Key Characteristics

• Average output voltage: \(V_{dc} = \frac{2V_m}{\pi} \cos \alpha\)

• Average power output: \(P_{av} = V_{dc} \times I_{dc} = \frac{2V_m}{\pi} \cos \alpha \times I_{dc}\)

• Peak inverse voltage (PIV): Each thyristor withstands \(2V_m\)

• Maximum power output: 7643.31 W at \(\alpha = 0°\) and \(I_{dc} = 50\) A

Key Characteristics

• Peak inverse voltage (PIV): Each thyristor withstands \(V_m\)

• Transformer requirement: No center-tap needed, better transformer utilization

• Average output voltage: \(V_{dc} = \frac{2V_m}{\pi} \cos \alpha\)

• Maximum power output: 15286.624 W at \(\alpha = 0°\) and \(I_{dc} = 50\) A

• Advantage: Same power rating as mid-point converter but with lower PIV requirements

Key Differences

Half Controlled: Two SCRs + Two Diodes (Unidirectional power flow)
Full Controlled: Four SCRs (Bidirectional power flow)

Key Parameters

• Average output voltage:

  \(V_{dc} = \frac{2\sqrt{2}V}{\pi} \cos \alpha\)

• Average load current:

  \(I_{dc} = \frac{V_{dc}}{R} = \frac{2\sqrt{2}V}{\pi R} \cos \alpha\)

• RMS output voltage:

  \(V_{rms} = V\sqrt{\frac{\pi - \alpha + \frac{1}{2}\sin(2\alpha)}{\pi}}\)

• Form factor: \(FF = \frac{V_{rms}}{V_{dc}}\), Ripple factor: \(RF = \sqrt{FF^2 - 1}\)

Performance Equations - Continuous Current

• Average output voltage:

  \(V_{dc} = \frac{2\sqrt{2}V}{\pi} \cos \alpha\)

• Average load current:

  \(I_{dc} = \frac{V_{dc}}{R} = \frac{2\sqrt{2}V}{\pi R} \cos \alpha\)

• RMS output voltage:

  \(V_{rms} = V\sqrt{\frac{\pi + \alpha}{\pi}}\) (for highly inductive load)

• Form factor: \(FF = \sqrt{\frac{\pi + \alpha}{\pi}} \times \frac{\pi}{2\cos \alpha}\)

Performance Equations - Discontinuous Current

• Average output voltage:

  \(V_{dc} = \frac{\sqrt{2}V}{\pi} (\cos \alpha - \cos \beta)\)

• Average load current:

  \(I_{dc} = \frac{V_{dc}}{R} = \frac{\sqrt{2}V}{\pi R} (\cos \alpha - \cos \beta)\)

• RMS output voltage:

  \(V_{rms} = V\sqrt{\frac{1}{2\pi}[(\beta - \alpha) + \frac{1}{2}(\sin 2\beta - \sin 2\alpha)]}\)

• Extinction angle \(\beta\): Determined by \(i(\beta) = 0\) condition

Modified Performance Equations

• Average output voltage (\(0 \leq \alpha \leq \pi/2\)):

  \(V_{dc} = \frac{\sqrt{2}V}{\pi}(1 + \cos \alpha)\)

• Average output voltage (\(\pi/2 \leq \alpha \leq \pi\)):

  \(V_{dc} = 0\) (Free wheeling diode conducts continuously)

• Input displacement factor:

  \(DF = \cos \alpha\)

• Power factor improvement: Better than without free wheeling diode

Current Expression

For \(\alpha \leq \omega t \leq \beta\) (extinction angle):

where: \(Z = \sqrt{R^2 + (\omega L)^2}\), \(\phi = \tan^{-1}\left(\frac{\omega L}{R}\right)\)

Key Parameters

• Average voltage: \(V_{dc} = \frac{\sqrt{2}V}{\pi}[(\cos \alpha - \cos \beta) + \sin\theta(\pi - \beta + \alpha)]\)

• Average current: \(I_{dc} = \frac{V_{dc} - E}{R}\)

• Boundary condition: Current extinction when \(i(\beta) = 0\)

Steady-State Operation

• Periodic boundary condition: \(i(\alpha) = i(\pi + \alpha)\)

• Average output voltage: \(V_{dc} = \frac{2\sqrt{2}V}{\pi} \cos \alpha\)

• Average current: \(I_{dc} = \frac{V_{dc} - E}{R}\)

Current Expression

where \(K\) is determined from periodic boundary condition

Motor Operation

• Motoring mode (\(V_{dc} > E\)): Power delivered to motor

• Regenerative braking (\(V_{dc} < E\)): Power returned to supply

• Speed control through firing angle adjustment

Boundary Conditions

• Transition occurs when current just touches zero

• Depends on circuit parameters: \(R\), \(L\), \(E\), and firing angle \(\alpha\)

• Critical inductance concept applies

Key Design Insight

The firing angle \(\alpha\) controls both magnitude and polarity of average output voltage, enabling seamless transition between motoring and regenerative braking.

Fourier Series Analysis

For continuous conduction with highly inductive load:

Harmonic Components

• Input current contains only odd harmonics

• Fundamental component: \(I_1 = \frac{4I_{dc}}{\pi}\)

• Fifth harmonic: \(I_5 = \frac{4I_{dc}}{5\pi}\)

• Seventh harmonic: \(I_7 = \frac{4I_{dc}}{7\pi}\)

Key Performance Indicators

• Total harmonic distortion (THD):

  \(THD = \sqrt{\left(\frac{I_{rms}}{I_1}\right)^2 - 1} = \frac{\sqrt{\pi^2 - 8}}{\sqrt{8}} = 48.34\%\)

• Displacement power factor: \(DPF = \cos\alpha\)

• Current distortion factor: \(CDF = \frac{I_1}{I_{rms}} = \frac{2\sqrt{2}}{\pi}\)

• Input power factor: \(PF = CDF \times DPF = \frac{2\sqrt{2}}{\pi}\cos\alpha = 0.9\cos\alpha\)

Power Calculations

• Active power: \(P = V \times \frac{2\sqrt{2}I_{dc}}{\pi} \times \cos\alpha\)

• Reactive power: \(Q = V \times \frac{2\sqrt{2}I_{dc}}{\pi} \times \sin\alpha\)

Selection Criteria

• Cost vs. Performance: Bridge offers better utilization

• Power Rating: Higher power applications favor bridge topology

• Ripple Requirements: Free wheeling diode reduces ripple

• Bidirectional Power: Full bridge required for inverter operation

Key Takeaways

• Single-phase controlled bridge rectifiers offer versatile AC-DC conversion

• Firing angle control enables output voltage regulation and power flow control

• Load characteristics significantly influence converter performance

• Four-quadrant operation capability makes them ideal for motor drive applications

• Trade-offs exist between control flexibility and power quality