})} - e^{\frac{in\mathcal{C}}{2}}) n \neq 0 \\\ \underbrace{\mathcal{Q}\mu\_{00} + (\frac{2\pi}{\mathcal{Q}} - \mathcal{J}) \sum\_{k=1}^{\mathcal{Q}} \mu\_{\text{inv}k}}\_{2\pi} n = 0 \end{cases} \tag{28}$$

$$\begin{split} \mu\_{n}^{rec} &= \begin{cases} \sum\_{k=1}^{\mathbb{Q}} \frac{1}{2\pi i m \mu\_{0}} e^{j n \left(\frac{2k\pi}{\mathbb{Q}} + \theta\_{0}\right)} \left(e^{\frac{i\pi\theta}{2}} - e^{-\frac{i\pi\theta}{2}}\right) + \sum\_{k=1}^{\mathbb{Q}} \frac{1}{2\pi i m \mu\_{imk}} e^{j n \left(\frac{2k\pi}{\mathbb{Q}} + \theta\_{0}\right)} \left(e^{j n \left(\frac{2\pi}{\mathbb{Q}} - \frac{\theta}{2}\right)} - e^{\frac{i\pi\theta}{2}}\right) n \neq 0\\ \frac{\frac{\sqrt{\mu}}{\mathbb{Q}} + \left(\frac{2\pi}{\mathbb{Q}} - \theta\right)}{2\pi} \frac{1}{\frac{1}{2\pi i m \mu\_{k}}} n = 0 \end{cases} \tag{29} \\ \end{split} \tag{20}$$

The convolution matrices μ*r*,cov and μθ,cov can then be obtained by substituting Equations (28) and (29) into (25) and (26).

Combining Equations (18)–(26) together, the VMP satisfies:

$$\frac{\partial^2 \mathbf{A}\_z^k}{\partial r^2} + \frac{1}{r} \frac{\partial \mathbf{A}\_z^k}{\partial r} - \frac{\mathbf{V} \mathbf{A}\_z^k}{r^2} = -\mu\_{\partial, \text{av}} \mathbf{J}\_z - \frac{\mu\_0}{r} (\mathbf{M}\_\partial + i \mathbf{U} \mathbf{M}\_r) \tag{30}$$

where **V** = μθ,cov**K**μ−1 *r*,cov*c***K** and **U** = μθ,cov**K**μ−1 *r*,cov*c*. The derivation process of Equation (30) is given in Appendix A.

Equation (30) is a Cauchy–Euler differential equation system [35]. The general solution of a single differential equation in Equation (30) is given by:

$$y = c\_1 r^{\nu \frac{1}{2}} + c\_2 r^{-\nu \frac{1}{2}} \tag{31}$$

where *c*1, *c*2 are unknown coefficients. Similarly, the general solution of the differential equation system in Equation (30) can be written in a matrix form, where the new element (*i*, *j*) in matrix *r***V** is defined as:

$$r^{\mathbf{V}}(i,j) = r^{\mathbf{V}(i,j)} \tag{32}$$

Hence, the complementary solution of Equation (30) **A***k z com* can be written as:

$$\left. \mathbf{A}\_z^k \right|\_{\rm com} = r^{\mathbf{V}^\frac{\tilde{l}}{2}} \mathbf{C}\_1 + r^{-\mathbf{V}^\frac{\tilde{l}}{2}} \mathbf{C}\_2 \tag{33}$$

where **C**1 and **C**2 are unknown coefficient vectors. Matrix *r***V** 1 2 can be factorized as:

$$r^{\mathbf{V}^{\frac{1}{2}}} = \mathbf{P}r^{\lambda}\mathbf{P}^{-1} \tag{34}$$

where λ is the eigenvalue matrix of **V**1/2, and matrix **P** is the eigenvector matrix of **V**1/2. Therefore, Equation (33) can be simplified as:

$$\left. \mathbf{A}\_z^k \right|\_{\rm com} = \mathbf{Pr}^\lambda(\mathbf{P}^{-1} \cdot \mathbf{C}\_1) + \mathbf{Pr}^{-\lambda}(\mathbf{P}^{-1} \cdot \mathbf{C}\_2) = \mathbf{Pr}^\lambda \mathbf{D} + \mathbf{Pr}^{-\lambda} \mathbf{E} \tag{35}$$

As for the particular solution of Equation (30), **A***k z par* is given by:

$$\mathbf{A}\_z^k \Big|\_{par} = r^2 \mathbf{F} + r \mathbf{G} \tag{36}$$

where **F** = (**V** − 4**I**) −1 μθ,*cov***J***z*, **G** = μ0(**V** − **I**) −1 (**M**θ + *i***UM***r*).

The general solution of VMP in Equation (30) is the sum of the complementary solution of Equation (35), and particular solution Equation (36):

$$\mathbf{A}\_z^k = \mathbf{A}\_z^k \big|\_{com} + \mathbf{A}\_z^k \big|\_{par} \tag{37}$$

The expression of VMP in each subdomain is given in Appendix A.

According to the geometrical characteristic of SCP-MGMs and DCP-MGMs, the magnetization intensity only exists in region III and region V, and the current density distribution only exists in region VIII. Their distribution waveforms and Fourier series coefficients can be seen in Table 1. Additionally, for regions I, II, IV, VI, IX, and X, there only exists one material type, so the coefficients within the convolution matrix are a constant. However, the permeability distributions are different in region III, V, VII, and VIII, as shown in Table 2; their coefficients can be obtained by substituting *a*, *b*, and *c* in Table 3 into the following equations:

$$\hat{\mu}\_{\text{lt}} = \frac{1}{2\pi} (\sum\_{k=1}^{c} \int\_{a}^{a+b} \mu\_0 \varepsilon^{in0} d\theta + \sum\_{k=1}^{c} \int\_{a+b}^{a+2\pi/c} \mu\_{\text{iron}}(k) \varepsilon^{in0} d\theta) \tag{38}$$

$$\mu\_n^{\text{rac}} = \frac{1}{2\pi} (\sum\_{k=1}^c \int\_a^{a+b} \frac{e^{in\theta}}{\mu\_0} d\theta + \sum\_{k=1}^c \int\_{a+b}^{a+2\pi/c} \frac{e^{in\theta}}{\mu\_{iron,k}} d\theta) \tag{39}$$

**Table 2.** Mathematical modeling of magnetic sources within SCP-MGMs and DCP-MGMs.

**Table 3.** Mathematical modeling of the permeability distribution of different regions within SCP-MGMs and DCP-MGMs.

#### *3.3. Bondary Condition Application*

At the interfaces between two different subdomains, the radial component of → *B* and the tangential component of → *H* should be continuous across the boundary. Due to their ring-like shapes, each subdomain only interfaces with two subdomains at most, and region I and X have only one interface. Hence, all the boundary conditions can be written as:

$$\mathbf{B}\_r^k(\mathcal{R}\_k) = \mathbf{B}\_r^{k+1}(\mathcal{R}\_k) \tag{40}$$

$$\mathbf{H}\_{\mathcal{O}}^{k}(\mathcal{R}\_{k}) = \mathbf{H}\_{\mathcal{O}}^{k+1}(\mathcal{R}\_{k}) \tag{41}$$

where *k* represents the *k*-th subdomain of the proposed machine, and 2 ≤ *k* ≤ 9.

Suppose the subdomain number of the proposed machine is L, and the harmonic order to be calculated is N. By applying the above boundary conditions to each interface of the machines, a system of 2\*(L − 1) linear equations with 2\*(L − 1) unknowns can be obtained, and each unknown is an (N × 1) vector. The system of linear equations can be further written in matrix form, as below:

$$\mathbf{S}\mathbf{X} = \mathbf{T} \tag{42}$$

where the expressions of **S**, **X**, and **T** in this paper are given in Appendix A.

As long as the coefficient matrix **S** is invertible, the unknown vector **X** can be acquired. The numerical solution of Equation (42) can be obtained in the MATLAB software.

#### *3.4. Saturation Consideration of Soft-Magnetic Material*

For the SMM in the consequent-pole part, modulator, and stator teeth, the flux line is concentrated, thus, the saturation of the SMM must be considered. The nonlinear B-μ curve of 50JN1300 is given in Figure 4. In HMM, the relative permeability μ is obtained by an iterative method, as shown by the dot lines in Figure 5, where the number "1, 2, 3, 4" means the iteration number. The detailed iterative process is shown in Figure 6. First, an initial value, namely μ0 = 1500, is assigned to a given region with SMM for the first iteration. In the *k*-th iteration, the average flux density → *B* of a specific region can be obtained by substituting μ*i*,*k* and solving the matrix Equation (42). A new average relative permeability μ*i*,*cal* in region *i* can then be acquired. Where *i* belongs to {III, V, VII, VIII}. The average relative permeability μ*i,k*+1 for the (*k* + 1)-th iteration in region *i* is given as:

$$
\mu\_{i,k+1} = \frac{\overline{\mu}\_{i,k} + \overline{\mu}\_{i,\text{kcal}}}{2} \tag{43}
$$

The iteration will stop only if the iteration time *ni* exceeds the maximum number of iterations *Ni* (*Ni* is set to be 50 here), or the maximum error Δ in all these regions is below the error requirement ξ, Δ is defined as:

$$\Delta = \max \left\{ \frac{\left| \overline{\mu}\_{i,k} - \overline{\mu}\_{i,\text{cal}} \right|}{\overline{\mu}\_{i,k}} \right\}\_{\prime} \text{ } i \in \{III\_{\prime} \\ V\_{\prime} \, VII\_{\prime} \, VIII \} \tag{44}$$

where ξ is set to be 0.05 in this paper. The saturations of SMM in SCP-MGMs and DCP-MGMs are calculated respectively using this method. The relative permeabilities of each region under on-load conditions are shown in Table 4. It can be observed that the saturation of regions V and VII is more severe in DCP-MGMs, since there are PMs inserted in the modulator, thus, there are more flux lines in the modulator and stator teeth.

**Figure 6.** Fast Fourier transform (FFT) of no-load outer-air-gap radial magnetic flux density for SCP-MGMs and DCP-MGMs.

**Table 4.** Relative permeabilities of different regions within SCP-MGMs and DCP-MGMs.

#### *3.5. Electromagnetic Parameters Calculation*

Once the VMP is solved in Equation (42), the related electromagnetic parameters of the two machines can be calculated. The electromagnetic torque of the two rotors of SCP-MGMs and DCP-MGMs can be calculated by using the Maxwell stress tensor.

The flux linkage of each coil side can be given by [36]:

$$q\rho\_k = \frac{N\_{\rm turn}L}{S\_{\rm coil}} \int\_{-\frac{\gamma}{2} + \frac{2k\pi}{P}}^{\frac{\gamma}{2} + \frac{2k\pi}{P}} \int\_{R\_7}^{R\_8} A\_z^{VIII}(r, \theta) r dr d\theta \tag{45}$$

where *Nturn* is the number of coil turns in each slot, and *Scoil* is the cross section area of a single slot. When all the coils in each phase are in series, the three-phase flux linkage can be written as:

$$\boldsymbol{\upmu} = \begin{bmatrix} \psi\_A \\ \psi\_B \\ \psi\_C \end{bmatrix} = \mathbf{C}\_{\text{turn}} \begin{bmatrix} \boldsymbol{\uprho}\_1 & \boldsymbol{\uprho}\_2 & \cdots & \boldsymbol{\uprho}\_{24} \end{bmatrix} \tag{46}$$

where **C***turn* is coil-connecting matrix of the proposed machine, the coil number is given in Figure 2, and **C***turn* can be expressed as:

**C***turn* = ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ 1 0 0 00 −1 −100 0 0 11 0 0 00 −1 −100 0 0 1 0 0 0 11 0 0 00 −1 −100 0 0 11 0 0 00 −1 −1 0 0 −1 −100 0 0 11 0 0 00 −1 −100 0 0 11 0 0 0 ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (47)

The back electromotive force (EMF) is computed by the derivative of ψ with respect to time:

$$\mathbf{E}\_{\rm ABC} = \frac{d\boldsymbol{\Psi}}{dt} = \frac{d\boldsymbol{\Psi}}{d\boldsymbol{\Theta}} \cdot \boldsymbol{\omega} \tag{48}$$

where ω is the rotating speed of the magnetic field in the outer air gap.

The electromagnetic torque is calculated by using a Maxwell stress tensor. Thus, the electromagnetic torque of the inner rotor *Tin* equals the calculus of the Maxwell stress tensor of the inner air gap along the circumferential direction, and the electromagnetic torque of the outer rotor *Tout* equals the algebraic sum of the calculus of the Maxwell stress tensor of both the inner air gap and the outer air gap along the circumferential direction. They can be expressed as:

$$T\_{\dot{m}} = \frac{LR\_i^2}{\mu\_0} \int\_0^{2\pi} B\_r^{IV}(\mathcal{R}\_{i\dot{\nu}}\theta) B\_0^{IV}(\mathcal{R}\_{i\dot{\nu}}\theta) d\theta \tag{49}$$

$$T\_{\rm out} = \frac{LR\_o^2}{\mu\_0} \int\_0^{2\pi} B\_r^{VI}(\mathbf{R}\_{o\prime}, \theta) B\_0^{VI}(\mathbf{R}\_{o\prime}, \theta) d\theta - \frac{LR\_i^2}{\mu\_0} \int\_0^{2\pi} B\_r^{IV}(\mathbf{R}\_{i\prime}, \theta) B\_0^{IV}(\mathbf{R}\_{i\prime}, \theta) d\theta \tag{50}$$

where *Ri* and *Ro* are the middle radius of the inner air gap and outer air gap, respectively.

#### **4. Validation and Comparison**

#### *4.1. Simulation Environment and Machine Parameters*

To make quantitative comparison between the SCP-MGMs and DCP-MGMs, all the geometrical parameters of these two machines should be set as the same, and other parameters, such as the slot filling factor and root mean square value of the winding current should be also set as the same, as shown in Table 5. The analytical prediction of the HMM was carried out using MATLAB, and the FEA model was constructed and run in JMAG software. The FEA model had 35,597 elements and 25,368 nodes; the element size near the air gap was set as 1 mm, to maintain calculation accuracy. It took 4.6 s for the computer to obtain the magnetic field distribution for one step. The computer system configuration was as follows: Processor: Intel Core i7-4790 CPU @ 3.60 GHz; Installed Memory (RAM): 28.0 GB-System type: 64-bit Windows Operating System. Additionally, the mean error percentage ε1 and maximum difference ε2 of the two methods was defined as:

$$\varepsilon\_1 = \frac{1}{N\_c} \sum\_{n=1}^{N\_c} \left| \frac{V\_{FEA,n} - V\_{HMM,n}}{V\_{FEA,n}} \right| \times 100\% \tag{51}$$

$$\varepsilon\_2 = \max \{ \left| V\_{FEA, \mathbf{u}} - V\_{HMM, \mathbf{u}} \right| \}, \mathbf{n} \in \{ 1, 2, \dots, N\_{\mathbf{c}} \} \tag{52}$$

where *VHMM,n* is the *n*-th value, calculated using the HMM, and *VFEA*,n is *n*-th the value obtained by FEA. N*c* is the total number of calculated points.

#### *4.2. Comparison between HMM and FEA*

Based on the calculation and simulation, a good agreement between the HMM and FEA can be observed in Figures 7–12 for SCP-MGMs and DCP-MGMs. It was also found that the difference between HMM and FEA at a no-load condition was less compared to that at a load condition. This is due to the truncation of the infinite Fourier series; the high-order components take up a greater proportion at a no-load condition, leading to a larger error for HMM. Additionally, the numerical values calculated via MATLAB were constrained by the computational accuracy. Values exceeding the computational accuracy were ignored, which led to errors of the magnetic field calculation. Generally, if the harmonic order and computational accuracy is improved, the error will decrease. However, the computational time increases rapidly with the increase of harmonic order and computational accuracy. Thus, there is a tradeoff between calculation accuracy and calculation time. In this paper, the computational accuracy of MATLAB was set as 32 bits.

The mean error percentage ε1 and maximum difference ε2 for the SCP-MGM and DCP-MGM are listed in Table 6.

**Figure 7.** On-load inner and outer air-gap radial magnetic flux density distribution of the SCP-MGM: (**a**) outer air gap; (**b**) inner air gap.

**Figure 8.** On-load inner and outer air-gap radial magnetic flux density distribution of the SCP-MGM: (**a**) outer air gap; (**b**) inner air gap.

**Figure 9.** Comparison of on-load magnetic flux density distribution of the SCP-MGM, drawn by harmonic modeling method (HMM) and finite element analysis (FEA): (**a**) HMM; (**b**) FEA.

**Figure 10.** No-load inner and outer air-gap radial magnetic flux density distribution of the DCP-MGM: (**a**) outer air gap; (**b**) inner air gap.

**Figure 11.** On-load inner and outer air-gap radial magnetic flux density distribution of the DCP-MGM: (**a**) outer air gap; (**b**) inner air gap.

**Figure 12.** Comparison of on-load magnetic flux density distribution of the DCP-MGM drawn by HMM and FEA: (**a**) HMM; (**b**) FEA.

**Table 5.** Geometrical parameters of the SCP-MGM and DCP-MGM.

**Table 6.** Mean error percentage and maximum difference of the magnetic flux density distributions between the FEA and HMM of the SCP-MGM and DCP-MGM.

A fast Fourier transform (FFT) was executed on the no-load radial component of the magnetic flux density of the outer air gap for both the SCP-MGM and DCP-MGM, as shown in Figure 13. It can be seen that the second harmonic component was much higher for the DCP-MGM, since the inserted PM on the outer rotor produced a second magnetic field after the modulation of the consequent-pole iron part of the inner rotor. Hence, the consequent-pole structure of the inner rotor and outer rotor of the DCP-MGM could modulate the magnetic field generated by its counterpart. The electromagnetic torque of the DCP-MGM was expected to be larger than that of the SCP-MGM under the same working conditions. Additionally, the difference of each frequency component between the HMM and FEA was very small. Specifically, the error percentages of the second harmonic component for the SCP-MGM and the DCP-MGM using HMM and FEA were 4.14% and 2.15%, respectively.

**Figure 13.** FFT of no-load outer-air-gap radial magnetic flux density for the SCP-MGM and the DCP-MGM: (**a**) SCP-MGM; (**b**) DCP-MGM.

#### *4.3. Electromagnetic Performance Analysis under Di*ff*erent Working Conditions*

The electromagnetic performances of the two MGMs are predicted by HMM and simulated in FEA software under different operating conditions, where the rotating speed of two rotors and the current frequency of stator windings are set to cooperate with the practical driving conditions of the HEC, as shown in Table 1. Since the ICE is connected to the inner rotor to provide the power; the outer rotor is connected to the differential, which is further connected to the wheels; the battery is connected to the stator windings, and there can be two-way power flow between the battery and the stator windings. Thus, the power transmission relation among inner rotor, outer rotor and stator windings is equal to the power transmission relation among ICE, wheels and battery. Assume the anti-clock direction is positive, the torque is positive if it's on the anti-clock direction, otherwise it's negative. The power of a component is defined as a positive one when this component is inputting energy; when a component is outputting energy, its power is defined as a negative one.

#### 4.3.1. Back EMF under No-Load Condition

The amplitude of back EMF was determined by the rotating speed of both the inner rotor and the outer rotor. Figure 14 shows the no-load back EMF of the SCP-MGM and the DCP-MGM under the same operating conditions, namely ωi = 1200 r/min, ωo = 1500 r/min. It can be seen that there was an error between the predicted back EMF and FEA result; because the back EMF was calculated as the derivative of flux linkage with respect to time, a small error of flux linkage will be amplified on the back EMF. Additionally, the back EMF of the DCP-MGM was larger than that of the SCP-MGM, due to the flux enhancing effect of the PMs inserted into the modulator. The maximum errors for the SCP-MGM and DCP-MGM were 165 V and 520 V, respectively. The average errors of the SCP-MGM and DCP-MGM were 3.7% and 6.9%, respectively.

**Figure 14.** Analytically predicted and FEA simulated back electromotive force (EMF) of the SCP-MGM and DCP-MGM: (**a**) SCP-MGM; (**b**) DCP-MGM.

#### 4.3.2. Pure Electric Mode (Mode 1)

Under this mode, the HEV is at a low driving speed, the inner rotor is locked, and the ICE does not come into service; stator windings only work to provide the power that the HEV needs. Thus, ωi = 0 r/min, ωo = 500 r/min. The torque waveforms of the power distribution of the two machines are shown in Figures 15 and 16.

**Figure 15.** Analytically predicted and FEA simulated torque waveforms of the SCP-MGM and DCP-MGM under pure electric mode: (**a**) SCP-MGM; (**b**) DCP-MGM.

**Figure 16.** Power distribution among the internal combustion engine (ICE), battery, and wheels under pure electric mode: (**a**) SCP-MGM; (**b**) DCP-MGM.

4.3.3. Pure Mechanical Mode (Mode 2)

Under this mode, the HEV is running at a medium speed, so the battery does not output power anymore and the ICE comes into use. However, to maintain the magnetic field, the stator windings are electrified with DC current [14]. The rotating speeds of two rotors are: ωi = 1200 r/min, ωo = 1015 r/min. The torque waveforms of the power distribution of the two machines are shown in Figures 17 and 18.

**Figure 17.** Analytically predicted and FEA simulated torque waveforms of SCP-MGM and DCP-MGM under pure mechanical mode: (**a**) SCP-MGM; (**b**) DCP-MGM.

**Figure 18.** Power distribution among ICE, battery and wheels under pure mechanical mode: (**a**) SCP-MGM; (**b**) DCP-MGM.

### 4.3.4. Hybrid Mode (Mode 3)

When the HEV needs to further accelerate, the ICE alone is not enough to provide the power that the HEV needs. The SCP-MGM and DCP-MGM can then switch into hybrid mode. In this mode, both ICE and battery provide the energy for the wheels. The rotating speeds of two rotors are: ωi = 1200 r/min, ωo = 2000 r/min. The torque waveforms of the power distribution of the two machines are shown in Figures 19 and 20.

**Figure 19.** Analytically predicted and FEA simulated electromagnetic torque of SCP-MGM and DCP-MGM under hybrid mode: (**a**) SCP-MGM; (**b**) DCP-MGM.

**Figure 20.** Power distribution among ICE, battery, and wheels under hybrid mode: (**a**) SCP-MGM; (**b**) DCP-MGM.

4.3.5. Regenerative Braking Mode (Mode 4)

When the HEV deaccelerates, the ICE stops working. The magnetic field generated via the stator windings provides a resistance for the wheels, and thus the power flows from the wheels to the battery. At this time, the power flows from the stator windings to the battery. Thus, the battery can be charged under this mode. The rotating speeds of two rotors are: ωi = 0 r/min, ωo = 1000 r/min. The torque waveforms of the power distribution of the two machines are shown in Figures 21 and 22.

**Figure 21.** Analytically predicted and FEA simulated back EMF of SCP-MGM and DCP-MGM: (**a**) SCP-MGM; (**b**) DCP-MGM.

**Figure 22.** Power distribution among ICE, battery, and wheels under hybrid mode: (**a**) SCP-MGM; (**b**) DCP-MGM.

#### 4.3.6. Quantitative comparison between HMM and FEA

Table 7 shows the mean error percentage and maximum difference of the torque waveforms between FEA and HMM of the SCP-MGM and DCP-MGM under different operating modes. It can be observed that the mean error percentage was below 8%, which is acceptable for the electromagnetic torque prediction of SCP-MGMs and DCP-MGMs. The error of torque prediction was mainly caused by the error of the air-gap magnetic flux density prediction on both the radial and tangential direction, since the torque was calculated using the calculus of the product of the radial component and tangential component of air-gap magnetic flux density.

**Table 7.** Mean error percentage and maximum difference of the torque waveforms between FEA and HMM of the SCP-MGM and DCP-MGM under different operating modes.

From the above comparisons between the SCP-MGM and DCP-MGM under various working conditions, it can be seen that the torque ratio between outer rotor and inner rotor was about 1.18, which is equal to the pole-pair ratio of outer rotor to inner rotor. The ripple rate of the outer rotor was larger than that of the inner rotor, which can be alleviated by adopting a skewed stator when these machines are applied in practical applications. Additionally, the electromagnetic torque of the DCP-MGM was about 1.88 times of that of SCP-MGM. However, the torque per unit weight of the PM of the DCP-MGM was only about 0.72 of that of SCP-MGM. Considering the fact that the consequent–pole structure already saves half the PM material that would otherwise have been used, the total usage of PM material for DCP-MGM is reasonable. Thus, the DCP-MGM is more suitable to be used in the propulsion systems of HEVs.

#### 4.3.7. Discussion of HMM

The greatest advantage of HMM is that the modeling process is simpler compared to the conventional subdomain method [30]. Since it unitizes Fourier series of relative permeability, the boundary conditions become simpler and the number of unknowns becomes less. For instance, the consequent-pole structure leads to a very complex general solution if the subdomain method is used [29]. Additionally, magnetic saturation can be taken into account to an extent in HMM, and a magnetic flux density distribution figure can be obtained. However, the HMM also has some drawbacks. First, HMM can only provide a simplified magnetic saturation model, since the magnetic saturation varies from point to point in some SMM parts. The relative permeability in the radial direction is also regarded as a constant, but, in reality, it could change. To the authors' knowledge, no existing analytical method can really reflect the magnetic saturation at every point within an electric machine. Secondly, the use of convolution sometimes leads to an extremely large or small value, which further leads to a large error in the final result due to the accuracy limit of the numerical calculation.

#### **5. Conclusions**

In this paper, an analytical modeling method for consequent-pole MGMs was proposed and elaborated. Two machine topologies, namely the SCP-MGM and DCP-MGM, were analyzed and compared quantitatively using HMM and FEA. A good agreement was achieved for these two methods. Furthermore, these two machines were embedded into the propulsion system of HEVs under different operating conditions. By inserting extra PMs in the modulator, the electromagnetic torque of the DCP-MGM increased greatly compared to its counterpart. Therefore, the DCP-MGM has the potential to be applied in the propulsion systems of HEVs. Additionally, the proposed HMM can also be applied in the mathematical modeling of other consequent-pole electric machines.

**Author Contributions:** The work presented in this paper is the output of the research projects undertaken by C.L. In specific, H.Z. and C.L. developed the topic. H.Z. carried out the calculation and simulation, analyzed the results, and wrote the paper. Z.S. gave some suggestions on the calculation process. J.Y. helped to carried out a part of the simulation.

**Funding:** This research was supported by a grant (Project No. 51677159) from the Natural Science Foundation of China (NSFC), China. Also, the work is supported by a grant (Project No. JCYJ20180307123918658) from the Science Technology and Innovation Committee of Shenzhen Municipality, Shenzhen, China. Moreover, it was also supported by a grant (Project No. CityU 21201216) from Research Grants Council of HKSAR, China.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A**

The expressions of **H***r* and **H**θ are given by:

$$\mathbf{H}\_{\rm r} = -i\frac{1}{r}\mu\_{r,\rm cov}^{-1}\mathbf{K}\mathbf{A}\_{\rm z} - \mu\_{0}\mu\_{r,\rm cov}^{-1}\mathbf{M}\_{\rm r} \text{ and } \mathbf{H}\_{\rm 0} = -\mu\_{0,\rm cov}^{-1}\frac{\partial\mathbf{A}\_{\rm z}}{\partial r} - \mu\_{0}\mu\_{0,\rm cov}^{-1}\mathbf{M}\_{\rm 0} \tag{A1}$$

In polar coordinate, (15) can be simplified as:

$$\nabla \times \mathbf{H} = \frac{1}{r} \frac{\partial}{\partial r} (r \mathbf{H}\_{\partial}) + i \frac{\mathbf{K} \mathbf{H}\_{r}}{r} = \mathbf{J}\_{z} \tag{A2}$$

By substituting (A1) into (A2), one can obtain that:

$$\frac{\partial^2 \mathbf{A}\_z^k}{\partial r^2} + \frac{1}{r} \frac{\partial \mathbf{A}\_z^k}{\partial r} - \frac{b\_{\partial, \text{cov}} \mathbf{K} \mu\_{r, \text{cov}}^{-1} \mathbf{K} \mathbf{A}\_z^k}{r^2} = -\mu\_{\partial, \text{cov}} \mathbf{J}\_z - \frac{\mu\_0}{r} (\mathbf{M}\_\partial + i \mu\_{\partial, \text{cov}} \mathbf{K} \mu\_{r, \text{cov}}^{-1} \mathbf{M}\_r) \tag{A3}$$

The vector magnetic potential in region I~X are given in Table A1.

**Table A1.** The Vector Magnetic Potential Expressions.

Where **D**I , **D**II, ... , **D**IX and **E**I , **E**II, ... , **E**X are (N\*1) vector.

The expressions of **X**, **S**, **T** are:

**X** = (**D***I* ) *T* (**D***II*) *T* (**E***II*) *T* (**D***III*) *T* (**E***III*) *T* ··· (**D***IX*) *T* (**E***IX*) *T* (**E***X*) *T T* (A4) **S** = ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ **K**1,1 **K**1,2 **K**1,3 0000 0 0 0 0 0 0 0 0 0 0 0 **K**2,1 **K**2,2 **K**2,3 0000 0 0 0 0 0 0 0 0 0 0 0 0 **K**3,2 **K**3,3 **K**3,4 **K**3,5 00 0 0 0 0 0 0 0 0 0 0 0 0 **K**4,2 **K**4,3 **K**4,4 **K**4,5 00 0 0 0 0 0 0 0 0 0 0 0 000 **K**5,4 **K**5,5 **K**5,6 **K**5,7 00 0 0 0 0 0 0 0 0 0 000 **K**6,4 **K**6,5 **K**6,6 **K**6,7 00 0 0 0 0 0 0 0 0 0 00000 **K**7,6 **K**7,7 **K**7,8 **K**7,9 000000000 00000 **K**8,6 **K**8,7 **K**8,8 **K**8,9 000000000 0000000 **K**9,8 **K**9,9 **K**9,10 **K**9,11 0000000 0000000 **K**10,8 **K**10,9 **K**10,10 **K**10,11 0000000 0000000 0 0 **K**11,10 **K**11,11 **K**11,12 **K**11,13 00000 0000000 0 0 **K**12,10 **K**12,11 **K**12,12 **K**12,13 00000 0000000 0 0 0 0 **K**13,12 **K**13,13 **K**13,14 **K**13,15<PHONE_NUMBER> 0 0 0 0 **K**14,12 **K**14,13 **K**14,14 **K**14,15<PHONE_NUMBER> 0 0 0 0 0 0 **K**15,14 **K**15,15 **K**15,16 **K**15,17 0 0000000 0 0 0 0 0 0 **K**16,14 **K**16,15 **K**16,16 **K**16,17 0 0000000 0 0 0 0 0000 **K**17,16 **K**17,17 **K**17,18 0000000 0 0 0 0 0000 **K**18,16 **K**18,17 **K**18,18 ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (A5) **T** = 0 0 (*R*2**G**1) *T* (*R*2μ*II***G**1) *T* (−*R*3**G**1) *T* (−*R*3μ*IV***G**1) *T* (*R*4**G**2) *T* (*R*4μ*IV***G**2) *T* (−*R*5**G**2) *T T* 0 0 (*R*2 *T* (2*R*2 *T* (−*R*2 *T* (−2*R*2 *T* 0 0 *T* (A6)

$$\text{where:}$$

(−μ*VIR*5**G**2)

$$\begin{bmatrix} \begin{array}{cc} \mathbf{K}\_{1,1} & \mathbf{K}\_{1,2} & \mathbf{K}\_{1,3} \end{array} \end{bmatrix} = \begin{bmatrix} \begin{array}{cc} \mathbf{I} & -\begin{pmatrix} \frac{\mathbf{R}\_1}{\|\mathbf{R}\_2\|} & -\mathbf{I} \end{pmatrix} \end{bmatrix} \end{bmatrix} \tag{A7}$$

8μ*IX***F**)

8**F**)

$$\begin{bmatrix} \begin{array}{cc} \mathbf{K}\_{2,1} & \mathbf{K}\_{2,2} & \mathbf{K}\_{2,3} \end{array} \end{bmatrix} = \begin{bmatrix} \begin{array}{cc} \boldsymbol{\mu}\_{II} & -\boldsymbol{\mu}\_{I} \begin{pmatrix} \frac{\boldsymbol{R}\_{1}}{\boldsymbol{R}\_{2}} \end{pmatrix}^{\|\mathbf{K}\|} & \boldsymbol{\mu}\_{I} \end{bmatrix} \end{bmatrix} \tag{A8}$$

7μ*open*,θ**F**)

7**F**)

$$\begin{bmatrix} \mathbf{K}\_{3,2} & \mathbf{K}\_{3,3} & \mathbf{K}\_{3,4} & \mathbf{K}\_{3,5} \end{bmatrix} = \begin{bmatrix} \mathbf{I} & \begin{pmatrix} \frac{\mathbf{R}\_1}{\mathbf{R}\_2} \end{pmatrix}^{|\mathbf{K}|} & -\mathbf{P}^{III} \begin{pmatrix} \frac{\mathbf{R}\_1}{\mathbf{R}\_2} \end{pmatrix}^{|\mathbf{K}|} & -\mathbf{P}^{III} \end{bmatrix} \tag{A9}$$

$$\begin{bmatrix} \mathbf{K}\_{4,2} & \mathbf{K}\_{4,3} & \mathbf{K}\_{4,4} & \mathbf{K}\_{4,5} \end{bmatrix} = \begin{bmatrix} \begin{array}{cc} \mu\_{\rm PM,0} & -\mu\_{\rm PM,0} \lVert \mathbf{K} \rVert \begin{pmatrix} \frac{R\_1}{R\_2} \end{pmatrix}^{\lVert \mathbf{K} \rVert} & -\mu\_{\rm II} \mathbf{P}^{\textup{III}} \lambda^{\textup{III}} \begin{pmatrix} \frac{R\_2}{R\_3} \end{pmatrix}^{\lambda^{\textup{III}}} & \mu\_{\rm II} \mathbf{P}^{\textup{III}} \lambda^{\textup{III}} \end{bmatrix} & \begin{array}{cc} \text{(A10)} \end{array}$$

$$
\begin{bmatrix}
\begin{array}{cccc}
\mathbf{K}\_{5,4} & \mathbf{K}\_{5,5} & \mathbf{K}\_{5,6} & \mathbf{K}\_{5,7}
\end{array}
\end{bmatrix} = \begin{bmatrix}
\begin{array}{cccc}
\mathbf{p}^{\mathrm{III}} & \mathbf{p}^{\mathrm{III}}
\end{array}
\begin{array}{cccc}
\mathbf{p}^{\mathrm{III}} & \begin{array}{cccc}
\mathbf{p}^{\mathrm{III}} & \begin{array}{c}
\mathbf{p}^{\mathrm{III}}
\end{array}
\end{array}
\begin{array}{cccc}
\mathbf{k}^{\mathrm{III}} & \begin{array}{c}
\mathbf{k}^{\mathrm{I}\mathbb{I}}
\end{array}
\end{array}
\begin{array}{cccc}
\mathbf{k}^{\mathrm{I}\mathbb{I}}
\end{array}
\end{bmatrix}
\end{bmatrix}
\tag{A11}
$$

$$
\begin{bmatrix}
\mathbf{K}\_{6,4} & \mathbf{K}\_{6,5} & \mathbf{K}\_{6,6} & \mathbf{K}\_{6,7}
\end{bmatrix} = \begin{bmatrix}
\begin{array}{ccccc}
\boldsymbol{\mu}\_{\rm IV} \mathbf{P}^{\rm II} \boldsymbol{\lambda}^{\rm III} & -\boldsymbol{\mu}\_{\rm IV} \mathbf{P}^{\rm III} \boldsymbol{\lambda}^{\rm III}
\end{array}
\begin{array}{ccccc}
\boldsymbol{\mu}\_{\rm PL} |\mathbf{K}| \begin{pmatrix}
\boldsymbol{R}\_{3} \\
\boldsymbol{R}\_{3}
\end{pmatrix}^{\rm IV} & \boldsymbol{\mu}\_{\rm PL} |\mathbf{K}| \begin{pmatrix}
\boldsymbol{R}\_{3} \\
\boldsymbol{R}\_{3}
\end{pmatrix}^{\rm IV} & \boldsymbol{\mu}\_{\rm PL} |\mathbf{K}|
\end{bmatrix}
\end{bmatrix} \tag{A12}
$$

$$
\begin{bmatrix}
\boldsymbol{\nu} & \boldsymbol{\nu} & \boldsymbol{\nu} & \boldsymbol{\nu} & \boldsymbol{\nu}
\end{bmatrix} \begin{bmatrix}
\boldsymbol{\nu}\_{\rm L} (|\mathbf{K}| & \boldsymbol{R}\_{\rm PL}) \boldsymbol{\lambda}^{\rm IV} & \boldsymbol{\nu}\_{\rm PL}
\end{bmatrix}
\tag{A13}
$$

 **K**7,6 **K**7,7 **K**7,8 **K**7,9 = **I** *R*3 *R*4 |**K**| −**P***VR*4 *R*5 λ*V* −**P***V* (A13)

$$
\begin{bmatrix}
\begin{bmatrix}
\mathbf{K}\_{8,6} & \mathbf{K}\_{8,7} & \mathbf{K}\_{8,8} & \mathbf{K}\_{8,9} \\
\end{bmatrix}
\end{bmatrix} = \begin{bmatrix}
\begin{bmatrix}
\mu\_{\mathrm{Mod},\beta} \mathrm{|}\mathbf{K}\rangle & -\mu\_{\mathrm{Mod},\beta} \mathrm{|}\mathbf{K}\big{\big{(}}^{\frac{\mathrm{R}}{\mathrm{R}\_{4}}\big{)}\mathbf{K}\big{\big{(}}^{\frac{\mathrm{R}}{\mathrm{R}\_{5}}\big{)}} -\mu\_{\mathrm{IV}} \mathrm{P}\big{\mathrm{V}}\big{\mathrm{V}}^{\mathrm{V}}\_{\mathrm{R}\_{5}}\big{)}^{\mathrm{V}} & \mu\_{\mathrm{IV}} \mathrm{P}\big{\mathrm{V}}^{\mathrm{V}}
\end{bmatrix}
\end{bmatrix}
\tag{A14}
$$

$$
\begin{bmatrix}
\begin{bmatrix}
\mathbf{K}\_{0,8} & \mathbf{K}\_{0,0} & \mathbf{K}\_{0,0} & \mathbf{K}\_{0,0}
\end{bmatrix}
\mathbf{n}^{\mathrm{V}} & \mathbf{n}^{\mathrm{V}}\_{1} \mathrm{(}\boldsymbol{\mathbb{R}}^{\mathrm{V}}\_{4}\big{)}^{\mathrm{V}} & \left(\boldsymbol{\mathbb{R}}\_{8}\right)^{\mathrm{[}}\mathrm{K}\big{\big{}}^{\mathrm{R}}
\end{bmatrix}
\end{bmatrix}
\tag{A15}
$$

$$
\begin{bmatrix}
\mathbf{K}\_{9,8} & \mathbf{K}\_{9,9} & \mathbf{K}\_{9,10} & \mathbf{K}\_{9,11} \\
\end{bmatrix} = \begin{bmatrix}
\mathbf{p}^V & \mathbf{p}^V \begin{pmatrix}
\frac{R\_4}{R\_5} \end{pmatrix}^V & -\begin{pmatrix}
\frac{R\_5}{R\_6} \end{pmatrix}^{|\mathbf{K}|} & -\mathbf{I} \\
& \dots & \\
\end{bmatrix} \tag{A15}
$$

$$
\begin{bmatrix}
\begin{bmatrix}
\mathbf{K}\_{10,8} & \mathbf{K}\_{10,9} & \mathbf{K}\_{10,10} & \mathbf{K}\_{10,11} \\
\end{bmatrix}
\end{bmatrix} = \begin{bmatrix}
\begin{bmatrix}
\boldsymbol{\mu}\_{\mathrm{VI}}\mathbf{P}^{\mathrm{V}}\boldsymbol{\lambda}^{\mathrm{V}} & -\boldsymbol{\mu}\_{\mathrm{VI}}\mathbf{P}^{\mathrm{V}}\boldsymbol{\lambda}^{\mathrm{V}}
\end{bmatrix}
\begin{bmatrix}
\frac{\boldsymbol{\mathcal{R}}\_{\mathrm{S}}}{\boldsymbol{\mathcal{R}}\_{\mathrm{S}}}
\end{bmatrix}
\begin{bmatrix}
\frac{\boldsymbol{\mathcal{R}}\_{\mathrm{S}}}{\boldsymbol{\mathcal{R}}\_{\mathrm{S}}}
\end{bmatrix}
\begin{bmatrix}
\mathbf{K}\_{1}
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
\mathbf{K}
\end{bmatrix}
\end{bmatrix}
\tag{A16}
$$

$$
\begin{bmatrix}
\begin{bmatrix}
\mathbf{K}\_{1}\boldsymbol{\omega}\dots\mathbf{K}\_{10} & \mathbf{K}\_{10,10} & \mathbf{K}\_{10,10}
\end{bmatrix}
\end{bmatrix} = \begin{bmatrix}
\begin{bmatrix}
\mathbf{I}\_{1}
\end{bmatrix}
\begin{bmatrix}
\mathbf{K}
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
\mathbf{K}
\end{bmatrix}
\mathbf{J}^{\mathrm{V}\mathrm{I}}
\end{bmatrix}
\begin{bmatrix}
\begin{bmatrix}
\mathbf{K}
\end{bmatrix}
\end{bmatrix}
\tag{A17}
$$

$$\begin{bmatrix} \mathbf{K}\_{11,10} & \mathbf{K}\_{11,11} & \mathbf{K}\_{11,12} & \mathbf{K}\_{11,13} \end{bmatrix} = \begin{bmatrix} \mathbf{I} & \begin{pmatrix} \frac{R\varsigma}{R} \end{pmatrix}^{\vert \mathbf{K} \vert} & -\mathbf{P}^{VII} \begin{pmatrix} \frac{R\varsigma}{R} \end{pmatrix}^{\lambda^{\prime III}} & -\mathbf{P}^{VIII} \end{bmatrix} \tag{A17}$$

$$\begin{bmatrix} \begin{array}{cccc} \mathbf{K}\_{12,10} & \mathbf{K}\_{12,11} & \mathbf{K}\_{12,12} & \mathbf{K}\_{12,13} \end{bmatrix} \end{bmatrix} = \begin{bmatrix} \begin{array}{cccc} \mu\_{\text{Opra},0}|\mathbf{K}| & \mu\_{\text{Opra},0}|\mathbf{K} \Big(\frac{\mathbf{R}\_{\text{S}}}{\mathbf{R}\_{\text{S}}}\Big)^{|\mathbf{K}|} & -\mu\_{VI}\mathbf{P}^{VII}\lambda^{VI}\Big(\frac{\mathbf{R}\_{\text{S}}}{\mathbf{R}\_{\text{T}}}\Big)^{\lambda^{VI}} & \mu\_{VI}\mathbf{P}^{VII}\lambda^{VII} \end{bmatrix} \end{bmatrix} \tag{A18}$$

$$
\begin{bmatrix}
\begin{bmatrix}
\mathbf{K}\_{13,12} & \mathbf{K}\_{13,13} & \mathbf{K}\_{13,14} & \mathbf{K}\_{13,15}
\end{bmatrix}
\end{bmatrix} =
\begin{bmatrix}
\mathbf{p}^{\text{VII}}\lambda^{\text{VIII}} & \mathbf{p}^{\text{VIII}}\left(\frac{\mathbf{R}\_{8}}{\mathbf{R}\_{7}}\right)^{\lambda^{\text{II}}} & -\mathbf{p}^{\text{VIII}}\left(\frac{\mathbf{R}\_{7}}{\mathbf{R}\_{8}}\right)^{\lambda^{\text{III}}} & -\mathbf{p}^{\text{VIII}}
\end{bmatrix}
\tag{A19}
$$

$$
\begin{bmatrix}
\begin{array}{cccc}
\mathbf{K}\_{112} & \mathbf{K}\_{113} & \mathbf{K}\_{114} & \mathbf{K}\_{115}
\end{bmatrix}
\end{bmatrix} -
\begin{bmatrix}
\begin{bmatrix}
\mathbf{K}\_{12} & \mathbf{p}^{\text{III}}\lambda^{\text{III}} & \mathbf{p}^{\text{III}}\lambda^{\text{III}}
\end{bmatrix}
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
\begin{array}{cccc}
\mathbf{k}^{\text{VII}} & \mathbf{k}^{\text{VIII}} \\
\mathbf{k}^{\text{II}}
\end{bmatrix}
\end{bmatrix}
\tag{A19}
$$

$$\begin{bmatrix} \mathbf{K}\_{4\cup 2} & \mathbf{K}\_{4\cup 3} & \mathbf{K}\_{4\cup 4} & \mathbf{K}\_{4\cup 5} \end{bmatrix} - \begin{bmatrix} \boldsymbol{\mu}\_{\text{Sat}\beta} \mathbf{P}^{\text{VII}} \boldsymbol{\lambda}^{\text{VII}} & -\boldsymbol{\mu}\_{\text{Sat}\beta} \mathbf{P}^{\text{III}} \boldsymbol{\lambda}^{\text{III}} \left(\frac{\mathbf{R}}{\mathbf{R}}\right)^{\mathbf{A}^{\text{IV}}} & -\boldsymbol{\mu}\_{\text{Opm}\beta} \mathbf{P}^{\text{III}} \boldsymbol{\lambda}^{\text{III}} \left(\frac{\mathbf{R}}{\mathbf{R}}\right)^{\mathbf{A}^{\text{IV}}} & \boldsymbol{\mu}\_{\text{Opm}\beta} \mathbf{P}^{\text{III}} \boldsymbol{\lambda}^{\text{III}} \end{bmatrix} \tag{A20}$$

$$\begin{bmatrix} \mathbf{K}\_{15,14} & \mathbf{K}\_{15,15} & \mathbf{K}\_{15,16} & \mathbf{K}\_{15,17} \end{bmatrix} = \begin{bmatrix} \mathbf{P}^{\text{VIII}} & \mathbf{P}^{\text{VIII}} \left(\frac{\mathbf{R}}{\mathbf{R}}\right)^{\mathbf{A}^{\text{III}}} & -\left(\frac{\mathbf{R}\_{\mathbf{q}}}{\mathbf{R}\mathbf{q}}\right)^{\mathbf{K} \mathbf{I}} & -\mathbf{I} \end{bmatrix} \tag{A21}$$

$$\begin{bmatrix} \begin{bmatrix} \mathbf{K}\_{16,14} & \mathbf{K}\_{16,15} & \mathbf{K}\_{16,16} & \mathbf{K}\_{16,17} \end{bmatrix} \end{bmatrix} = \begin{bmatrix} \begin{array}{cc} \boldsymbol{\mu}\_{\text{IX}} \mathbf{P}^{\text{VII}} \boldsymbol{\lambda}^{\text{VIII}} & -\boldsymbol{\mu}\_{\text{IX}} \mathbf{P}^{\text{VIII}} \boldsymbol{\lambda}^{\text{VIII}} \left(\frac{\boldsymbol{\mu}\_{\text{7}}}{\boldsymbol{\lambda}\_{6}}\right)^{\text{IX}\text{II}} & \boldsymbol{\mu}\_{\text{Sil},0} \|\mathbf{K}\left(\frac{\boldsymbol{\mu}\_{\text{8}}}{\boldsymbol{\lambda}\_{6}}\right)^{\text{IK}} & \boldsymbol{\mu}\_{\text{Sil},0} \|\mathbf{K}\| \end{bmatrix} \end{bmatrix} \tag{A.22}$$

$$\begin{bmatrix} \mathbf{K}\_{17,16} & \mathbf{K}\_{17,17} & \mathbf{K}\_{17,18} \end{bmatrix} = \begin{bmatrix} \mathbf{I} & -\begin{pmatrix} \frac{R\_8}{R\_9} \end{pmatrix}^{\vert \mathbf{K} \rangle} & -\mathbf{I} \end{bmatrix} \tag{A23}$$

$$\begin{bmatrix} \begin{array}{cc} \mathbf{K}\_{18,16} & \mathbf{K}\_{18,17} & \mathbf{K}\_{18,18} \end{array} \end{bmatrix} = \begin{bmatrix} \begin{array}{cc} \boldsymbol{\mu}\_{\boldsymbol{X}} & -\boldsymbol{\mu}\_{\boldsymbol{X}} \begin{pmatrix} \frac{\boldsymbol{R}\_{8}}{\mathbf{K}\_{9}} \end{pmatrix}^{|\mathbf{K}|} & \boldsymbol{\mu}\_{\boldsymbol{I}\boldsymbol{X}} \end{bmatrix} \end{bmatrix} \tag{A24}$$

#### **References**

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).


**1. Introduction**

Caffeine (1,3,7-trimethylxanthine), a type of methylxanthine series alkaloid [1], is commonly found in coffee, tea, and soft drinks, and also exists in cocoa, chocolate, and a number of dietary supplements [2]. Caffeine is commonly taken orally in the form of coffee or tea, and 99% of it is absorbed into the bloodstream from the gastrointestinal tract, reaching peak concentrations 30–60 min after ingestion and circulation throughout the body [3]. In the USA, adults consume an average of 179 mg of caffeine daily, which is equivalent to 2 cups (100 mg/240 mL) of ground coffee [4]. Caffeine action is thought to be mediated via several mechanisms: the antagonism of adenosine receptors, the inhibition of phosphodiesterase, the release of calcium from intracellular stores, and the antagonism of benzodiazepine receptors [5]. There are also reports that caffeine changes estrogen levels in women [6]. Since estrogen has a neuroprotective or neurotrophic effect and regulates the dopamine system of the black striatum [7], estrogen regulates the effect of caffeine on the dopamine system and suggests that a complex interaction between caffeine, estrogen, and dopamine exists in the basal ganglia system [8]. Furthermore, caffeine is a stimulant for the central nervous system that can penetrate biological membranes, including the blood–brain barrier and placental barrier, and it maintains arousal function in the brain as a nonspecific potent inhibitor of the A1 and A2A adenosine receptors that promote drowsiness [9]. Caffeine also has psychostimulant effects via modulation of the dopaminergic neuron [10], contributing to an attenuated risk for depression in coffee drinkers [7].  However, excessive caffeine consumption (1~1.5 g/day) can cause caffeine poisoning

(caffeinism), which includes symptoms such as anxiety, agitation, insomnia, gastrointestinal disorders, tremors, and mental disorders [12]. Furthermore, depending on the sensitivity, in rare cases, it can also cause death [13]. Caffeine resistance and the rate of caffeine metabolism vary greatly from person to person, especially depending on the activity of the cytochrome P450 1A2 (CYP1A2) gene, encoding an enzyme that breaks down caffeine [6]. CYP1A2 is a major enzyme responsible for the metabolism of purine alkaloid (1,3,7-trimethylxanthine), a caffeine that occurs naturally in coffee beans, and plays an important role in the metabolism of estrogen and coffee [14]. Additionally, depending on endogenous and exogenous factors, the half-life of caffeine is 2 to 10 h (average 3.7 h), and is mainly excreted by urine after being metabolized in the liver [7]. Accumulating evidence has shown sex-specific differences in the activity and expression of many CYP isoforms [15–17]. Recent studies demonstrate that CYP1A2 and CYP2E1 activities are higher in men than in women, while the activity of CYP3A, one of the most clinically relevant CYP isoforms, is greater in women [15]. The activity of several other CYP (CYP2C16, CYP2C19, and CYP2D6) isozymes and the conjugation (glucuronidation) activity involved in drug metabolism are higher in men than in women [17]. According to the World Health Organization (WHO), about 6.8 million people worldwide die each year from various neurological and psychiatric disorders, including stroke, Alzheimer's disease (AD), Parkinson's disease (PD), and depression. Neurological and psychiatric disorders are not only expensive to treat, but patients have experienced serious stigma, social exclusion, and poor quality of life as a result of their affliction [18]. Over the years, caffeine has been investigated as a potential risk or protective factor for neurological and psychiatric disorders [19]. Some studies have shown that by drinking more than three cups of coffee a day, caffeine reduces the risk of developing AD and PD [20]. Meanwhile, the risk of developing anxiety and panic disorder has been reported to increase after consumption of more than six cups of coffee a day [21]. Interestingly, there are significant sex/gender differences in the prevalence or incidence of neurological and psychiatric disorders. Moreover, a recent study found that individuals with reduced CYP2D6 activity due to the mutated CYP2D6 \* 4A (allelic variants of CYP2D6) genotype had a 2.5 times higher risk of PD than those with wild type, which was higher in men [22]. In addition, the presence of high-risk alleles in both CYP17 and CYP19 increased the risk of AD in menopausal women by almost four times [23,24]. From these studies, it has been suggested that the effects of caffeine on the prevalence/incidence of neurological and psychiatric disorders may vary depending on sex, but it is still not thoroughly understood. In the present review, we first review the sex differences in the prevalence and/or incidence of several neurological and psychiatric disorders. Further, this review summarizes the effect of caffeine intake as a risk or protection factor for these disorders in men and women.

### **2. Sex Di**ff**erences in the Prevalence**/**Incidence of Neurological and Psychiatric Disorders**

Millions of people worldwide are affected by neurological and psychiatric disorders. More than six million people die each year from stroke, and there are 7.7 million new cases of dementia each year [25]. The prevalence/incidence of several neurological and psychiatric disorders in men and women are discussed in the following sections and summarized in Table 1.

#### *2.1. Stroke*

Stroke is a disease in which the vessels that supply blood to the brain develop abnormalities and suddenly cause local brain dysfunction, accompanied by various neurological deficits such as consciousness disorders, unilateral paralysis, and/or speech disorders [26]. There are two main types of stroke: ischemic stroke, due to lack of blood flow (85%), and hemorrhagic stroke, due to bleeding (15%) [27–30]. It has been identified that the symptoms of stroke are different between men and women. Fatigue (women vs. men = 31.2% vs. 21.1%), disorientation (44.4% vs. 34.7%), and fever (12.1% vs. 5.3%) appear predominantly in women, while paresthesia (24.2% vs. 37.9%) and ataxia (61.4% vs. 74.7%) are more common in men [31]. In a systematic review by Appelros et. al, it was shown that there was high variance between age groups and countries, but on average, both the incidence and the prevalence of stroke were higher in men than in women [32]. On the other hand, mortality from stroke

was greater in women (24.7%) than in men (19.7%). A prospective study from the USA conducted on 505 patients with first ischemic stroke (ischemic stroke genetics study) found that 270 patients (55%) were men and 229 (45%) were women [33]. In their study, no sex differences were found in stroke severity, stroke subtype, or infarct size and location, but a higher percentage of mortality was shown in women [27–30,33]. In 2009, a study conducted by the Beth Israel Deaconess Medical Center in the USA analyzed 1107 inpatients aged 21 and older who were diagnosed with neuro-ischemic stroke [28]. This study revealed no difference in the prevalence of stroke between men and women, but when comparing the age of patients, women were older than men and were more likely to have heart embolism [28]. Taken together, at most ages, women have a lower or similar risk of stroke than men. However, possibly due to the longer lifespan of women, the incidence of stroke in women gradually increases with age, and as a result, mortality rates in women are higher.

NA, not analyzed; Ref., reference; AD, Alzheimer's disease; MCI, mild cognitive impairment; PD, Parkinson's disease; HCV, hepatitis C virus; MG, myasthenia gravis; MNCS, motor nerve conduction studies; CMT1X, Charcot–Marie–Tooth type 1X; ALS, amyotrophic lateral sclerosis; F, female; M, male.

#### *2.2. Sleep Disorder*

Sleep mediates changes in various physiological functions, including brain activity, breathing, and heart rate, and sufficient sleep improves attention, creativity, memory, and learning [55]. Insufficient sleep or poor sleep quality can act as a risk factor for a variety of diseases, including dementia, psychosis, and diabetes [56,57]. Prevalence of sleep disorders is high, with about 25–30% of population worldwide having some form of inadequate sleep. Sleep disorders degrade the quality of life due to secondary psychological stress as well as promoting physical illness [58]. A recent study found that there was a difference in the prevalence of sleep disorders between sexes. Insomnia, the most common type of sleep disorder, is defined as a condition where it is difficult to initiate and maintain sleeping, resulting in difficulty in early rising [59]. Regarding sex differences in the prevalence of insomnia, many studies have reported that insomnia occurs more frequently in women [37]. In the USA, insomnia diagnosis is double in women compared with that in men, and insomnia symptoms for two nights or more per week have been reported to occur in 30.5% of women and 24.5% of men. In the case of chronic insomnia, the incidence rate was higher in women (12.9%) than in men (6.2%) [36]. In particular, women over 65 have the highest risk for insomnia and have been reported to have an increasing risk with age [35,38]. One of the reasons that women are more susceptible to insomnia than men is the changes in body hormones due to menstruation and menopause [60]. This is because estrogen, an important female hormone, decreases and body symptoms such as hot flashes and sweating are caused by an imbalance of hormones in the body.

#### *2.3. Dementia*

Dementia is a pathological neurodegenerative process characterized by a gradual decrease in cognitive, memory, and functional capacity that is severe enough to affect daily functioning [61]. Other symptoms include emotional problems, speech problems, and decreased motivation [62]. AD is the most common form of dementia and most studies do not distinguish AD from all-cause dementia [63]. Global estimates on the prevalence of dementia are up to 7% of the population aged 65 and over, and in developed countries with a longer lifespan, the prevalence is slightly higher still (8–10%) [64]. According to the World Alzheimer's Report 2015, there are currently 46.8 million people with dementia worldwide, with an estimated increase to 74.7 million by 2030 and 131.5 million by 2050 [65]. Age is a major risk factor for AD, and on average, women live longer than men. However, the difference in lifespan between men and women does not fully explain why two-thirds of Alzheimer's patients are women. Even after accounting for differences in longevity, some studies have found that women are still at a higher risk [66]. Recently, sex-related differences in neuroanatomy and function are being considered in patient diagnostics, and sex can be an important factor in stratified and personalized treatment in AD patients [39]. Consistent with this finding, analysis of longitudinal data from the Alzheimer's Disease Neuroimaging Initiative cohort showed that women had greater hippocampal atrophy and faster cognitive decline in the presence of AD biomarkers (Cerebrospinal fluid levels Aβ1-42 and total tau) compared to men [67]. Similarly, a study published in 2017 showed that in dementia patients who were classified as fast progressors, there was a faster rate of dementia in women than men, even when the diagnostic biomarker levels were similar [68]. Sex-related differences and treatment responses related to disease progression after AD diagnosis were also reported [40]. According to a Mayo Clinic study on aging, the progression from mild cognitive impairment (MCI) to AD was similar in men and women in the ages of 70–79, but higher in women than men after 80 years of age. This is likely due to the difference in brain anatomy between men and women, and it is reported that men are expected to withstand more pathologies because their heads are about 10% larger and have more brain volume compared to women, a hypothesis that was supported by autopsy. At the same level of pathology, the probability of clinical diagnosis of AD was found to be significantly higher in women than in men [69]. In the Framingham Study cohort, a study conducted in individuals aged between 65 and 100 years old, incidence of AD in women was twice as high as men [41], and another study reported that two-thirds of AD patients are women [42]. Overall, women

showed higher incidence and prevalence of dementia than men, possibly due to various factors, such as longer life expectancy of women and different neuroanatomical function [42].

#### *2.4. PD*

PD is one of the neurodegenerative disorders with characteristic features, such as hand tremor, muscle stiffness, and postural instability [70]. PD is the second most frequent age-related neurodegenerative disorder, affecting about three percent of people over 65 and five percent over 85 years old [71]. The formation of the Lewy body (α-synuclein accumulation in neurons) in the stromal nigra pars compacta leads to basal ganglion circuit degeneration [46]. Patients under 40 years of age are rare, and prevalence increases with age, approaching three percent of the population over the age of 80 [72]. Increasing evidence has suggested that sex is an important factor in the development of PD. In PD, the onset age, severity, and type of symptoms vary by sex. According to several studies, the onset of PD in men occurs, on average, two years earlier than in women, and the incidence rate in men is twice as high as that in women [73]. It has also been reported that sex differences in PD are determined by the nigrostriatal dopamine system arising from genetic, environmental, and hormonal effects. Sex itself is a variable that can affect the manifestation of non-motor symptoms in PD patients [46]. Women have better cognitive performance than men in two measures: the Symbol Digit Modalities Test, a screening test for cognitive impairment, and Scales for Outcomes of Parkinson's disease-cognition, a measure of memory and learning, attention, executive function, and virtual space function [45]. Despite the higher incidence of PD in men at all ages, the difference in PD risk between men and women is reduced with age. In those aged 65 to 69, the incidence of PD was shown to be similar between men and women [44]. The reason is likely that women have a longer lifespan than men, and men are at greater risk of dying at a younger age [43]. In addition, motor improvement after deep brain stimulation is similar in men and women, but women are likely to show better improvement in daily living activities compared to men [74]. One of the reasons why the onset of PD is higher in men than in women may be due to the effect of estrogen on dopaminergic neurons and pathways in the brain [75].

#### *2.5. Depression*

Depression is a common and serious mental disorder that can have long-term consequences and affects all aspects of life. People with depression tend to feel sad, anxious, hopeless, irritable, and ashamed [76]. Severe cases of depression can lead to loss of appetite, sudden weight loss, sleeping problems, and frequent thoughts of death or suicide [77]. It is commonly comorbid with other chronic illnesses and/or mood disorders that make it a complicated disorder difficult to properly diagnose and treat [78]. Depression is more frequently experienced by women compared to men, with a peak in prevalence occurring in middle age. Gender differences in depression are known to be affected by several factors, such as biological, psychological, and environmental factors [79]. In 2018, in Canada, the THINC-integrated tool (THINC-it), a newly developed cognitive tool, was used to evaluate cognitive impairment in patients suffering from major depressive disorder (MDD). It was reported that women had a higher rate of severe depression than men [47]. Additionally, patients with chronic liver disease have a higher incidence of depression than the general population and depression is a common psychiatric comorbidity among individuals with hepatitis C virus (HCV) [80]. Studies have shown that 23% of women, but only 4.1% of men, with chronic HCV have depression. In conclusion, in those with chronic HCV infection, anxiety and depression were more common in women than in men [48]. The University of Michigan's survey center conducted a community-based study named "American Changing Lives (ACL)" that included two sets of data collected in 1986 and 1989. These data revealed that stressed women were more prone to depression than stressed men [49]. Recent evidence suggests that changes in ovarian hormone levels, especially biological factors such as decreased estrogen, may contribute to increased risk for depression in women [81].

#### *2.6. Anxiety*

Anxiety, which manifests as a sudden increase in alertness, excessive fear, and worry, is the most common mental health disorder and 1 in 9 individuals have experienced anxiety for a year. It is also known that women have a higher prevalence of anxiety than men [82,83]. Results of the State Trait Anxiety Inventory (STAT) score, a psychological inventory that determines individual anxiety and trait anxiety among healthy men and women volunteers at Utretch University Campus, confirmed that women have a high level of trait anxiety [50]. In 2017, a research team at Yale University in the USA conducted an experiment on stress-induced anxiety disorder in healthy adults between the ages of 19 and 50. Their results show that women are more susceptible to stress-induced anxiety [51]. That is, as a result of various anxiety measurement experiments, the incidence of anxiety was found likely to be higher in women than in men. Women have a higher incidence of anxiety disorders not only because they are more sensitive to the lower levels of hormones that make up the stress response, but also because women experience residual anxiety from sexual abuse/violence more often than men [84].

#### *2.7. Neuromuscular Disease*

Neuromuscular diseases are a broadly defined group of disorders that involve injury or dysfunction of the peripheral nerve or muscle and include wide variety of disorders, such as multiple sclerosis (MS), Charcot–Marie–Tooth (CMT) disease, amyotrophic lateral sclerosis (ALS), myasthenia gravis (MG), and neuropathic pain [85]. The most common of these diseases is MG, which is an autoimmune disease where the immune system produces antibodies that attach themselves to the neuromuscular junction and prevent transmission of the nerve impulse to the muscle [86]. The onset of MG occurs at any age, but significantly earlier in women than men. The incidence of MG has been reported to be significantly higher in women under age 40, but higher in men over age 50 [52]. CMT disease encompasses a group of disorders called hereditary sensory and motor neuropathies, which damage the peripheral nerves [87]. The highest prevalence of CMT disease occurs at ages 50–64, with men having a higher prevalence than women [53,88]. A study by Nivedita U. Jerath et al. reviewed the results of electrodiagnostic retrospectively in 45 women and 31 men. As a result, women with CMT1X have less severe outcomes for almost all parameters of motor nerve conduction studies (MNCS) (compound motor action potential amplitude, delay time on exercise, and conduction rate) compared to men with CMT1X [53]. ALS is a highly debilitating disease caused by progressive degeneration of motoneurons [89]. Both the incidence and the prevalence of ALS are greater in men than women. The reasons for the difference in the incidence of ALS between men and women is known as the differences in biological responses to exogenous toxins, various exposures to environmental toxins, and fundamental differences between male and female nervous systems and their ability to repair damage [54]. The prevalence/incidence of neuromuscular disease varies according to the age of men and women, but in most cases, it is higher in men than women. The difference in the incidence of ALS between men and women may be explained by differences in the biological response to exogenous toxins.

#### **3. E**ff**ect of Ca**ff**eine Consumption on the Risk for Neurological and Psychiatric Disorders in Men and Women**

According to a paper published in British Medical Journal (BMJ) in 2017, drinking 3–4 cups of coffee per day is associated with a reduced risk of various neurological disorders, including AD, PD, and depression [90]. However, few studies have been reported on sex differences in the effects of caffeine on neurological and psychiatric disorders. This review investigates the differential effects of caffeine on the incidence of symptoms of several neurological and psychiatric disorders in men and women (summarized in Table 2).

#### *3.1. Stroke*

According to the WHO, 15 million people worldwide suffer from stroke each year, five million of these people die, and another five million are permanently disabled [91]. The causative or protective effect of caffeine on stroke onset has been controversial, and moreover, sex differences have not been studied. In 2015, the National Health and Nutrition Examination Survey in the USA examined the association between coffee consumption and stroke in 19,994 participants (men 9374; women 10,620) over the age of 17. Multivariate analysis found that higher coffee consumption (≥3 cups/day) reduced the incidence of stroke [92]. In 2017, data from the Health Examinees study, a large, prospective, community-based cohort study, were used to analyze the association between coffee consumption and stroke. A survey of about 15,000 men and women between 40 and 69 years of age did not show any significant association between coffee consumption and stroke risk among men. However, in the case of young women, the inverse relationship between coffee consumption and stroke risk was prominent. In other words, higher coffee consumption was found to be inversely proportional to the incidence rate of stroke in women [93]. Some studies show that coffee consumption temporarily increases the risk of ischemic stroke. Mostofsky's study showed that the incidence of stroke temporarily doubled in those who drank seven or more cups of coffee per week compared with non-drinkers, but there were no gender differences [94]. In summary, the effect of coffee intake on the risk for stroke showed controversial results, but more studies have shown that women have a lower risk of stroke incidence by caffeine intake, than men. As indirect evidence of the preventive effect of coffee consumption on stroke occurrence, there are papers reporting the preventive effect of coffee consumption on the onset of diabetes by maximizing insulin sensitivity, which is a risk factor for stroke, but no differences between sexes were revealed [95].

*Nutrients* **2020**, *12*, 3080

Alzheimer's disease; PD, Parkinson's disease; MS, multiple sclerosis; ALS, amyotrophic

 lateral sclerosis.

#### *3.2. Sleep Disorder*

Caffeine overdose can delay sleep onset, reduce total sleep time, change normal sleep stages, and reduce sleep quality. Caffeine-induced sleep disorder is known as a psychiatric disorder caused by excessive caffeine consumption [120]. A double-blind cross-design study of 22 young participants (10 men, 12 women; 20–30 years old) and 25 middle-aged participants (12 men, 13 women; 40–60 years old) showed that in terms of sleep volume and efficiency, middle-aged participants in good health were more susceptible to increased caffeine doses compared to young adults [97]. In a study of 12 young and 12 middle-aged subjects who consumed one to three cups of coffee per day, caffeine intake was found to reduce sleep efficiency, sleep time, slow-wave sleep (SWS), and rapid eye movement (REM) sleep in both age groups. However, during the weekly recovery, middle-aged participants had significantly reduced sleep time and sleep efficiency compared to younger participants [98]. In addition, for 66 boys and 125 girls who consumed similar amounts of caffeine, the average sleep time decreased from 528.8 min (8.8 h) on Saturday nights, to 448.5 min on Sunday nights (7.5 h). Perhaps not surprisingly, teenagers who consumed large quantities of caffeine experienced interfered sleep [99]. Furthermore, a survey on the quality of sleep in 26 adult men and 54 women with an average daily caffeine intake of 164.9 mg, showed that 80% of respondents suffered from sleep disturbances once a week [100]. Finally, among 162 cognitively healthy Koreans aged 60–94 (85 men, 77 women), people who consumed more than 60 cups of coffee per year had a 20% lower volume of pineal parenchyma, a melatonin-producing region, than those who consumed less than 60 cups of coffee per year [101]. In summary, caffeine intake negatively affected the quality of sleep and the amount of sleep with age, with no differences seen between men and women.

#### *3.3. Dementia*

A number of studies report that caffeine consumption tends to decrease the incidence of dementia. Increased caffeine intake in white women aged 65–80 has been reported to lower the likelihood of dementia or cognitive impairment [105]. In addition, drinking a moderate amount of coffee (3~5 cups/day) lowers the incidence of dementia compared with not drinking coffee, and among coffee-drinkers, the incidence of dementia is lower in women than in men. Moreover, a later-life survey found that low consumers of coffee were more likely to develop depression (based on the Beck depression scale) compared to moderate coffee consumers [104].  A study of 587 people in a California retiree community found that those who consumed more than 200 mg of caffeine per day at the age of 90 and took extra vitamin C significantly reduced their risk for dementia [103]. Some studies have reported neuroprotective effects of caffeine by showing that women with high caffeine intake (more than three cups per day) have fewer speech retrieval and decreased spatiotemporal memory problems than women who consumed less than one cup of coffee per day [106]. The psycho-stimulating component of caffeine appears to reduce cognitive decline in women without dementia, especially in the elderly [106]. Retrospective observational studies have shown that lifetime coffee consumption tends to increase cognitive performance in aged women, but this is not the case in aged men [20]. Taken together, these studies show that caffeine intake did not help improve cognitive abilities in either men or women, although steady caffeine intake seems to reduce the risk of developing dementia for both men and women, with a greater effect in women.

#### *3.4. PD*

The possible relationship between caffeine intake and PD risk has attracted considerable attention since the early 1970s, and more and more observational studies have been conducted on this [107]. An inverse association between coffee consumption and PD risk has been found in several epidemiological studies [110,121–123]. Even though the evidence is increasing that caffeine intake can reduce the risk of PD, the number of cohort studies is still relatively small and is almost exclusively limited to the USA [109]. A meta-analysis of the Hui Qi research team found that consuming less than three cups of coffee per day significantly reduced the risk of PD, while consuming more than three cups of coffee per day did not significantly alter the risk for PD [107]. The link between caffeine consumption and the risk of developing PD was more pronounced in men than women. A study by Ascherio et al. reported that men who consumed a moderate amount of coffee had a significantly lower risk of PD than men who did not drink coffee at all [110]. For women, there is a U-shaped relationship between coffee consumption and Parkinson's disease risk, with women drinking 1–3 cups of coffee per day having the lowest risk. These results support the protective effect of moderate amounts of caffeine on Parkinson's disease risk. A cohort study conducted in the USA in 1992 found that in men, regular coffee consumption was associated with a reduced risk of PD [108]. In the case of women, the risk of PD was significantly reduced in the group with the highest caffeine intake (four or more cups per day) compared to the group with the lowest caffeine intake (less than one drink per day), but the decrease was lower than in men. According to a study published in 2000 by Ross et al., the incidence of PD observed among Japanese men participants aged 45 to 68 was two to three times higher in non-coffee drinkers than in coffee drinkers [111]. In summary, high caffeine intake is associated with a protective effect that suppresses PD incidence in men and women, significantly reduces the risk of PD in men, and only slightly reduces the risk in women. These results suggest that men and women respond differently to caffeine administration and that these gender differences may be mediated by changes in circulating steroid hormones [124].

#### *3.5. Depression*

Adequate caffeine intake has a positive effect on depression, but excessive caffeine intake can exacerbate depression by stimulating sympathetic nerves [125]. Moderate amounts of caffeine also help prevent an imbalance in brain neurotransmitters, such as serotonin and dopamine, that cause depression. The effect of caffeine intake on depression was investigated in students aged 11 to 17 years old (a total of 2307 students). As a result, consuming less than 1000 mg of caffeine per week increased the incidence of depression in girls compared to boys [115]. Unlike the above results, according to a survey conducted by the Centers for Disease Control and Prevention, the prevalence of depression decreased as caffeine intake increased. The incidence of depression among participants who drank more than two cups of coffee per day was reduced by 24% compared to those who didn't drink coffee [112]. In addition, as a result of analysis of data from the National Health and Nutrition Examination Survey conducted in 2019, it was found that the incidence of depression decreased as the amount of caffeine intake increased, but gender differences were not analyzed [114]. A 2011 cohort result from the US Nurses' Health Study found an inverse age-adjusted dose-response relationship between caffeine-containing coffee and depression risk in women. Compared with the group with the lowest caffeine consumption (<100 mg/d), the relative risk for depression was lower in the group with the highest caffeine consumption (≥550 mg/d). In other words, women who consumed more caffeine had a lower risk of depression than women who consumed less caffeine [113]. In summary, the risk of developing depression was decreased by caffeine intake to a greater extent in women than in men, and the effect of caffeine intake on depression incidence was different according to the age of women. In particular, in adolescence, caffeine decomposition ability is lower than that of adults, so the staying time in the body is relatively long and it can increase the risk of depression by inducing sleep disorders [126].

#### *3.6. Anxiety*

Generalized anxiety disorder is a serious mental illness that affects up to 6% of population in the world. Symptoms are complicated by the consequences of accompaniment with other mental disorders, such as MDD, panic disorder, and alcohol/substance abuse, resulting in worsening of symptoms and poor treatment responses [127]. Excessive caffeine can cause symptoms ranging from general anxiety to compulsive disorders [120]. However, few studies have been conducted on the incidence of anxiety in men and women, by caffeine. As a result from a cohort study of 3323 students aged 11–17 years (boys 48.5%, girl 51.5%), the effect of caffeine on anxiety was not significant in girls, but in boys, anxiety increased with caffeine intake [115]. Consistent with the above results, at the University of Valencia, Spain, a STAT test of 39 men and 60 women between 18 and 31 years of age showed that men had higher state anxiety than women [116]. As shown above, the effects of caffeine on anxiety were more pronounced in men than in women. However, very little research has been conducted to assess the association between caffeine intake and anxiety.

#### *3.7. Neuromuscular Disease*

There are only a few studies about the effect of caffeine on neuromuscular diseases, and little is known about its sex differences and mechanisms. A case-controlled study from the European ALS Consortium (EURALS Group) reported that people who drink more than one cup of coffee per day for at least six months have a lower risk of ALS than those who didn't drink coffee at all [119]. Similar findings of caffeine's impact on the risk of developing MS were found in two cohort studies conducted in the USA and Sweden in 2016 [117]. Compared to those who never had coffee, those with high coffee intakes in excess of six cups per day had a significantly reduced risk of MS in both men and women. However, a recent meta-analysis of five large cohort studies conducted in the USA showed no association between coffee consumption and ALS risk, in both males and female. Another large prospective study conducted by Massa et al. also reported no association between caffeine consumption and MS risk in white women [118].

#### **4. Conclusions**

This review has shown that the beneficial and/or risky effects of caffeine on several neurological and psychiatric disorders may vary depending on sex. In the case of stroke, caffeine intake has a greater protective effect in women than in men, and for sleep disorders, caffeine intake tends to increase the risk to a similar extent in both men and women. This review also shows that the risk for developing dementia is reduced to a greater extent in women than in men. In contrast, the protective effect of caffeine against PD was found to be greater in men than in women. Notably, in the case of anxiety and depression, the effect of caffeine on the risk for their incidence tends to be age dependent. In fact, the risk of depression has shown to decrease in adult women but not in men, while in adolescence, women have a much higher risk of depression than men. For anxiety, the risk seems to be increased primarily in adult men but not in women, while during adolescence, the risk increases in both men and women, but to a much greater extent in men. In other words, caffeine consumption not only has a positive effect of reducing the risk of stroke, dementia, and depression in women and reducing the risk of PD in men, but also has a negative effect of increasing sleep disorders and anxiety disorders in adolescence in both men and women. Moreover, there are not many research articles that analyzed individual sex/gender differences in the effect of caffeine on neurological disorders. Therefore, further studies focusing on sex/gender differences are needed to fully understand the positive and negative effects of coffee intake on neurological and psychiatric disorders in men and women, and to develop new strategies for sex-specific caffeine use.

**Author Contributions:** All authors participated in the literature review. S.G.L. and K.J.B. searched for the data. H.J.J. wrote the first draft. Y.-S.J. edited and revised the manuscript. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was supported by the Support Program for Women in Science, Engineering and Technology through the Center for Women In Science, Engineering and Technology (WISET) and funded by the Ministry of Science and ICT (No. WISET202003GI01); the Korea Health Technology R&D Project through the Korea Health Industry Development Institute (KHIDI), funded by the Ministry of Health and Welfare (HI18C0920); the Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Education (2018R1D1A1B07048729), Republic of Korea.

**Conflicts of Interest:** The authors declare no conflict of interest.