## 2 Methods
### Theory Analysis of Combiner Branch
#### 2.1.4 High-Power Capability Analysis  
The high-power capability is dependent on the safe thermal accumulation, while excessive thermal accumulation is suppressed by the lower IL level, thereby enhancing power capacity. The high-power capability is primarily constrained by thermal safety considerations [18; 19].  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multicolumn{1}{c}{**Group.**} & \multicolumn{1}{c}{**Zs (\(\Omega\))**} & \multicolumn{1}{c}{**2\(\Omega\)(\({}^{\circ}\))**} & \multicolumn{1}{c}{**C (pF)**} & \multicolumn{1}{c}{**Z\({}_{0}\) (\(\Omega\))**} & \multicolumn{1}{c}{**Z\({}_{1}\) (\(\Omega\))**} & \multicolumn{1}{c}{**Angle (\({}^{\circ}\))**} & \multicolumn{1}{c}{**h (mm)**} & \multicolumn{1}{c}{**w (mm)**} & \multicolumn{1}{c}{**IL (dB)**} \\ \hline
1 & 50 & 60 & 30 & 0 & 70.7 & 100 & 90 & 2.5 & 2.50 & 0.10 \\
2 & 50 & 50 & 25 & 35 & 51 & 101 + \(\dagger\)4.3 & 75 & 2.5 & 4.55 & 0.083 \\
3 & 50 & 40 & 20 & 35 & 51 & 71 - \(\dagger\)100 & 60 & 2.5 & 4.55 & 0.071 \\
4 & 50 & 40 & 20 & 48 & 51 & 105.3 – \(\dagger\)15.8 & 60 & 2.5 & 4.55 & 0.071 \\
5 &The combiner surface temperature must be controlled below Tmax. Subsequently, the combiner surface temperature must maintain compliance with Tmax limitations, whether considering Tpeak or Tavg--specifically, Tpeak \(\leq\) Tmax and Tavg \(\leq\) Tmax. Here, Tpeak represents the transient peak temperature, while Tavg denotes the average temperature.  
The transient peak power is expressed as:  
\[P_{peak,max}=\frac{C_{total}.(T_{max}-T_{amb})}{T_{pulse}} \tag{11}\]  
The average power is defined as:  
\[P_{avg,max}=\frac{(T_{max}-T_{amb})}{R_{\phi,total}} \tag{12}\]  
where R\({}_{\phi,total}\) is the system's thermal resistance, which is relevant to the thermal cooling; in general, it is around 0.1 \({}^{\circ}\)C/W.  
To address the high-power capability, the single-branch microstrip of Table 1 was analyzed. The ambient temperature was assumed to be 20 \({}^{\circ}\)C, with a maximum operating temperature Tmax = 50 \({}^{\circ}\)C. The design could withstand a large peak power Ppeak = 5 KW at a 5% duty cycle, yielding an average power of 250 W.  
The FR4 material's properties were R0,total = 0.1 \({}^{\circ}\)C/W, C\({}_{\rm p}\) = 1, and \(\rho\) = 1.85 g/cm\({}^{3}\), and the heat capacity--Cvol--is described below:  
\[C_{Vol}=\rho.C_{p}=\frac{1.85\rm J}{\rm cm^{3}} \tag{13}\]  
The volume of the substrate of FR4 was 7.56 cm\({}^{3}\) (l \(\times\) w1 \(\times\) h: w\({}_{1}\) = 20 mm, h = 2.5 mm, and L = 2\(l_{1}\) + \(l_{2}\) = 201.5 mm); thus, the heat capacity Ctotal is as follows:  
\[C_{total}=V.C_{vol}=\frac{10.08\rm J}{\rm cm^{3}} \tag{14}\]  
The power limits were calculated using Equations (11) and (12).  
The single-branch \(P_{\rm peak,max}\) was 6.05 kW per branch. With four input ports combined into two output ports, the total peak power \(P_{\rm peak,max}\) reached 12.10 kW, while the average power \(P_{\rm avg,max}\) for the two output ports was approximately 300 W.  
Typically, a two-way combiner comprises two branches. When operating under high-power transmission conditions with a peak power of 6.05 kW (and an average power exceeding 300 W), the signal branch design can achieve approximately 6 kW high-power transmission provided that the internal temperature rise remains within 30 \({}^{\circ}\)C.