# Equations in TEX

List of Equations that are in TEX Format. Not all equations are listed since some are in MathML and some are images.

$$start$$

$$\frac{H}{H_{constant }}=1+0.033 \cos \left(\frac{360(n-2)}{365}\right)$$

$$B=\frac{360}{365}(d-81)$$

$$T C=4(\text {Longitude}-LSTM)+EoT$$

$$LST=L T+\frac{T C}{60}$$

$$HRA=15^{\circ}(LST-12)$$

$$sin\ \delta=\sin{\left({-23.45}^0\right)}\cos{\left[\frac{360}{365}\left(d+10\right)\right]}$$

$$S_{horizontal}=S_{incident} \sin \alpha$$

$$S_{module}=S_{incident} \sin (\alpha+\beta)$$

$$\alpha=90-\phi+\delta$$

$$\delta=23.45^{\circ} \sin \left[\frac{360}{365}(284+d)\right]$$

$$S_{module}=\frac{S_{Shorizontal} \sin (\alpha+\beta)}{\sin \alpha}$$

$$n_{0} p_{0}=n_{i}^{2}$$

$$\text { n-type: } n_{0}=N_{D}, p_{0}=\frac{n_{i}^{2}}{N_{D}}$$

$$\text { p-type: } p_{0}=N_{A}, n_{0}=\frac{n_{i}^{2}}{N_{A}}$$

$$\alpha=\frac{4 \pi k}{\lambda}$$

$$I=I_{0} e^{-\alpha x}$$

$$G=\alpha N_{0} e^{-\alpha x}$$

$$\tau=\frac{\Delta n}{R}$$

$$\frac{1}{\tau_{\text {bulk}}}=\frac{1}{\tau_{\text {Band}}}+\frac{1}{\tau_{\text {Auger}}}+\frac{1}{\tau_{S R H}}$$

$$\tau_{\text {Auger}}=\frac{1}{C N_{A}^{2}}$$

$$L=\sqrt{D \tau}$$

$$I=I_{0}\left(e^{\frac{q V}{k T}}-1\right)$$

$$I=I_{0}\left(e^{\frac{q V}{n k T}}-1\right)$$

$$\frac{d n}{d t}=\frac{1}{q} \frac{d J_{n}}{d x}-(U-G)$$

$$\frac{d p}{d t}=-\frac{1}{q} \frac{d J_{p}}{d x}-(U-G)$$

$$\frac{1}{q} \frac{d J_{n}}{d x}=U-G$$

$$\frac{1}{q} \frac{d J_{p}}{d x}=-(U-G)$$

$$\frac{1}{q} \frac{d J_{n}}{d x}=U-G \quad \frac{1}{q} \frac{d J_{p}}{d x}=-(U-G)$$

$$J_{L}=q \int_{0}^{W} G(x) C P(x) d x=q \int_{0}^{W}\left[\int \alpha(\lambda) H_{0} \exp (-\alpha(\lambda) x) d \lambda\right] C P(x) d x$$

$$SR=\frac{q \lambda}{h c} QE$$

$$I=I_{0}\left[\exp \left(\frac{q V}{n k T}\right)-1\right]-I_{L}$$

$$V = \frac{n k T}{q} ln \left(\frac{I_L – I}{I_0}\right)$$

$$J_{SC}=q G\left(L_{n}+L_{p}\right)$$

$$I_{SC}=J_{SC} A$$

$$V_{OC}=\frac{n k T}{q} \ln \left(\frac{I_{L}}{I_{0}}+1\right)$$

$$V_{OC}=\frac{k T}{q} \ln \left[\frac{\left(N_{A}+\Delta n\right) \Delta n}{n_{i}^{2}}\right]$$

$$J_{0}=\frac{q}{k} \frac{15 \sigma}{\pi^{4}} T^{3} \int_{u}^{\infty} \frac{x^{2}}{e^{x}-1} d x$$

$$u=\frac{E_{G}}{k T}$$

$$FF= \frac{P_{MP}}{V_{OC}\times I_{SC}}$$

$$\frac{d(I V)}{d V}=0$$

$$V_{MP} = V_{OC} - \frac{nkT}{q}ln(\frac{q V_{MP}}{nkT}+1)$$

$$I = I_L-I_0\left[\exp\left(\frac{V}{nV_t}\right)-1\right]$$

$$P = V I_L- V I_0\exp\left(\frac{V}{nV_t}\right)$$

$$0 = I_L- I_0\exp\left(\frac{V_{MP}}{nV_t}\right)\left(1+\frac{V_{MP}}{nV_t}\right)$$

$$\frac{I_L}{I_0} =\exp\left(\frac{V_{MP}}{nV_t}\right)\left(\frac{V_{MP}}{nV_t}\right)$$

$$Y = Xe^x \Leftrightarrow X = W (Y)$$

$$\frac{V_{MP}}{nV_t} =W\left(\frac{I_L}{I_0}\right)$$

$$V_{MP} = nV_t W\left(\frac{I_L}{I_0}\right)$$

$$V_{MP} = nV_t W\left(\exp\left(\frac{V_{OC}}{nV_t}\right)\right)$$

$$R^{\prime}\left(\Omega cm^{2}\right)=\frac{V}{J}$$

$$I_{SC}(total) = I_{SC}(cell)\times M$$

$$I_{MP}(total) = I_{MP}(cell)\times M$$

$$V_{OC}(total) = V_{OC}(cell)\times N$$

$$V_{MP}(total) = V_{MP}(cell)\times N$$

$$R_{b}=\frac{\rho l}{A}=\frac{\rho_{b} W}{A}$$

$$\rho_{\square}=\frac{\rho}{t}$$

$$\rho_{\square}=\frac{1}{\int_{0}^{t} \frac{1}{\rho(x)} d x}$$

$$d P_{\text {loss}}=I^{2} d R$$

$$d R=\frac{\rho}{b} d y$$

$$I(y)=J b y$$

$$P_{loss}=\int I(y)^{2} d R=\int_{0}^{S / 2} \frac{J^{2} b^{2} y^{2} \rho_{\square} d y}{b}=\frac{J^{2} b \rho_{\square} S^{3}}{24}$$

$$P_{g e n}=J_{M P} b \frac{S}{2} V_{M P}$$

$$P_{\% \text { lost }}=\frac{P_{\text {loss}}}{P_{\text {gen}}}=\frac{\rho_{\mathrm{D}} S^{2} J_{M P}}{12 V_{M P}}$$

$$\int_{0}^{L} \frac{\left(x J_{M P} S_{f}\right)^{2} \rho_{f}}{w_{f} d_{f}} d x=\frac{1}{3} L^{3} J_{M P}^{2} S_{f}^{2} \frac{\rho_{f}}{w_{f} d_{f}}$$

$$S = \sqrt{D \left(\frac{1}{\tau_{eff}}- \frac{1}{\tau_b}\right)} \tan \left[\frac{W}{2} \sqrt{\frac{1}{D}\left(\frac{1}{\tau_{eff}}- \frac{1}{\tau_b}\right)}\right]$$

$$Azimuth=cos^{-1}[\frac{sin\delta cos\varphi-cos\delta sin \varphi cos(HRA)}{cos\alpha}]$$

$$V_{MP}=V_{OC}-\frac{nkT}{q}\ln{\left(\frac{V_{MP}}{nkT/q}+1\right)}$$

$$f\left(x\right)=a_0+\sum_{n=1}^{\infty}\left(a_n\cos{\frac{n\pi x}{L}}+b_n\sin{\frac{n\pi x}{L}}\right)$$

$$E=\frac{hc}{\lambda}$$

$$E=\frac{1.24}{\lambda\left(\mu m\right)}$$

$$\Phi=\frac{#\ of\ photons}{sec\ m^2}$$

$$H\left(\frac{W}{m^2}\right)=\Phi\times\frac{hc}{\lambda}\left(J\right)=q\Phi\frac{1.24}{\lambda\left(\mu m\right)}$$

$$F=\left(\frac{W}{m^2\mu m}\right)=q\Phi E\frac{1.24}{\lambda^2\left(\mu m\right)}=q\Phi\frac{1.24}{\lambda^2\left(\mu m\right)}=q\Phi\frac{E^2\left(eV\right)}{1.24}$$

$$H=\int_{0}^{\infty}{F(\lambda)d\lambda}=\sum_{i=0}^{\infty}{F(\lambda)\Delta\lambda}$$

$$H\left(\frac{W}{m^2}\right)=\Phi\times\frac{E\left(J\right)}{photon}=q\Phi E\left(eV\right)=q\Phi\frac{1.24}{\lambda}$$

$$F\left(\lambda\right)=\frac{2\pi h c^2}{\lambda^5\left(\exp{\left(\frac{hc}{k\lambda T}\right)}-1\right)}$$

$$H=\sigma T^4$$

$$H\left(\frac{W}{m^2}\right)=\varepsilon\sigma T^4$$

$$\lambda_p\left(\mu m\right)=\frac{2900}{T}$$

$$H_0=\frac{R_{sun}^2}{D^2}H_{sun}$$

$$\frac{H}{H_{constant}}=1+0.033\cos{\left[\frac{360\left(n-2\right)}{365}\right]}$$

$$AM=\frac{1}{\cos{\theta}}$$

$$AM=\sqrt{1+\left(\frac{s}{h}\right)^2}$$

$$AM=\frac{1}{\cos{\theta}+0.50572\left(96.07995-\theta\right)^{-1.6364}}$$

$$I_D=1.353\times{0.7}^{AM^{0.678}}$$

$$I_G=1.1I_D$$

$$LSTM={15}^0∆TUTC$$

$$EoT=9.87sin\left(2B\right)-7.53\cos{\left(B\right)}-1.5\sin{\left(B\right)}$$

$$B=\frac{360}{365}\left(d-81\right)$$

$$TC=4\left(Longitude-LSTM\right)+EoT$$

$$LST=LT+TC$$

$$HRA={15}^0(LST-12)$$

$$\delta={23.45}^0sin\left[\frac{360}{365}\left(d-81\right)\right]$$

$$\delta=\sin^{-1}{\left\{\sin{\left({23.45}^0\right)}\sin{\left[\frac{360}{365}\left(d-81\right)\right]}\right\}}$$

$$Elevation=sin-1[sinδsinφ+cosδcosφcos(HRA)]$$

$$Sunrise=12-\frac{1}{{15}^0}\cos^{-1}{\left(\frac{-\sin{\varphi}\sin{\delta}}{\cos{\varphi}\cos{\delta}}\right)-\frac{TC}{60}}$$

$$Sunset=12+\frac{1}{{15}^0}\cos^{-1}{\left(\frac{-\sin{\varphi}\sin{\delta}}{\cos{\varphi}\cos{\delta}}\right)-\frac{TC}{60}}$$

$$Sunrise=12-\frac{1}{{15}^0}\cos^{-1}{\left(\frac{-\sin{\varphi}\sin{\delta}}{\cos{\varphi}\cos{\delta}}\right)}$$

$$Sunset=12+\frac{1}{{15}^0}\cos^{-1}{\left(\frac{-\sin{\varphi}\sin{\delta}}{\cos{\varphi}\cos{\delta}}\right)}$$

$$P_{MAX} = V_{OC} \times I_{SC} \times FF$$

$$V_{corrected} = V_{measured} + 0.0022 \times N \times (T_{measured} - 25)$$

$$R_{load} = \frac{V_{MP}}{I_{MP}}$$

$$FF = \frac{P_{MAX}}{V_{OC} \times I_{SC}}$$

$$E=E_{clearsky} \times \frac{100 - \alpha}{100}$$