Introduction to Single-Phase Drives
Basic Concept
Single-Phase Phase-Controlled DC Drive
A DC motor drive where the armature circuit is connected to the output of a single-phase controlled rectifier using thyristors (SCRs).
Key Operating Principles:
- Armature voltage controlled by varying the delay angle \(\alpha_a\) of the converter
- Field current also controlled using a converter with delay angle \(\alpha_f\)
- Phase-controlled converters use line-commutated thyristors
- For improved power factor and reduced harmonics, forced-commutated converters (choppers) can be used
Basic Circuit Arrangement
Important Components:
- Smoothing inductor \(L_m\) reduces ripple current to acceptable magnitude
- Separate converters for armature and field circuits
- Essential at low delay angles and high speeds to prevent discontinuous current
- Free-wheeling diode (in semiconverter) improves performance
Motor Speed-Torque Relationship
DC Motor Fundamental Equations:
\[V_a = E_b + I_a R_a\]
where \(E_b = K_b \phi \omega_m\) is the back EMF.
Speed Equation:
\[\omega_m = \frac{V_a - I_a R_a}{K_b \phi}\]
Torque Equation:
\[T_e = K_t \phi I_a\]
Speed control is achieved by controlling \(V_a\) (armature voltage) and \(\phi\) (field flux).
Single-Phase Semiconverter Drives
Semiconverter Drive Configuration
Characteristics:
- One-quadrant drive (forward motoring only)
- Limited to applications up to 15 kW
- Two thyristors and two diodes
- Free-wheeling diode improves performance and reduces ripple
- Current waveforms for highly inductive load
Semiconverter – Voltage Equations
Average Armature Voltage:
\[V_a = \frac{V_m}{\pi}(1 + \cos \alpha_a) \quad \text{for } 0 \leq \alpha_a \leq \pi\]
where \(V_m\) is the peak value of the AC supply voltage.
Average Field Voltage (if semiconverter used):
\[V_f = \frac{V_m}{\pi}(1 + \cos \alpha_f) \quad \text{for } 0 \leq \alpha_f \leq \pi\]
Control Range
- Delay angle \(\alpha_a\) varies from 0 to \(\pi\) radians
- Voltage can be controlled from \(\frac{2V_m}{\pi}\) (at \(\alpha_a = 0\)) to 0 (at \(\alpha_a = \pi\))
- Output voltage is always positive (unidirectional)
Semiconverter – Current Analysis
RMS Armature Current:
For continuous conduction mode:
\[I_{a,rms} = \sqrt{\frac{1}{2\pi}\int_{\alpha_a}^{\pi + \alpha_a} i_a^2 \, d\omega t}\]
Current Ripple:
- Peak-to-peak ripple depends on \(L_m\), load, and \(\alpha_a\)
- Critical inductance to maintain continuous conduction
- Free-wheeling diode reduces negative voltage period
Important Note
For discontinuous conduction, analysis becomes more complex and average voltage decreases.
Power Quality and Harmonics
Harmonics in Single-Phase Drives
Output Voltage Harmonics:
- Fundamental frequency: \(2f\) (for full-wave converters)
- Dominant harmonics: \(2f\), \(4f\), \(6f\), etc.
- Amplitude decreases with harmonic order
- Smoothing inductor filters high-frequency harmonics
Supply Current Harmonics:
- Odd harmonics predominate: 3rd, 5th, 7th, 11th, 13th, etc.
- \(n\)-th harmonic current: \(I_n \approx \frac{I_1}{n}\)
- Total Harmonic Distortion: \(\text{THD} = \frac{\sqrt{\sum_{n=2}^{\infty} I_n^2}}{I_1}\)
- Typical THD: 40–80% depending on operating point
Mitigation Techniques
- AC line filters
- Multi-pulse converters (for higher power ratings)
- Active power filters
Power Factor in Phase-Controlled Drives
Displacement Power Factor:
\[\text{DPF} = \cos\phi_1 \approx \cos\alpha_a\]
Distortion Factor:
\[\text{DF} = \frac{I_1}{I_{s,rms}} = \frac{1}{\sqrt{1 + \text{THD}^2}}\]
Total Power Factor:
\[\text{PF} = \text{DPF} \times \text{DF}\]
Power Factor Issues
- Power factor decreases with increasing \(\alpha_a\)
- At \(\alpha_a = 90°\), DPF = 0, no real power transfer
- Poor power factor leads to reactive power penalties
- Capacitor banks may be required for PF correction
