DC to AC Converters

Comprehensive Guide to Inverter Circuits and Analysis

1. Introduction to Inverters

An inverter is a power electronic circuit that converts DC (Direct Current) power into AC (Alternating Current) power. Inverters are essential in applications such as UPS systems, motor drives, renewable energy systems, and induction heating.

Classification of Inverters

  • Voltage Source Inverter (VSI): DC voltage source with small impedance
  • Current Source Inverter (CSI): DC current source with high impedance
  • Single-phase Inverters: For low power applications
  • Three-phase Inverters: For high power and industrial applications

2. Single-Phase Half-Bridge Inverter

The half-bridge inverter uses two switches and a center-tapped DC source. The switches operate alternately to produce an AC output.

Output Voltage

Instantaneous output voltage:

$$V_o = \frac{V_{dc}}{2} \text{ (when S1 is ON)}$$ $$V_o = -\frac{V_{dc}}{2} \text{ (when S2 is ON)}$$

RMS Output Voltage

$$V_{o(rms)} = \frac{V_{dc}}{2}$$

Fourier Series

$$V_o(t) = \sum_{n=1,3,5...}^{\infty} \frac{2V_{dc}}{n\pi} \sin(n\omega t)$$

For fundamental component (n=1):

$$V_{o1} = \frac{2V_{dc}}{\pi}$$

3. Single-Phase Full-Bridge Inverter

The full-bridge inverter uses four switches arranged in an H-bridge configuration, providing better voltage utilization than the half-bridge.

Output Voltage

Instantaneous output voltage:

$$V_o = V_{dc} \text{ (when S1, S2 are ON)}$$ $$V_o = -V_{dc} \text{ (when S3, S4 are ON)}$$

RMS Output Voltage

$$V_{o(rms)} = V_{dc}$$

Fourier Series Expansion

$$V_o(t) = \sum_{n=1,3,5...}^{\infty} \frac{4V_{dc}}{n\pi} \sin(n\omega t)$$

Fundamental component:

$$V_{o1(rms)} = \frac{4V_{dc}}{\pi\sqrt{2}} = \frac{2\sqrt{2}V_{dc}}{\pi} \approx 0.9V_{dc}$$
Note: The full-bridge inverter provides twice the output voltage of the half-bridge for the same DC input voltage.

4. Pulse Width Modulation (PWM) Control

PWM techniques are used to control the output voltage and reduce harmonics in the inverter output.

Single-Pulse Width Modulation

Output voltage with pulse width \(2d\):

$$V_{on} = \frac{4V_{dc}}{n\pi}\sin\left(\frac{nd}{2}\right)\cos(n\omega t)$$

Fundamental RMS voltage:

$$V_{o1(rms)} = \frac{2\sqrt{2}V_{dc}}{\pi}\sin\left(\frac{d}{2}\right)$$

Multiple-Pulse Width Modulation

For m pulses per half cycle:

$$V_{on} = \frac{4V_{dc}}{n\pi}\sum_{i=1}^{m}\cos(n\alpha_i)\sin(nd_i)$$

where \(\alpha_i\) is the center angle and \(d_i\) is the width of pulse i

Sinusoidal PWM (SPWM)

Modulation index:

$$m_a = \frac{V_{control}}{V_{carrier}}$$

Fundamental output voltage:

$$V_{o1} = m_a \cdot V_{dc} \quad \text{(for } m_a \leq 1\text{)}$$

Frequency modulation ratio:

$$m_f = \frac{f_{carrier}}{f_{reference}}$$
Note: For \(m_a > 1\), the inverter operates in overmodulation mode, and the relationship becomes nonlinear.

5. Harmonic Analysis

Total Harmonic Distortion (THD)

$$THD = \frac{\sqrt{V_{o(rms)}^2 - V_{o1(rms)}^2}}{V_{o1(rms)}} = \frac{\sqrt{\sum_{n=2}^{\infty}V_{on}^2}}{V_{o1}}$$

Alternatively:

$$THD = \sqrt{\left(\frac{V_{o(rms)}}{V_{o1(rms)}}\right)^2 - 1}$$

Distortion Factor (DF)

$$DF = \frac{V_{o1(rms)}}{V_{o(rms)}} = \frac{1}{\sqrt{1+THD^2}}$$

Harmonic Factor (HF)

$$HF_n = \frac{V_{on}}{V_{o1}}$$

For square wave output:

$$HF_n = \frac{1}{n} \quad \text{(for odd harmonics)}$$

6. Three-Phase Inverters

Three-phase inverters use six switches to generate three-phase AC output from DC input. They are widely used in motor drives and high-power applications.

180° Conduction Mode

Phase voltage (line to neutral):

$$V_{an} = \sum_{n=1,3,5...}^{\infty}\frac{2V_{dc}}{n\pi}\sin(n\omega t)$$

Line voltage (line to line):

$$V_{ab} = V_{an} - V_{bn} = \sum_{n=1,3,5...}^{\infty}\frac{2\sqrt{3}V_{dc}}{n\pi}\sin\left(n\omega t - \frac{\pi}{6}\right)$$

RMS line voltage:

$$V_{LL(rms)} = \sqrt{\frac{2}{3}}V_{dc} \approx 0.816V_{dc}$$

120° Conduction Mode

Fundamental line voltage:

$$V_{LL1(rms)} = \frac{\sqrt{6}V_{dc}}{\pi} \approx 0.78V_{dc}$$

Space Vector PWM (SVPWM)

Maximum fundamental line voltage:

$$V_{LL1(max)} = \frac{V_{dc}}{\sqrt{3}} \approx 0.577V_{dc}$$

DC bus utilization improvement over SPWM:

$$\text{Improvement} = \frac{2}{\sqrt{3}} \approx 1.15 \text{ (15% better)}$$

7. Voltage Control Methods

External Control

  • AC Voltage Control: Using autotransformer or series inductor at output
  • DC Voltage Control: Varying DC input using controlled rectifier

Internal Control

  • Pulse Width Modulation: Varying duty cycle
  • Multiple PWM: Several pulses per half cycle
  • Sinusoidal PWM: Sine-triangle comparison
  • Selected Harmonic Elimination: Specific notch angles

8. Performance Parameters

Output Power

$$P_o = V_{o(rms)} \cdot I_{o(rms)} \cdot \cos\phi$$

where \(\phi\) is the power factor angle

Efficiency

$$\eta = \frac{P_{out}}{P_{in}} = \frac{P_{out}}{P_{out} + P_{losses}} \times 100\%$$

Crest Factor (CF)

$$CF = \frac{V_{peak}}{V_{rms}}$$

For ideal sine wave: \(CF = \sqrt{2} \approx 1.414\)

Form Factor (FF)

$$FF = \frac{V_{rms}}{V_{avg}}$$

For ideal sine wave: \(FF = \frac{\pi}{2\sqrt{2}} \approx 1.11\)

9. Multilevel Inverters

Multilevel inverters generate output voltage with multiple levels, reducing harmonics and improving power quality.

Diode-Clamped (Neutral Point Clamped)

Number of voltage levels:

$$m = n + 1$$

where n is the number of DC sources

Peak output voltage:

$$V_{peak} = n \cdot V_{dc}$$

Capacitor-Clamped (Flying Capacitor)

Number of switches required:

$$N_s = 2(m-1)$$

Number of capacitors:

$$N_c = \frac{(m-1)(m-2)}{2}$$

Cascaded H-Bridge

For s H-bridge cells:

$$m = 2s + 1$$

Number of switches:

$$N_s = 4s$$

10. Current Source Inverter (CSI)

CSI uses a DC current source (typically a large inductor) and operates with controlled switches that regulate output current.

Output Current

Fourier series of output current:

$$i_o(t) = \sum_{n=1,3,5...}^{\infty}\frac{4I_{dc}}{n\pi}\sin(n\omega t)$$

Fundamental component RMS:

$$I_{o1(rms)} = \frac{2\sqrt{2}I_{dc}}{\pi}$$
Note: CSI requires capacitive filtering at output and is suitable for high power applications with high inductance loads.

11. Comparison of Inverter Types

Parameter Half-Bridge Full-Bridge Three-Phase
Number of Switches 2 4 6
Peak Output Voltage Vdc/2 Vdc 2Vdc/3
RMS Fundamental (180°) 0.45Vdc 0.9Vdc 0.78Vdc
THD (Square Wave) ~48% ~48% ~31%
Complexity Low Medium High

12. Applications of Inverters

  • Uninterruptible Power Supplies (UPS): Backup power for critical loads
  • Motor Drives: Variable frequency drives for AC motors
  • Renewable Energy Systems: Solar PV and wind energy conversion
  • Induction Heating: High-frequency inverters for heating applications
  • HVDC Transmission: DC to AC conversion for power grids
  • Electric Vehicles: Traction motor control
  • Active Filters: Harmonic compensation in power systems