7.3 DC-to-DC Switch-Mode Converters (Buck, Boost, Buck-Boost) & DC-AC Inverters

7.3 DC-to-DC Switch-Mode Converters (Buck, Boost, Buck-Boost) & DC-AC Inverters

Executive Overview: DC-DC switch-mode converters and DC-AC inverters constitute the power conversion core for solar photovoltaic (PV) string/central inverters, Battery Energy Storage Systems (BESS), electric vehicle (EV) powertrains, and industrial Variable Frequency Drives (VFDs). On the NCEES PE Electrical: Power exam, questions focus on steady-state volt-second balance derivations, duty cycle ($D$) relationships, inductor ripple current ($\Delta I_L$), critical boundary inductance ($L_{min}$) for Continuous Conduction Mode (CCM), output voltage ripple ($\Delta V_{out}$), and inverter Sinusoidal PWM (SPWM) vs. Space Vector PWM (SVPWM) modulation index voltage scaling.

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1. Governing Principles of Switch-Mode Power Conversion

High-efficiency DC-DC conversion relies on solid-state switches (MOSFETs or IGBTs) pulsing at switching frequencies ($f_{sw} = 1/T_s$, typically $20\text{ kHz} - 500\text{ kHz}$) with duty ratio $D = \frac{t_{on}}{T_s}$. In steady-state periodic operation, energy balance is enforced by two fundamental conservation laws:

+---------------------------------------------------------------------------------------------------+
|                         STEADY-STATE ENERGY CONSERVATION LAWS                                     |
+---------------------------------------------------------------------------------------------------+
| 1. Inductor Volt-Second Balance:                                                                  |
|    $$\int_0^{T_s} v_L(t) dt = 0 \implies \langle v_L \rangle = V_{L,on} \cdot D + V_{L,off} \cdot (1 - D) = 0$$ |
|    (The average voltage across an ideal inductor over any complete switching cycle is zero).       |
|                                                                                                   |
| 2. Capacitor Charge (Ampere-Second) Balance:                                                      |
|    $$\int_0^{T_s} i_C(t) dt = 0 \implies \langle i_C \rangle = I_{C,on} \cdot D + I_{C,off} \cdot (1 - D) = 0$$ |
|    (The average current through an ideal capacitor over any complete switching cycle is zero).     |
+---------------------------------------------------------------------------------------------------+

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2. Continuous Conduction Mode (CCM) DC-DC Topologies

+---------------------------------------------------------------------------------------------------+
|                         DC-DC CONVERTER TOPOLOGY & DESIGN MATRIX (CCM)                            |
+---------------------------------------------------------------------------------------------------+
| Topology | Voltage Conversion ($V_{out}$) | Inductor Ripple Current ($\Delta I_L$) | Critical Inductance ($L_{min}$) | Voltage Ripple Ratio ($\frac{\Delta V_{out}}{V_{out}}$) |
| :---     | :---                          | :---                                  | :---                           | :---                                                     |
| **Buck** (Step-Down) | $V_{out} = D \cdot V_{in}$ | $\Delta I_L = \frac{V_{in} D (1 - D)}{f_{sw} L}$ | $L_{min} = \frac{(1 - D) R}{2 f_{sw}}$ | $\frac{\Delta V_{out}}{V_{out}} = \frac{1 - D}{8 L C f_{sw}^2}$ |
| **Boost** (Step-Up)  | $V_{out} = \frac{V_{in}}{1 - D}$ | $\Delta I_L = \frac{V_{in} D}{f_{sw} L}$ | $L_{min} = \frac{D (1 - D)^2 R}{2 f_{sw}}$ | $\frac{\Delta V_{out}}{V_{out}} = \frac{D}{R C f_{sw}}$ |
| **Buck-Boost** (Inverting) | $V_{out} = -V_{in} \frac{D}{1 - D}$ | $\Delta I_L = \frac{V_{in} D}{f_{sw} L}$ | $L_{min} = \frac{(1 - D)^2 R}{2 f_{sw}}$ | $\frac{\Delta V_{out}}{V_{out}} = \frac{D}{R C f_{sw}}$ |
+---------------------------------------------------------------------------------------------------+

1. Buck Converter (Step-Down)

                         BUCK CONVERTER CIRCUIT TOPOLOGY
                       Switch (Q1)           Inductor (L)
                +---------[ / ]--------+--------[ZZZZ]--------+--------+ (+) V_out
                |                      |                      |        |
             +  |                      |                      |        |
           (~) V_in                 +--+--+                 +---+    +---+ 
             -  |              (D1)  /\   Freewheeling      |   | C  |   | R_load
                |                   /--\  Diode             +---+    +---+ 
                |                      |                      |        |
                +----------------------+----------------------+--------+ (-) Common

• Switch ON Interval ($0 \le t \le D T_s$): $Q_1$ closed, $D_1$ reverse-biased. Voltage across inductor is $v_L = V_{in} - V_{out}$. Current ramps up linearly: $\frac{di_L}{dt} = \frac{V_{in} - V_{out}}{L}$
• Switch OFF Interval ($D T_s < t \le T_s$): $Q_1$ open, $D_1$ forward-biased by inductor inductive kick. Voltage across inductor is $v_L = -V_{out}$. Current ramps down: $\frac{di_L}{dt} = -\frac{V_{out}}{L}$
• Volt-Second Balance Derivation: $(V_{in} - V_{out}) D T_s + (-V_{out}) (1 - D) T_s = 0 \implies V_{in} D - V_{out} D - V_{out} + V_{out} D = 0 \implies \mathbf{V_{out} = D \cdot V_{in}}$

2. Boost Converter (Step-Up)

                         BOOST CONVERTER CIRCUIT TOPOLOGY
                         Inductor (L)          Diode (D1)
                +-----------[ZZZZ]-----+----------|>|---------+--------+ (+) V_out
                |                      |                      |        |
             +  |                      |                      |        |
           (~) V_in                 +--+--+                 +---+    +---+ 
             -  |                   |     | Switch (Q1)     |   | C  |   | R_load
                |                   +--+--+                 +---+    +---+ 
                |                      |                      |        |
                +----------------------+----------------------+--------+ (-) Common

• Switch ON Interval ($0 \le t \le D T_s$): $Q_1$ closed, $D_1$ reverse-biased. Inductor connects directly across source: $v_L = V_{in}$. Inductor stores magnetic energy ($\Delta I_L = \frac{V_{in} D T_s}{L}$).
• Switch OFF Interval ($D T_s < t \le T_s$): $Q_1$ open, $D_1$ conducts. Inductor discharges in series with source: $v_L = V_{in} - V_{out}$.
• Volt-Second Balance Derivation: $(V_{in}) D T_s + (V_{in} - V_{out}) (1 - D) T_s = 0 \implies V_{in} D + V_{in} - V_{out} - V_{in} D + V_{out} D = 0$ $V_{in} = V_{out} (1 - D) \implies \mathbf{V_{out} = \frac{V_{in}}{1 - D}}$

3. Buck-Boost Converter (Inverting Step-Up/Down)

• Switch ON: $v_L = V_{in}$; Switch OFF: $v_L = V_{out}$ (with reversed output polarity).
• Voltage conversion: $V_{out} = -V_{in} \frac{D}{1 - D}$.
  • When $D < 0.5$, $|V_{out}| < V_{in}$ (step-down).
  • When $D > 0.5$, $|V_{out}| > V_{in}$ (step-up).
  • When $D = 0.5$, $|V_{out}| = V_{in}$.

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3. Discontinuous Conduction Mode (DCM) & Boundary Conditions

When load current decreases (or load resistance $R$ increases) such that the minimum inductor current hits zero before the end of the switching cycle, the converter enters Discontinuous Conduction Mode (DCM).

                     INDUCTOR CURRENT MODES OF OPERATION
        Inductor Current (i_L)
           ^
           |       /\          /\          /\ 
           |      /  \        /  \        /  \       CONTINUOUS CONDUCTION MODE (CCM)
     I_min +-----+----+------+----+------+----+----+ (i_L > 0 at all times)
           |    /      \    /      \    /      \ 
           0---+--------+--+--------+--+--------+--> Time
           |
           |       /\          /\          /\ 
           |      /  \        /  \        /  \       DISCONTINUOUS CONDUCTION MODE (DCM)
           0-----+----+---+--+----+---+--+----+---+-> Time (i_L reaches zero and
                 |<-DT->|   |                       stays at zero until next cycle)

[!WARNING] DCM Voltage Regulation Loss: In DCM, the simple linear voltage conversion ratios ($V_{out} = D V_{in}$) no longer hold! The output voltage becomes a non-linear function of load resistance $R$, inductance $L$, switching frequency $f_{sw}$, and duty cycle $D$. At no load ($R \to \infty$), the output voltage in a Buck or Boost converter floats up to $V_{in}$ or dangerously high levels unless clamped by a feedback control loop.

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4. DC-to-AC Inverter Architectures

Inverters synthesize alternating AC voltage waveforms from a fixed DC source ($V_{dc}$). Industrial systems deploy single-phase H-bridges and three-phase six-switch Voltage Source Inverters (VSI).

                         THREE-PHASE VOLTAGE SOURCE INVERTER (VSI)
                                    +--- (+) DC Bus (V_dc)
                                    |
                         Q1        Q3        Q5
                      +--[ / ]--+--[ / ]--+--[ / ]--+
                      |         |         |         |
                      | Phase A | Phase B | Phase C |
                      +----+    +----+    +----+    |
                      |    |    |    |    |    |    |
                      |    +----+----+----+    |    |
                      |         |   (Motor)    |    |
                      +--[ / ]--+--[ / ]--+--[ / ]--+
                         Q4        Q6        Q2
                                    |
                                    +--- (-) DC Bus Return

Six-Step ($180^\circ$ Conduction) Inverter Mode

Each switch conducts for $180^\circ$ electrical degrees. The resulting line-to-line output voltage is a six-step waveform:

• Total RMS Line-to-Line Voltage: $V_{LL,rms} = \sqrt{\frac{2}{3}} V_{dc} \approx 0.8165 \cdot V_{dc}$
• Fundamental RMS Line-to-Line Voltage ($V_{LL1,rms}$): $V_{LL1,rms} = \frac{\sqrt{6}}{\pi} V_{dc} \approx 0.7797 \cdot V_{dc}$

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5. Sinusoidal Pulse-Width Modulation (SPWM)

To eliminate low-order harmonics and provide linear output voltage control, inverters use Sinusoidal PWM (SPWM), comparing three low-frequency reference control sinusoids ($v_{control}$) against a high-frequency triangular carrier wave ($v_{tri}$).

                          SPWM MODULATION PRINCIPLE
        Voltage
           ^
     +V_tri|   /\  /\  /\  /\  /\  /\  /\  /\  /\     High-Frequency Triangle Carrier (f_sw)
           |  /  \/  \/  \/  \/  \/  \/  \/  \/  \ 
           | /  .-'"""'-.   /  \  /  \   .-'"""'-. 
           |/  /         \ /    \/    \ /         \   Low-Frequency Sine Reference (f_fund)
     -V_tri+-----------------------------------------> Time
           |
     PWM   | +---+       +-----+     +---+       +---+
     Gate  | |   |       |     |     |   |       |   |
     Out   0-+---+-------+-----+-----+---+-------+---+-> Time

SPWM Modulation Parameters:

1. Amplitude Modulation Index ($m_a$): $m_a = \frac{V_{control,pk}}{V_{tri,pk}}$
2. Frequency Modulation Index ($m_f$): $m_f = \frac{f_{tri}}{f_{control}} = \frac{f_{sw}}{f_1}$ (For 3-phase systems, $m_f$ is chosen as an odd multiple of 3, e.g., $15, 21, 27, 33$, to cancel triplen harmonics in line-to-line voltages and eliminate even harmonics).

Linear Modulation Range ($0 \le m_a \le 1.0$)

In the linear region, the fundamental output voltage scales strictly proportionally with $m_a$:

• Fundamental Peak Line-to-Neutral Voltage: $V_{LN1,pk} = m_a \frac{V_{dc}}{2}$
• Fundamental RMS Line-to-Neutral Voltage: $V_{LN1,rms} = \frac{m_a V_{dc}}{2\sqrt{2}} \approx 0.3535 \cdot m_a V_{dc}$
• Fundamental RMS Line-to-Line Voltage ($V_{LL1,rms}$): $V_{LL1,rms} = \sqrt{3} \cdot V_{LN1,rms} = \frac{\sqrt{3}}{2\sqrt{2}} m_a V_{dc} = \frac{\sqrt{6}}{4} m_a V_{dc} \approx 0.61237 \cdot m_a V_{dc}$

Overmodulation ($1.0 < m_a \le 3.24$) and Six-Step Limit

When $m_a > 1.0$, the sine wave peaks exceed the triangle carrier, causing pulse-dropping and harmonic distortion until the output degenerates into the square-wave six-step limit ($V_{LL1,rms} = 0.7797 V_{dc}$).

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6. Space Vector PWM (SVPWM)

Space Vector PWM (SVPWM) treats the 3-phase inverter as a single unit with 8 possible discrete switching states ($V_0 [000]$ to $V_7 [111]$), forming a hexagon in the complex $\alpha-\beta$ stationary reference plane.

                    SPACE VECTOR VOLTAGE HEXAGON (SVPWM)
                                  V2 (010)
                                  (120°)
                         +-----------+-----------+
                        / \         |         / \
                       /   \ Sector | Sector /   \
                      /     \  II   |   I   /     \
                     /       \      |      /       \
           V3 (011) + Sector  \     |     / Sector  + V1 (100)
           (180°)   |  III     \    |    /    VI    | (0°)
                    |           \   |   /           |
                    |------------+-(0)-+------------| Reference Vector V_ref
                    |           /   |   \           | rotates smoothly 360°
           V4 (001) + Sector   /    |    \ Sector   + V6 (101)
           (240°)    \  IV    /     |     \   V    /
                      \      /      |      \      /
                       \    / Sector| Sector\    /
                        \  /   III  |   IV   \  /
                         +-----------+-----------+
                                  V5 (001)
                                  (300°)
                         (Zero Vectors: V0 [000], V7 [111])

The SVPWM DC-Bus Advantage

In standard SPWM, the maximum fundamental peak line-to-line voltage in the linear region is bounded by $V_{dc}$. In SVPWM, the reference vector can reach the inner circle inscribed within the hexagon without overmodulation:

$V_{ref,max} = \frac{V_{dc}}{\sqrt{3}}$

$V_{LL1,rms,max(SVPWM)} = \frac{V_{dc}}{\sqrt{2}} \approx 0.7071 \cdot V_{dc}$

$\text{Voltage Utilization Improvement Ratio: } \frac{0.7071}{0.6124} = \frac{2}{\sqrt{3}} \approx 1.1547 \implies \mathbf{+15.47\% \text{ Higher DC Bus Utilization}}$

This $15.5%$ boost allows driving standard $460\text{ V}$ rated AC motors from a lower DC bus voltage ($650\text{ V}$ instead of $751\text{ V}$), significantly lowering switch voltage ratings and inverter losses.

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7. Comprehensive Step-by-Step Worked Mathematical Example

Problem Statement

A solar PV and battery storage system utilizes a DC-DC Boost Converter feeding a 3-phase Voltage Source Inverter (VSI) to drive an industrial 3-phase water pump motor:

• Boost Converter Input Voltage: $V_{in} = 280.0\text{ V DC}$
• Boost Converter Switching Frequency: $f_{sw} = 25.0\text{ kHz}$
• Target Inverter DC Bus Voltage: $V_{dc} = 700.0\text{ V DC}$
• Full-Load DC Power Delivered: $P_o = 35.0\text{ kW}$
• Design Requirement 1: Inductor peak-to-peak ripple current must not exceed $20.0%$ of average input current ($\Delta I_L \le 0.20 I_{in}$).
• Design Requirement 2: DC bus output voltage ripple must not exceed $1.0%$ ($\frac{\Delta V_{dc}}{V_{dc}} \le 0.010$).
• The 3-phase VSI drives the motor using linear Sinusoidal PWM (SPWM) with an amplitude modulation index of $m_a = 0.850$.

Calculate:

1. The required duty cycle ($D$) of the Boost converter in CCM.
2. The average input current ($I_{in}$) and the minimum required inductance ($L$) to satisfy the $20%$ ripple limit.
3. The minimum required DC bus filter capacitance ($C$) to satisfy the $1.0%$ voltage ripple limit.
4. The fundamental line-to-line RMS voltage ($V_{LL1,rms}$) delivered to the motor terminals under SPWM.
5. The fundamental line-to-line RMS voltage ($V_{LL1,rms,SVPWM}$) if the inverter is upgraded to Space Vector PWM (SVPWM) operating at maximum linear modulation ($m_a = 1.0$).

=========================================================================================
CALCULATION WORKFLOW & DETAILED STEP-BY-STEP SOLUTION:
=========================================================================================

Step 1: Compute Required Boost Duty Cycle (D)
  For a CCM Boost converter:
    V_out = V_in / (1 - D)
    1 - D = V_in / V_out = 280.0 V / 700.0 V = 0.4000
    D = 1.0 - 0.4000 = 0.6000 (60.0% duty cycle)

Step 2: Calculate Input Current and Minimum Boost Inductance (L)
  Average DC output current:
    I_out = P_o / V_out = 35,000 W / 700.0 V = 50.00 A

  Average input inductor current (assuming 100% ideal efficiency):
    I_in = P_o / V_in = 35,000 W / 280.0 V = 125.00 A
    (Verification: I_in = I_out / (1 - D) = 50.0 A / 0.40 = 125.00 A)

  Maximum allowable peak-to-peak inductor ripple current:
    Delta_I_L = 0.20 * I_in = 0.20 * 125.00 A = 25.00 A

  Inductance formula for Boost converter:
    Delta_I_L = (V_in * D) / (f_sw * L)
    L = (V_in * D) / (f_sw * Delta_I_L)
      = (280.0 V * 0.6000) / (25,000 Hz * 25.00 A)
      = 168.0 V / 625,000 A/s
      = 0.0002688 H = 268.8 μH ≈ 269 μH

  Critical Inductance for CCM Verification:
    Equivalent load resistance: R = V_out / I_out = 700 V / 50 A = 14.0 ohms
    L_min,crit = (D * (1 - D)^2 * R) / (2 * f_sw)
               = (0.60 * (0.40)^2 * 14.0) / (2 * 25,000)
               = (0.60 * 0.16 * 14.0) / 50,000
               = 1.344 / 50,000 = 26.88 μH
    Since L = 268.8 μH >> 26.88 μH, the converter operates deeply in CCM! (CONFIRMED)

Step 3: Calculate Minimum Filter Capacitance (C)
  Output voltage ripple formula for Boost converter:
    Delta_V_out / V_out = D / (R * C * f_sw)
    C = D / (R * f_sw * (Delta_V_out / V_out))
      = 0.6000 / (14.0 ohms * 25,000 Hz * 0.010)
      = 0.6000 / 3,500
      = 0.00017143 F = 171.43 μF ≈ 172 μF

Step 4: Compute Motor Terminal Voltage with SPWM (m_a = 0.850)
  Fundamental line-to-line RMS voltage:
    V_LL1,rms = (sqrt(3) / (2 * sqrt(2))) * m_a * V_dc
              = 0.612372 * 0.850 * 700.0 V
              = 0.612372 * 595.0 V
              = 364.36 V ≈ 364.4 V

Step 5: Compute Motor Terminal Voltage with SVPWM (m_a = 1.0)
  Fundamental line-to-line RMS voltage at maximum linear SVPWM:
    V_LL1,rms,SVPWM = V_dc / sqrt(2)
                    = 700.0 V / 1.414214
                    = 494.97 V ≈ 495.0 V

  Comparison with SPWM at m_a = 1.0:
    V_LL1,rms,SPWM(m_a=1.0) = 0.612372 * 700.0 V = 428.66 V
    SVPWM Voltage Advantage: 494.97 V / 428.66 V = 1.1547 (+15.47% increase!)
=========================================================================================

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8. Common PE Exam Traps & Tactical Pitfalls

• Inverting the Duty Cycle Ratio in Boost Converters: Using $V_{out} = (1 - D) V_{in}$ instead of $V_{out} = \frac{V_{in}}{1 - D}$. Remember that as $D \to 1$, Boost output voltage approaches infinity ($V_{out} \to \infty$).
• Applying Buck Voltage Ripple Equations to Boost Converters: The Buck converter has an inductor continuously connected to the output, yielding an 8-factor second-order filter equation ($\Delta V / V = \frac{1-D}{8 L C f^2}$). The Boost converter's inductor is disconnected from the output during the ON-interval, discharging the capacitor directly into the load ($\Delta V / V = \frac{D}{R C f}$). Applying the Buck ripple formula to a Boost converter underestimates required capacitance by an order of magnitude!
• Modulation Index Voltage Scaling Confusion ($V_{LN}$ vs. $V_{LL}$): On the exam, remember:
  • Line-to-Neutral RMS: $V_{LN1,rms} = \frac{m_a V_{dc}}{2\sqrt{2}} \approx 0.3535 m_a V_{dc}$
  • Line-to-Line RMS: $V_{LL1,rms} = \sqrt{3} V_{LN1,rms} = \frac{\sqrt{6}}{4} m_a V_{dc} \approx 0.6124 m_a V_{dc}$
• Neglecting Inverting Polarity in Buck-Boost Converters: The output voltage of a classic single-switch Buck-Boost converter has a negative potential relative to the common input rail ($V_{out} = -V_{in} \frac{D}{1-D}$).