At the turn-off moment of the flyback circuit, the primary switch is turned off and the secondary diode conducts (with a voltage drop V_D). The voltage of the magnetizing inductance L_m is clamped by the output voltage, causing resonance between the primary leakage inductance L_k1, the primary capacitance C_p, and the switch’s drain-source capacitance C_ds, as shown in Figure 5 which represents the equivalent circuit during the turn-off period of the flyback circuit. Based on Figure 5, the Equation 1 for the turn-off moment can be derived. In solving for the magnetizing inductance voltage V_{Lm_off1}(t), the influence of the secondary side resistance R₂ and the magnetizing resistance R_m is neglected, assuming that the magnetizing current decreases linearly at this time.

By applying the Laplace transform to solve Equation 1, the time-domain equations for the switch capacitance voltage and magnetizing current at the turn-off moment can be obtained. Substituting these into Equation 2 allows further solution of the key circuit waveforms.

It is important to note that to completely solve Equation 1, the initial values of the switch junction capacitance voltage, the current through the switch, and the magnetizing current must be determined. At the turn-off moment, the current through the switch instantaneously transfers to the secondary side for “freewheeling,” meaning the magnetizing current remains constant (reaching its peak value I _peak) and the switch current becomes zero. The voltage across the switch junction capacitance C _ds) cannot change instantaneously and remains at zero. Therefore, at the turn-off moment, the initial values of the switch junction capacitance voltage, the current through the switch, and the magnetizing current satisfy Equation 3.

It should be noted that the oscillation frequencies corresponding to each switching process are different. Since resistance is frequency-dependent, a fixed resistance value cannot represent the actual winding resistance (such as R₁ and R₂) or core equivalent resistance (such as R_m) for each process. These values need to be estimated based on the oscillation frequency. At the turn-off moment, the resonance frequency of the primary circuit can be estimated using Equation 4. These resistances mainly act as damping during the LC resonance process. As long as their values do not affect the nature of the resonance (such as underdamped, critically damped, and overdamped resonance), they will not impact the analysis of the influence of other distributed parameters (such as transformer leakage inductance and parasitic capacitance) on the switch waveform. Therefore, the estimated value range for these resistances can be quite broad. L C 1 ⋅ d 2 d t 2 V c d s _ o f f 1 t + R C 1 ⋅ d d t V c d s _ o f f 1 t + V c d s _ o f f 1 t = V i n + V F 1 L m d d t i m _ o f f 1 t + R m ⋅ i m _ o f f 1 t = − V F 1 L C 1 = L 1 k ⋅ ( C p + C d s ) R C 1 = R 1 ⋅ ( C p + C d s ) V F 1 = n ⋅ V o + V D 1 + n 2 ⋅ L 2 k / L m (1) V c p _ o f f 1 t = V i n − V c d s _ o f f 1 t i c p _ o f f 1 t = C p d d t V c p _ o f f 1 t i s _ o f f 1 t = n ⋅ i c p _ o f f 1 t + i m _ o f f 1 t − i 1 _ o f f 1 t (2) V c d s _ o f f 1 0 = 0 d d t V c d s _ o f f 1 0 = 0 i m _ o f f 1 0 = I p e a k (3) f r 1 = 1 2 π L 1 k ⋅ ( C p + C d s ) (4)

During the discontinuous conduction period of the flyback circuit, the secondary side diode turns off, and the magnetizing inductance is no longer clamped by the output voltage. Instead, it participates in the resonance of the primary side circuit along with the primary leakage inductance L_1k, the primary winding capacitance C_p, and the switch drain-source capacitance C_ds. The resonant frequency can be calculated using Equation 5. As shown in Figure 6, the equivalent circuit during the discontinuous conduction period can be used to derive the corresponding circuit equations, as given in Equation 6. When the freewheeling time t _off of the secondary side diode is determined, the initial values of the voltages and currents across the switch and the secondary winding during this period satisfy Equation 7.

Using the Laplace transform to solve Equation 6, the time-domain equations for the voltages across the switch and the secondary winding during the discontinuous conduction period can be obtained. Substituting these time-domain equations into Equation 8 allows the key waveforms of the circuit during the discontinuous conduction period to be determined. f r 2 = 1 2 π L 1 k + L m ⋅ ( C p + C d s ) (5) L C 11 d 2 d t 2 V c d s _ o f f 2 t + R C 11 d d t V c d s _ o f f 2 t + V c d s _ o f f 2 t = V i n − L C 12 d 2 d t 2 V c s _ o f f 2 t − R C 12 d d t V c s _ o f f 2 t L C 21 d 2 d t 2 V c d s _ o f f 2 t + R C 21 d d t V c d s _ o f f 2 t + L C 22 d 2 d t 2 V c s _ o f f 2 t + R C 22 d d t V c s _ o f f 2 t + V c s _ o f f 2 t = 0 L C 11 = L 1 k + L m ⋅ ( C d s + C p ) L C 12 = L m ( C s + C j ) n L C 21 = L m ( C p + C d s ) n L C 22 = L 2 k + L m / n 2 ⋅ ( C s + C j ) R C 11 = R 1 + R m ⋅ ( C d s + C p ) R C 12 = R m ⋅ ( C s + C j ) n R C 21 = R m ⋅ ( C p + C d s ) n R C 22 = R 2 + R m / n 2 ⋅ ( C s + C j ) (6) V c d s _ o f f 2 0 = V c d s _ o f f 1 t o f f d d t V c d s _ o f f 2 0 = d d t V c d s _ o f f 1 t o f f V c s _ o f f 2 0 = V c s _ o f f 1 t o f f d d t V c s _ o f f 2 0 = d d t V c s _ o f f 1 t o f f (7) V c p _ o f f 2 t = V i n − V c d s _ o f f 2 t V D _ o f f 2 t = V c s _ o f f 2 t − V o i 1 _ o f f 2 t = C d s d d t V c d s _ o f f 2 t i s _ o f f 2 t = ( C s + C j ) d d t V c s _ o f f 2 t (8)

During turn on moment, the switch can be approximately represented as a conducting resistance R_on, and in view of its small value, it can be shorted. At this moment, to prevent a sudden change in the voltage across the primary parasitic capacitance of the transformer, a lead inductance L_1w, which can be extracted by PCB simulation, is introduced for analysis. As shown in Figure 7, this is the equivalent circuit at the turn-on moment. The charging current also exhibits a steep spike due to the rate of change of the V_cp potential approaching a sudden change. Additionally, since the magnetizing inductance is large at the turn-on instant and the magnetizing current remains essentially constant, the current changes occurring in the circuit can be considered not to flow through the magnetizing loop. Therefore, the magnetizing inductance is effectively disconnected, which is why it is not reflected in the equivalent circuit shown in Figure 7.

Based on Figure 7, the Equation 9 can be derived. When the duration t_d of the discontinuous stage is determined, the initial values of the voltage and current across the primary and secondary winding parasitic capacitors at the moment of turn-on satisfy Equation 10. By applying the Laplace transform to solve Equation 9, the time-domain equations for the port voltages (i.e., parasitic capacitor voltages) of the primary and secondary windings at the moment of turn-on can be obtained. Substituting these into Equation 11 allows further solution of other key waveforms in the flyback circuit at this moment.

At the turn-on moment, a peak in the primary current occurs, and the secondary side also experiences voltage and current oscillations. However, at this time, the oscillation frequencies of the primary and secondary sides are different. Their resonance frequencies are determined by their circuit parameters respectively and can be estimated using Equation 12, where f_rp represents the primary resonance frequency and f_rs represents the secondary resonance frequency. L 1 w d d t i 1 _ o n 1 t + V c p _ o n 1 t = V i n L 2 k d d t i s _ o n 1 t + R 2 ⋅ i s _ o n 1 t + V c s _ o n 1 t + V L m _ o n 1 t n = 0 V L m _ o n 1 t = L 1 k n d d t i s _ o n 1 t + R 1 n i s _ o n 1 t + V c p _ o n 1 t i c p _ o n 1 t = i 1 _ o n 1 t + i s _ o n 1 t n i s _ o n 1 t = ( C s + C j ) d d t V c s _ o n 1 t (9) V c p _ o n 1 0 = V c p _ o f f 2 t d d d t V c p _ o n 1 0 = d d t V c p _ o f f 2 t d V c s _ o n 1 0 = V c s _ o f f 2 t d d d t V c s _ o n 1 0 = d d t V c s _ o f f 2 t d (10) i s _ o n 1 t = ( C s + C j ) ⋅ d d t V c s _ o n 1 t i 1 _ o n 1 t = C p d d t V c p _ o n 1 t − i s _ o n 1 t n V D _ o n 1 t = V c s _ o n 1 t − V o V c d s _ o n 1 t = 0 (11) f r p = 1 2 π L 1 w ⋅ C p f r s = 1 2 π L 2 k + L 1 k n 2 ⋅ ( C s + C j ) (12)

This article suggests that during the conduction period, the voltage of primary-side capacitor V_cp is clamped by the input voltage, and the magnetizing current can be considered to steadily increase under constant voltage excitation, while the secondary-side current is zero. Therefore, the equivalent circuit during the conduction period needs to consider the influence of the magnetizing loop, as shown in Figure 8. Based on this, the corresponding circuit Equation 13 can be derived, which can be simplified to Equation 14. In DCM mode, it can be assumed that the initial value of the primary current i_{1_on2} (0) is zero. However, when the oscillation time t_rn at the moment of turn-on is determined, Equation 15 can also be used to estimate i_{1_on2} (0). L 1 k d d t i L 1 k _ o n 2 t + R 1 ⋅ i L 1 k _ o n 2 t + L m d d t i m _ o n 2 t + R m ⋅ i m _ o n 2 t = V i n i 1 _ o n 2 t = i c p _ o n 2 t + i L 1 k _ o n 2 t i m _ o n 2 t = i L 1 k _ o n 2 t + i s _ o n 2 t n (13) R 1 + R m ⋅ i 1 _ o n 2 t + L 1 k + L m ⋅ d d t i 1 _ o n 2 t = V i n (14) i 1 _ o n 2 0 = i 1 _ o n 1 t r n (15)

The previous sections have introduced the circuit equations and their solution methods for each switching process in the flyback circuit, which can be used to construct the main waveform characteristics of the circuit during each switching process. However, to fully draw the switch waveforms for the entire cycle, it is necessary to determine the duration of each switching process.

When analyzing the duration of each switching process, this article estimates using ideal primary and secondary current waveforms, as shown in Figure 9, ignoring the influence of winding resistances R₁ and R₂ on the switch waveforms. Although in the DCM mode of the flyback circuit, the circuit operation can be divided into four states: turn-off moment, discontinuous period, turn-on moment, and conduction period, considering that the waveform oscillation at the turn-on moment is relatively short, this article ignores its time proportion during analysis. Therefore, in Figure 9, only three time periods are shown: the conduction time ton, the diode freewheeling time t_off, and the discontinuous time t_d.

Given the output power P_o and output voltage V_o, the average output current I_o can be calculated using Equation 16. According to Figure 9, Io also satisfies Equation 17. During the period of t_off, when ignoring the effect of R₂, Equation 18 can be derived. During the period of t_on, when ignoring the effect of R₁, Equation 19 can be derived. The input peak current I_pk and the output peak current I_sk satisfy the rule of ampere-turns balance, as shown in Equation 20. By combining Equations 16, 20, the conduction time t_on and the output diode freewheeling time t_off can be determined, as shown in Equation 21. The discontinuous time t_d is then equal to the switching cycle minus t_on and t_off, as shown in Equation 22.

Based on the above analysis and combining the time-domain equations for each stage, the required circuit waveforms for the entire cycle can be written. Taking the voltage of the primary-side MOSFET and the secondary-side diode as examples, their equations for the entire cycle can be represented by Equations 23, 24, respectively. I o = P o V o (16) I o = I s k ⋅ t o f f 2 ⋅ T s (17) L 2 k + L m n 2 ⋅ I s k t o f f = V o + V D (18) L 1 k + L m ⋅ I p k t o n = V i n (19) n ⋅ I p k = I s k (20) t o f f = 2 ⋅ P o ⋅ T s ⋅ L 2 k + L m / n 2 V o ⋅ V o + V D t o n = 2 ⋅ I o ⋅ T s ⋅ L 1 k + L m n ⋅ V i n ⋅ t o f f (21) t d = T s − t o n − t o f f (22) V c d s t = V c d s _ o f f 1 t 0 ≤ t ≤ t o f f V c d s _ o f f 2 t t o f f ≤ t ≤ t o f f + t d 0 t o f f + t d ≤ t ≤ T s (23) V D t = V D _ o f f 1 t 0 ≤ t ≤ t o f f V D _ o f f 2 t t o f f ≤ t ≤ t o f f + t d V D _ o n 1 t t o f f + t d ≤ t ≤ t o f f + t d + t r n V D _ o n 2 t t o f f + t d + t r n ≤ t ≤ T s (24)

To specifically draw the switch waveforms, it is accepted to assume the following parameters: an output power P_o of 28 W, an input voltage V_in of 380 V, an output voltage V_o of 12 V, a switching frequency f_s of 65 kHz, primary and secondary winding turns ratio of 40:10, a load resistance R_o of 5.1Ω, a secondary-side diode forward voltage V_D of 0.5 V, and relevant distributed parameters as shown in Table 1.

According to the analysis in Section 3-1, the values of R₁, R₂, and R_m are affected by frequency. Based on the given parameters, the resonant frequency f_r1 of the primary circuit at the turn-off moment can be calculated as 5.18 MHz. Accordingly, it is assumed that at this time, R₁ = 30Ω, R_m = 2Ω, and R₂ = 5Ω. During the discontinuous period, the primary resonant frequency f_r2 is 741 kHz, and it is assumed that at this time, R₁ = 10Ω, R_m = 1Ω, and R₂ = 3Ω. At the turn-on moment, it is assumed that R₁ = 30Ω, R_m = 30Ω, R₂ = 5Ω, L_1w = 10nH, and R_1w = 5Ω. During the conduction period, it is assumed that R₁ = 30Ω, R_m = 30Ω, and R₂ = 1Ω.

Based on the given parameters and combining Equations 23, 24, the complete cycle waveforms of the switch transistor’s drain-source voltage and the secondary-side diode voltage can be plotted, as shown in Figure 10.

Based on the analysis in Section 3, different primary current waveforms corresponding to various distributed parameters can be constructed and subjected to FFT decomposition to obtain the corresponding noise spectrum, as shown in Figure 11. It can be concluded that an increase in capacitance C_p, C_ds, and primary leakage inductance L_1k will worsen the differential-mode noise in the low-frequency range but improve it in the high-frequency range. It is observed that due to the smaller value of C_p, its impact is much less than that of C_ds. Moreover, as C_p, C_ds, and L_1k increase, the frequency corresponding to the noise peaks in the mid-to-high frequency range of differential-mode noise decrease. This undoubtedly requires the filter to enhance its low-frequency filtering performance, increasing the difficulty of filter design. The distributed parameters L_2k and C_s of the secondary circuit have essentially no effect on differential-mode noise.

Similarly, by constructing the primary switch transistor voltage waveforms and secondary diode voltage waveforms corresponding to different distributed parameters, and performing FFT decomposition to obtain the corresponding common-mode noise spectrum, as shown in Figures 12, 13, the following conclusions can be drawn:

(1) Due to changes in the secondary winding leakage inductance L_2k and parasitic capacitance C_s, the FFT results of V_cds and V_D remain essentially unchanged. Therefore, the secondary circuit parameters have essentially no effect on common-mode noise.