What are Inverters?

• Definition: Power electronic circuits that convert DC power to AC power

• Function: Convert fixed DC voltage to variable AC voltage with variable frequency

• Classification:

  • Voltage Source Inverters (VSI)

  • Current Source Inverters (CSI)

• Based on Output:

  • Single-phase inverters

  • Three-phase inverters

• Applications: Motor drives, UPS systems, renewable energy systems

Single-Phase Half-Bridge VSI

• Configuration: Two switches, center-tapped DC supply

• Output Voltage: \(V_o = \pm \dfrac{V_{dc}}{2}\)

• RMS Output Voltage: \(V_{o,rms} = \dfrac{V_{dc}}{2}\)

• Fundamental Component: \(V_{o1} = \dfrac{2V_{dc}}{\pi}\)

• RMS Fundamental: \(V_{o1,rms} = \dfrac{V_{dc}\sqrt{2}}{\pi}\)

• Distortion Factor: \(DF = \dfrac{V_{o1,rms}}{V_{o,rms}} = \dfrac{2\sqrt{2}}{\pi} = 0.9\)

• THD: \(THD = \sqrt{\left(\dfrac{V_{o,rms}}{V_{o1,rms}}\right)^2 - 1} = 48.34\%\)

Single-Phase Full-Bridge VSI

• Configuration: Four switches in H-bridge configuration

• Output Voltage: \(V_o = \pm V_{dc}\)

• RMS Output Voltage: \(V_{o,rms} = V_{dc}\)

• Fundamental Component: \(V_{o1} = \dfrac{4V_{dc}}{\pi}\)

• RMS Fundamental: \(V_{o1,rms} = \dfrac{2\sqrt{2}V_{dc}}{\pi}\)

• Switching Modes:

  • Mode 1: S1, S4 ON \(\to\) \(V_o = +V_{dc}\)

  • Mode 2: S2, S3 ON \(\to\) \(V_o = -V_{dc}\)

• THD: \(48.34\%\) (same as half-bridge)

Single-Phase VSI with R Load

• Load Current: \(i_o(t) = \dfrac{v_o(t)}{R}\)

• RMS Current: \(I_{o,rms} = \dfrac{V_{dc}}{R}\) (full-bridge)

• Average Power: \(P_{avg} = \dfrac{V_{dc}^2}{R}\)

• Power Factor: \(pf = 1.0\) (resistive load)

• Fundamental RMS Current: \(I_{o1,rms} = \dfrac{2\sqrt{2}V_{dc}}{\pi R}\)

Single-Phase VSI with RL Load

• Load Impedance: \(Z = \sqrt{R^2 + (\omega L)^2}\)

• Current Waveform: Exponential rise and fall

• Fundamental RMS Current: \(I_{o1,rms} = \dfrac{2\sqrt{2}V_{dc}}{\pi Z}\)

• Phase Angle: \(\phi = \tan^{-1}\left(\dfrac{\omega L}{R}\right)\)

• Power Factor: \(pf = \cos\phi = \dfrac{R}{Z}\)

• Conduction Angle: \(> 180^{\circ}\) due to inductance

• Freewheeling Required: For continuous current flow

Three-Phase VSI Configuration

• Structure: Six switches arranged in three legs

• Switching States: \(2^3 = 8\) possible states

• Valid States: 6 active states + 2 zero states

• Line Voltages: \(V_{ab}, V_{bc}, V_{ca}\)

• Phase Voltages: \(V_{ao}, V_{bo}, V_{co}\) (with respect to neutral)

• Relationship: \(V_{line} = \sqrt{3} \times V_{phase}\)

Three-Phase VSI - \(180^{\circ}\) Conduction Mode

• Conduction: Each switch conducts for \(180^{\circ}\)

• Switching Sequence: Six-step operation (\(60^{\circ}\) intervals)

• Line Voltage RMS: \(V_{L,rms} = \dfrac{2\sqrt{3}V_{dc}}{\pi}\)

• Phase Voltage RMS: \(V_{ph,rms} = \dfrac{2V_{dc}}{\pi}\)

• Fundamental Line Voltage: \(V_{L1,rms} = \dfrac{\sqrt{6}V_{dc}}{\pi}\)

• Fundamental Phase Voltage: \(V_{ph1,rms} = \dfrac{\sqrt{2}V_{dc}}{\pi}\)

Three-Phase VSI - \(120^{\circ}\) Conduction Mode

• Conduction: Each switch conducts for \(120^{\circ}\)

• Operation: Only two switches ON at any time

• Line Voltage RMS: \(V_{L,rms} = \dfrac{\sqrt{6}V_{dc}}{\pi}\)

• Phase Voltage RMS: \(V_{ph,rms} = \dfrac{\sqrt{2}V_{dc}}{\pi}\)

• Fundamental Line Voltage: \(V_{L1,rms} = \dfrac{3V_{dc}}{2\pi}\)

• Advantages: Lower harmonic content

• Disadvantages: Reduced output voltage

Three-Phase VSI with Star/Delta Load

• Star Connected Load:

  • Phase voltage: \(V_{ph} = \dfrac{V_{line}}{\sqrt{3}}\)

  • Line current = Phase current: \(I_L = I_{ph}\)

  • Neutral current = 0 (for balanced load)

• Delta Connected Load:

  • Phase voltage = Line voltage: \(V_{ph} = V_L\)

  • Line current: \(I_L = \sqrt{3} \times I_{ph}\)

  • No neutral connection required

• Total Power: \(P = \sqrt{3} V_L I_L \cos\phi\)

Single-Phase CSI Characteristics

• Input: Stiff DC current source (high inductance)

• Output: Controlled AC current (square wave)

• Voltage: Determined by load: \(V_o = I_o \times Z_{load}\)

• Configuration: Four thyristors in bridge + commutation capacitors

• Commutation: Natural or forced commutation required

• Output Current: \(I_o = \pm I_{dc}\)

• RMS Output Current: \(I_{o,rms} = I_{dc}\)

• Fundamental Current: \(I_{o1} = \dfrac{4I_{dc}}{\pi}\)

Single-Phase CSI Commutation

• Commutation Capacitor: Pre-charged to reverse voltage

• Turn-off Process: Reverse voltage turns off conducting thyristor

• Current Transfer: From one thyristor to next

• Commutation Time: \(t_c = \dfrac{C \times V_c}{I_{dc}}\)

• Capacitor Sizing: \(C = \dfrac{I_{dc} \times t_c}{V_c}\)

• Voltage Rating: \(V_c \geq 2 \times V_{load,max}\)

Three-Phase CSI Configuration

• Structure: Six thyristors + DC inductor + commutation capacitors

• Switching Sequence: Six-step operation (\(60^{\circ}\) intervals)

• Line Current RMS: \(I_{L,rms} = \dfrac{\sqrt{6}I_{dc}}{\pi}\)

• Phase Current RMS: \(I_{ph,rms} = \dfrac{\sqrt{2}I_{dc}}{\pi}\)

• Fundamental Line Current: \(I_{L1,rms} = \dfrac{3I_{dc}}{2\pi}\)

• DC Inductor: \(L_{dc} = \dfrac{V_{dc}}{6f \times \Delta I_{dc}}\) (for \(<5\%\) ripple)

VSI vs CSI Comparison

SPWM Principle

• Concept: Compare sinusoidal reference with triangular carrier

• Switching Logic:

  • When \(V_{ref} > V_{carrier}\) \(\to\) Upper switch ON

  • When \(V_{ref} < V_{carrier}\) \(\to\) Lower switch ON

• Modulation Index: \(m_a = \dfrac{V_{ref,peak}}{V_{carrier,peak}}\)

• Frequency Ratio: \(m_f = \dfrac{f_{carrier}}{f_{ref}}\) (should be odd, \(>21\))

• Switching Frequency: \(f_{sw} = f_{carrier}\)

• Advantage: Low harmonic distortion, variable voltage control

SPWM Output Analysis

• Linear Region (\(m_a \leq 1\)):

  • Full-bridge fundamental: \(V_{o1} = m_a \times V_{dc}\)

  • Half-bridge fundamental: \(V_{o1} = m_a \times \dfrac{V_{dc}}{2}\)

  • Linear relationship between \(m_a\) and output

• Overmodulation (\(m_a > 1\)):

  • Non-linear relationship

  • Increased harmonic distortion

  • Maximum fundamental: \(V_{o1,max} = \dfrac{4V_{dc}}{\pi}\)

• Harmonic Locations: At \(m_f \pm 1, m_f \pm 3, 2m_f \pm 1\), etc.

Bipolar vs Unipolar SPWM

• Bipolar SPWM:

  • Output levels: \(+V_{dc}\) and \(-V_{dc}\)

  • Single reference signal

  • Fundamental: \(V_{o1} = m_a \times V_{dc}\)

• Unipolar SPWM:

  • Output levels: \(+V_{dc}\), \(0\), and \(-V_{dc}\)

  • Two reference signals (\(180^{\circ}\) apart)

  • Better harmonic performance

  • Effective switching frequency: \(2f_c\)

• Comparison: Unipolar has lower harmonic content but complex control

Three-Phase SPWM

• Three Reference Signals:

  • \(v_{ref,a}(t) = V_m \sin(\omega t)\)

  • \(v_{ref,b}(t) = V_m \sin(\omega t - 120^{\circ})\)

  • \(v_{ref,c}(t) = V_m \sin(\omega t + 120^{\circ})\)

• Common Triangular Carrier: at frequency \(f_c\)

• Phase Voltage Fundamental: \(V_{ph1} = \dfrac{m_a V_{dc}}{2}\)

• Line Voltage Fundamental: \(V_{L1} = \dfrac{\sqrt{3}}{2} m_a V_{dc}\)

• Balanced Output: Inherent three-phase balance

Single-Phase Inverter Formulas

• Half-Bridge VSI:

  • RMS output: \(V_{o,rms} = \dfrac{V_{dc}}{2}\)

  • Fundamental: \(V_{o1} = \dfrac{2V_{dc}}{\pi}\)

• Full-Bridge VSI:

  • RMS output: \(V_{o,rms} = V_{dc}\)

  • Fundamental: \(V_{o1} = \dfrac{4V_{dc}}{\pi}\)

• Common Values:

  • THD = 48.34%

  • Distortion Factor = 0.9

Three-Phase Inverter Formulas

• \(180^{\circ}\) Conduction Mode:

  • Line voltage RMS: \(V_{L,rms} = \dfrac{2\sqrt{3}V_{dc}}{\pi}\)

  • Phase voltage RMS: \(V_{ph,rms} = \dfrac{2V_{dc}}{\pi}\)

  • Fundamental line voltage: \(V_{L1,rms} = \dfrac{\sqrt{6}V_{dc}}{\pi}\)

• \(120^{\circ}\) Conduction Mode:

  • Line voltage RMS: \(V_{L,rms} = \dfrac{\sqrt{6}V_{dc}}{\pi}\)

  • Phase voltage RMS: \(V_{ph,rms} = \dfrac{\sqrt{2}V_{dc}}{\pi}\)

SPWM and CSI Formulas

• SPWM (Linear Region):

  • Modulation index: \(m_a = \dfrac{V_{ref}}{V_{carrier}}\)

  • Full-bridge output: \(V_{o1} = m_a \times V_{dc}\)

  • Three-phase line voltage: \(V_{L1} = \dfrac{\sqrt{3}}{2} m_a V_{dc}\)

• Current Source Inverter:

  • Single-phase fundamental: \(I_{o1} = \dfrac{4I_{dc}}{\pi}\)

  • Three-phase line current: \(I_{L,rms} = \dfrac{\sqrt{6}I_{dc}}{\pi}\)

  • Commutation time: \(t_c = \dfrac{C \times V_c}{I_{dc}}\)

GATE Problem-Solving Tips

• Key Steps:

  • Identify inverter type (VSI/CSI, single/three-phase)

  • Determine conduction mode (\(180^{\circ}/120^{\circ}\))

  • Apply appropriate formulas for RMS and fundamental

  • Consider load effects (R, RL, motor)

• Common Question Types:

  • Output voltage/current calculations

  • Harmonic analysis (THD, distortion factor)

  • SPWM modulation index problems

  • Power calculations with different loads

• Important Relations:

  • \(V_{line} = \sqrt{3} V_{phase}\) (star connection)

  • \(I_{line} = \sqrt{3} I_{phase}\) (delta connection)