## 2 Small-Signal Analysis  
The following procedure was followed to derive the small-signal model and converter transfer functions:  
1. Make assumptions
2. Define state variables
3. Write state equations for each interval of operation
4. Average the state equations over a switching cycle
5. Introduce perturbation in state variables
6. Equate AC and DC quantities and proceed with AC equations
7. Take Laplace transform
8. Prepare matrix small-signal model
9. Calculate desired transfer functions  
The following assumptions were made for the small-signal modeling of the converter: (1) all the power semiconductor devices are lossless and ideal; (2) inductors \(L_{1}\) and \(L_{2}\) are large enough to maintain constant current through them; (3) inductor \(L_{s}\) includes the leakage inductance of the HF transformer; steady-state operating waveforms are shown in Figure 2; (4) snubber/device capacitance charging and discharging intervals are short and neglected. The primary side devices \(S_{1}\) and \(S_{2}\) are operated with identical gating signals phase-shifted with each other by 180\({}^{\circ}\). The duty cycle of the primary switches is always kept above 50%. The operation during different intervals in a one-half switching cycle is explained with equivalent circuits shown in Figure 3.  
Figure 1: The zero current switching (ZCS) naturally clamped current-fed half-bridge isolated DC-DC converter. Reproduced with permission from [17], IEEE, 2020. Figure 3: _Cont._  
Figure 2: Operating waveforms of the ZCS current-fed converter shown in Figure 1. Reproduced with permission from [17], IEEE, 2020. State variables defined for the small-signal modeling of the converter are: (1) currents through the transformer leakage inductor \(i_{lk1}\) and \(i_{lk2}\); (2) input inductor current \(i_{in}\); (3) output voltage \(v_{o}\); (4) input voltage \(v_{in}\); (5) duty cycle loss \(d\). Interval 1 (Figure 3a; \(t_{1}<t<t_{2}\))  
During this interval, primary side switch \(S_{1}\) and anti-parallel body diodes \(D_{4}\) and \(D_{5}\) of secondary switches are conducting to rectify the HF AC waveform across the transformer secondary. Power is transferred to the load through an HF transformer. Transformer primary current is negative and constant. Switch \(S_{1}\) is carrying the entire input current. State equations of this interval are:  
\[L_{1}\frac{di_{L1}}{dt}=v_{in} \tag{1}\]  
\[L_{1}\frac{di_{L1}}{dt}=v_{in} \tag{2}\]  
\[C_{o}\frac{dv_{o}}{dt}=\frac{i_{e}}{n}-\frac{v_{o}}{R_{L}} \tag{3}\]  
\[L_{s}\frac{di_{Ls}}{dt}=0 \tag{4}\]  
Interval 2 (Figure 3b; \(t_{2}<t<t_{3}\))  
Primary side switch \(S_{2}\) is turned on and starts conducting. The current from \(S_{1}\) is transferred to the switch \(S_{2}\) through the transformer primary winding with a slope limited by transformer leakage inductance \(L_{s}\). State equations of this interval are:  
\[L_{2}\frac{di_{L2}}{dt}=v_{in} \tag{5}\]  
Figure 1:0 Equivalent circuits during different intervals of the operation for the waveform in Figure 2 Reproduced with permission from [17], IEEE, 2020. (**a**) \(S_{2}\) in blocking mode; (**b**) \(S_{2}\) is turned-on; (**c**) ZVS turn-on of \(S_{4}\) and \(S_{5}\); (**d**) ZCS turn-off of \(S_{1}\) and body diode conducts; (**e**) D\({}_{3}\) and D\({}_{6}\) start conduction for rectification; (**f**) \(S_{1}\) goes in blocking mode. Figure 1:1  
Interval 3 (Figure 3c; \(\rm{t_{3}<t<t_{4}}\))  
Secondary side devices \(S_{4}\) and \(S_{5}\) are turned on with ZVS. The current waveforms follow the same pattern and slope. State equations of interval 2 still hold good. Interval 4 (Figure 3d; \(\rm{t_{4}<t<t_{5}}\))  
Primary side device \(S_{1}\) turns off with ZCS, and its anti-parallel body diode starts conducting. State equations of interval 2 still hold good. Interval 5 (Figure 3e; \(\rm{t_{5}<t<t_{6}}\))  
Secondary side devices \(S_{4}\) and \(S_{5}\) are forced commutated, and anti-parallel body diodes \(D_{3}\) and \(D_{6}\) of secondary switches take over the current.