The Buck circuit is one of the most widely used DC-DC converter topologies. Its basic structure is illustrated in the figure below. This circuit operates by switching a power transistor to control the energy transfer from the input to the output, allowing for efficient voltage regulation.
When simplified, the equivalent circuit can be represented as shown in Figure 2. In this model, the output voltage appears as a pulsed waveform, with the high level corresponding to the input DC voltage and the low level at zero. The presence of diode D1 in the original circuit ensures unidirectional current flow, which is modeled here as a series diode D2 in the equivalent circuit.
### 2. Conventional Angle Analysis of the Buck Circuit
#### 2.1 Time Domain Analysis
Time domain analysis involves examining the charging and discharging behavior of the inductor and capacitor based on the input voltage’s high and low states. This method typically starts with the Continuous Conduction Mode (CCM), where the inductor current never drops to zero during the switching cycle.
In CCM mode, when the switch is on, the inductor current increases, providing energy to both the load and the capacitor. When the switch turns off, the inductor current decreases, but remains above the average load current. During this time, the capacitor continues to discharge until the next switching cycle begins. This results in a periodic waveform, as shown in the following figure:
In Discontinuous Conduction Mode (DCM), the inductor current eventually reaches zero after the switch turns off. This leads to a different waveform pattern, where the capacitor voltage gradually decreases before the next cycle begins, as seen in Figure 4.
In CCM mode, the output voltage is directly proportional to the input voltage, with the proportionality determined by the duty cycle (D). In DCM mode, the relationship becomes more complex, involving factors such as the circuit parameters, switching frequency, and duty cycle.
The specific derivation of the voltage transfer function is given below:
Where:
According to this formula, when the output is open (infinite resistance), the input voltage equals the output voltage.
#### 2.2 Phase Plane Analysis
In addition to time-domain analysis, the phase plane method offers an alternative way to understand the dynamic behavior of the circuit. By plotting the inductor current against the capacitor voltage, we can visualize the system's state transitions.
In CCM mode, the phase plane diagram shows a linear variation of the inductor current and capacitor voltage, as illustrated in Figure 5. However, in reality, the changes are not strictly linear but involve resonant effects between the inductor and capacitor.
In DCM mode, the phase plane includes a period where the inductor current drops to zero, leading to non-linear behavior. This is shown in Figure 6, where the red portion represents the change in state variables.
For both CCM and DCM modes, the graphics remain relatively simple, with the switching frequency much higher than the resonant frequency. Varying the duty cycle affects the size of the resonance, changing the position of the red area in the phase plane.
### 3. Filter Angle Analysis of the Buck Circuit
#### 3.1 Typical Second Order Filter
A second-order filter is commonly used in Buck circuits to smooth out the output voltage. The circuit structure is similar to the latter part of the Buck circuit, but with the key difference that the Buck circuit allows only one-way current flow.
The voltage transfer function of this filter is derived as follows:
The overall impedance is:
From the transfer function, it can be observed that the natural frequency corresponds to the resonant frequency. Under a fixed load, the capacitance value influences the damping coefficient of the system, affecting its response speed and overshoot. At low frequencies, the gain is 1, while at high frequencies, the attenuation is 40 dB/dec, making it effective for filtering high-frequency components.
#### 3.2 Current Unidirectional Second Order Filter
When the filter is modified to allow only unidirectional current flow, as shown in Figure 8, it no longer functions as a traditional filter but instead acts as a rectifier circuit.
Due to the presence of the diode, current flows only in one direction. When the voltage is positive, current flows forward, and when it is negative, the current gradually decreases to zero. This creates a phase difference between the voltage and current, as shown in the phasor diagram in Figure 9.
The capacitor voltage remains positive, effectively mimicking a rectifier circuit. Comparing the AC input and unidirectional output, the inductor current and output voltage show different phases. The output voltage stabilizes to a constant DC value, as shown in Figures 10 and 11.
This system functions as a typical second-order filter. A comparison with the Buck circuit reveals that the input voltage in the Buck circuit can be viewed as a combination of a DC component and an AC component. For the DC component, the filter provides a gain of 1, and the current flows in the positive direction. The analysis focuses mainly on the AC component and its effect on the output.
In CCM mode, the inductor current is continuous, similar to a classic filter. The AC component is superimposed on the output, resulting in some ripple. This ripple reflects the influence of the AC input through the filter.
In DCM mode, the AC component is not fully filtered, leading to partial current flow. The result is similar to a one-way current filter, with the average output current influenced by both AC and DC components. This explains why the output voltage rises in DCM mode.
When the output resistance is infinite, the average current is zero, and the DC component also disappears. The output voltage then matches the input voltage, as shown in Figure 14.
### 4. Buck Circuit and Parallel Load Resonance
#### 4.1 Parallel Load Resonant Circuit
Parallel load resonant circuits can take two forms: one with a voltage source on the output side and another with a current source. The specific circuit structures are shown in Figures 15 and 16.
Both forms can be equivalently represented as shown in Figure 17.
For the circuit in Figure 15, the output is equivalent to a voltage source. During normal operation, the voltage across the resonant capacitor is a sine wave with a clipped top, and the inductor current varies sinusoidally and linearly. This makes the equivalent analysis challenging.
For the current source form, the analysis is simpler. The output side is a current source, and the inductor current is continuous. The voltage across the resonant capacitor is sinusoidal, and current flows into the rectified output as long as the capacitor voltage is not zero, as shown in Figure 18.
As long as the capacitor voltage is not zero, the rectifier diode is forward-biased, and there is no freewheeling process. The inductance and capacitance of the latter stage act as a second-order filter, similar to the output of a Buck circuit. The output is the DC component of the rectified voltage.
Since the input voltage is the absolute value of the resonant capacitor voltage, the average value is calculated as:
The current through the resistor is:
The current on the rectifier input side is:
Taking the fundamental component:
The equivalent resistance is:
With this, the relationship between the output voltage and the peak voltage of the resonant capacitor, along with the equivalent resistance, is established. This allows for the derivation of the gain curve, leading to the equivalent circuit shown in Figure 17.
#### 4.2 Buck Circuit and Parallel Load Resonance
From Figure 17, it is clear that this is a second-order filter, not the output section of the Buck circuit. Even in cases where the resonant inductor current is interrupted, it differs from the current interrupt mode of the Buck circuit. The latter half of the circuit, where the output is a constant current, is analyzed in the continuous mode of the Buck circuit.
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