## 3 Joint Power Allocation for Co2P JDT under a Non-Unity TCPP
\[\lim_{\gamma_{2}\rightarrow\gamma_{1},P_{2}^{*}\to 0.5P_{S}}\frac{dP_{2}^{*}} {d\alpha_{2}}=\lim_{\gamma_{2}\rightarrow\gamma_{1},P_{2}^{*}\to 0.5P_{S}} \frac{d^{2}N_{T^{\prime}}/d{\alpha_{2}}^{2}}{d^{2}D_{T^{\prime}}/d{\alpha_{2} }^{2}}=\frac{N_{\beta^{\prime}}}{D_{\beta^{\prime}}}, \tag{11}\]  
with  
\[N_{\beta^{\prime}}= \left\{\left(\frac{dP_{2}^{*}}{d\alpha_{2}}\right)^{2}\frac{64}{ P_{S}^{3}}+\left(\frac{dP_{2}^{*}}{d\alpha_{2}}\right)^{2}\frac{64}{P_{S}^{4} \tilde{\gamma}_{1}}+\frac{dP_{2}^{*}}{d\alpha_{2}}\frac{16}{P_{S}^{2}}+\frac{ dP_{2}^{*}}{d\alpha_{2}}\frac{32}{P_{S}^{3}\tilde{\gamma}_{1}}+\frac{4}{P_{S}^{2} \tilde{\gamma}_{1}}\right\}\] \[\cdot e^{\frac{2}{P_{S}^{*}\gamma_{1}}}E_{1}\left(\frac{2}{P_{S} \tilde{\gamma}_{1}}\right)-\left(\frac{dP_{2}^{*}}{d\alpha_{2}}\right)^{2} \frac{16\tilde{\gamma}_{1}}{P_{S}^{2}}-\left(\frac{dP_{2}^{*}}{d\alpha_{2}} \right)^{2}\frac{32}{P_{S}^{3}}\] \[-\frac{dP_{2}^{*}}{d\alpha_{2}}\frac{16}{P_{S}^{2}}-\frac{dP_{2}^ {*}}{d\alpha_{2}}4\tilde{\gamma}_{1}^{2}-P_{S}\tilde{\gamma}_{1}^{2}+\tilde{ \gamma}_{1}-\frac{2}{P_{S}},\]  
\[D_{\beta^{\prime}}= -\left\{\left(\frac{dP_{2}^{*}}{d\alpha_{2}}\right)^{2}\frac{384} {P_{S}^{4}}+\left(\frac{dP_{2}^{*}}{d\alpha_{2}}\right)^{2}\frac{256}{P_{S}^{ 5}\tilde{\gamma}_{1}}+\frac{dP_{2}^{*}}{d\alpha_{2}}\frac{128}{P_{S}^{3}}+ \frac{dP_{2}^{*}}{d\alpha_{2}}\frac{128}{P_{S}^{4}\tilde{\gamma}_{1}}+\frac{8 }{P_{S}^{2}}+\frac{16}{P_{S}^{3}\tilde{\gamma}_{1}}\right\}\] \[\cdot e^{\frac{2}{P_{S}^{*}\gamma_{1}}}E_{1}\left(\frac{2}{P_{S} \tilde{\gamma}_{1}}\right)+\left(\frac{dP_{2}^{*}}{d\alpha_{2}}\right)^{2} \frac{128}{P_{S}^{4}}+\left(\frac{dP_{2}^{*}}{d\alpha_{2}}\right)^{2}\frac{128 \gamma_{1}}{P_{S}^{3}}-\left(\frac{dP_{2}^{*}}{d\alpha_{2}}\right)^{2}\frac{ 32\tilde{\gamma}_{1}^{2}}{P_{S}^{2}}\] \[+\frac{dP_{2}^{*}}{d\alpha_{2}}\frac{64}{P_{S}^{3}}+\frac{dP_{2}^ {*}}{d\alpha_{2}}\frac{32\tilde{\gamma}_{1}}{P_{S}^{2}}+\frac{8}{P_{S}^{2}}+2 \tilde{\gamma}_{1}^{2}.\]  
From Equation (11) and by defining \(\beta^{\prime}=dP_{2}^{*}/d\alpha_{2}|_{\tilde{\gamma}_{2}=\tilde{\gamma}_{1}, P_{2}^{*}=0.5P_{S}}\), we obtain Equation (12) of the cubic polynomial of \(\beta^{\prime}\) after factorization. \[-\frac{1}{\tilde{\gamma}_{1}}\left(\frac{4\beta^{\prime}}{P_{S}}+ 1\right)^{2}\left[\beta^{\prime}\left\{\left(\frac{24\tilde{\gamma}_{1}}{P_{S} ^{2}}+\frac{16}{P_{S}^{3}}\right)e^{\frac{2}{P_{S}^{*}\gamma_{1}}}E_{1}\left( \frac{2}{P_{S}\tilde{\gamma}_{1}}\right)+2\tilde{\gamma}_{1}^{3}-\frac{8 \tilde{\gamma}_{1}^{2}}{P_{S}}-\frac{8\tilde{\gamma}_{1}}{P_{S}^{2}}\right\}\right.\] \[\left.+\frac{4}{P_{S}^{2}}e^{\frac{2}{P_{S}^{*}\gamma_{1}}}E_{1} \left(\frac{2}{P_{S}\tilde{\gamma}_{1}}\right)-P_{S}\tilde{\gamma}_{1}^{3}- \frac{2\tilde{\gamma}_{1}}{P_{S}}+\tilde{\gamma}_{1}^{2}\right]=0. \tag{12}\]  
From the solid lines in Figure 2a-c, we note that \(\beta^{\prime}\) is a positive real value. Therefore, the valid slope of \(P_{2}^{*}\) with respect to \(\alpha_{2}\) at the balanced point, \(\beta^{\prime}=\beta^{\prime}\left(\tilde{\gamma}_{1}\right)\) is  
\[\beta^{\prime}\left(\tilde{\gamma}_{1}\right)=\frac{-\frac{4}{P_{S}^{2}}e^{ \frac{2}{P_{S}^{*}\gamma_{1}}}E_{1}\left(\frac{2}{P_{S}\tilde{\gamma}_{1}} \right)+P_{S}\tilde{\gamma}_{1}^{3}+\frac{2\tilde{\gamma}_{1}}{P_{S}}-\gamma_{1 }^{2}}{\left(\frac{24\tilde{\gamma}_{1}}{P_{S}^{2}}+\frac{16}{P_{S}^{3}} \right)e^{\frac{2}{P_{S}^{*}\gamma_{1}}}E_{1}\left(\frac{2}{P_{S}^{*}\gamma_{1}} \right)+2\tilde{\gamma}_{1}^{3}-\frac{8\tilde{\gamma}_{1}^{2}}{P_{S}}-\frac{8 \tilde{\gamma}_{1}}{P_{S}^{2}}}. \tag{13}\]  
The slope of \(P_{2}^{*}\) with respect to \(\tilde{\gamma}_{2}\)[dB] at the balanced point, \(\hat{\beta}\left(\tilde{\gamma}_{1}\right)\) is given by \(\hat{\beta}\left(\tilde{\gamma}_{1}\right)=\beta^{\prime}\left(\tilde{\gamma}_{ 1}\right)\ln 10/10=dP_{2}^{*}/d\tilde{\gamma}_{2}\)[dB]\(|_{\tilde{\gamma}_{2}=\tilde{\gamma}_{1},P_{2}^{*}=0.5P_{S}}\).