Visual
Key formulas
| Name | Formula | Variables / Notes |
|---|---|---|
| DC-DC Buck Converter Output Voltage | V_out = D x V_in | D = duty cycle (0 to 1) = t_on / T; V_in = input voltage; V_out = average output voltage; T = switching period |
| DC-DC Boost Converter Output Voltage | V_out = V_in / (1 - D) | D = duty cycle; V_in = input voltage; V_out = output voltage; valid for continuous conduction mode (CCM); D must be less than 1 |
| Single-Phase Half-Wave Rectifier Average Output | V_dc = (V_m / pi) x (1 + cos(alpha)) | V_m = peak supply voltage; alpha = firing angle in radians; for uncontrolled (alpha=0): V_dc = V_m / pi |
| Single-Phase Full-Wave Rectifier Average Output | V_dc = (2 x V_m / pi) x cos(alpha) | V_m = peak supply voltage; alpha = firing angle; for uncontrolled (alpha=0): V_dc = 2 x V_m / pi = 0.636 V_m |
| Three-Phase Full-Wave Rectifier Average Output | V_dc = (3 x sqrt(3) x V_m / pi) x cos(alpha) | V_m = peak phase voltage; alpha = firing angle; for uncontrolled: V_dc = 2.34 x V_rms(line) |
| Ripple Factor | RF = V_rms(ac) / V_dc = sqrt((V_rms / V_dc)^2 - 1) | V_rms = rms of output; V_dc = average output; V_rms(ac) = rms of ac ripple component; lower RF means better filtering |
Key concepts
Thyristor (SCR) Characteristics
A thyristor conducts when anode is positive relative to cathode and a gate pulse is applied. Once latched, gate loses control. It turns off only when anode current falls below holding current (I_H). Latching current (I_L) is the minimum current to sustain conduction after gate pulse is removed, I_L > I_H.
Converter Conduction Modes
Continuous Conduction Mode (CCM): inductor current never reaches zero; output voltage depends only on duty cycle. Discontinuous Conduction Mode (DCM): inductor current reaches zero each cycle; output voltage depends on load as well. CCM is preferred for tighter regulation.
Firing Angle and Power Factor
In a controlled rectifier, increasing firing angle alpha reduces average output voltage and also reduces displacement power factor. For a fully controlled bridge, PF = (2 sqrt(2) / pi) x cos(alpha) approximately 0.9 cos(alpha). Increasing alpha beyond 90 degrees puts the converter into inverter mode.
MOSFET vs IGBT in Power Circuits
MOSFETs switch faster (ns range) and suit high-frequency applications above 100 kHz. IGBTs have lower on-state voltage drop and handle higher currents, making them preferred in inverters and drives at 1 to 50 kHz. IGBTs have a current tail during turn-off due to minority carrier recombination.
Inverter Modulation
Sinusoidal PWM (SPWM) compares a sinusoidal reference with a triangular carrier. Modulation index M_a = V_ref / V_carrier. For M_a less than or equal to 1 (linear range), fundamental output voltage = M_a x (V_dc / 2) for a half bridge. Overmodulation occurs when M_a > 1.
Tables
DC-DC Converter Comparison
| Converter | V_out Formula | Output vs Input |
|---|---|---|
| Buck | D x V_in | Lower (step-down) |
| Boost | V_in / (1-D) | Higher (step-up) |
| Buck-Boost | -D/(1-D) x V_in | Inverted polarity |
| Cuk | -D/(1-D) x V_in | Inverted, low ripple |
Rectifier Output Voltage (Uncontrolled)
| Rectifier Type | V_dc | Ripple Frequency |
|---|---|---|
| 1-ph Half Wave | 0.318 V_m | f_supply |
| 1-ph Full Wave | 0.636 V_m | 2 x f_supply |
| 3-ph Half Wave | 0.827 V_m | 3 x f_supply |
| 3-ph Full Wave | 0.955 V_m | 6 x f_supply |
Power Semiconductor Comparison
| Device | Controlled Turn-Off | Frequency Range |
|---|---|---|
| SCR (Thyristor) | No (needs commutation) | Up to 1 kHz |
| GTO | Yes (gate turn-off) | Up to 10 kHz |
| MOSFET | Yes | Up to 10 MHz |
| IGBT | Yes | 1 kHz to 100 kHz |
Quick facts
- For a buck converter in CCM, critical inductance L_c = (1-D) x R / (2f), where R is load resistance and f is switching frequency.
- A full-wave bridge rectifier has a ripple factor of 0.482 with no filter capacitor.
- A half-wave rectifier has a ripple factor of 1.21, making it impractical for most applications without heavy filtering.
- The 3-phase full-wave bridge (6-pulse) produces output ripple at 6 times supply frequency, reducing filter requirements significantly.
- Snubber circuits (RC across a switch) limit dv/dt to protect thyristors from false triggering.
- The duty cycle D of a boost converter must remain below 0.9 in practice to avoid excessive inductor current and switch stress.
- A flyback converter is a buck-boost with transformer isolation; turns ratio n adjusts the voltage conversion ratio.
- IGBT safe operating area (SOA) is rectangular; MOSFET SOA is limited by on-resistance heating at high current.
Exam shortcuts
- For controlled rectifier problems: first write V_dc = (2 V_m / pi) cos(alpha) for full-wave, then substitute known V_dc and V_m to find alpha. Remember V_m = sqrt(2) x V_rms.
- Buck-boost output polarity is negative: if V_in = 12 V and D = 0.6, V_out = -(0.6/0.4) x 12 = -18 V. Many students forget the negative sign and lose marks.
- To identify CCM vs DCM in a problem: check if inductor current ripple (delta_iL = V_in x D / (L x f)) is less than twice the average inductor current. If delta_iL / 2 less than I_avg, operation is CCM.
- Form factor FF = V_rms / V_dc; Ripple Factor RF = sqrt(FF^2 - 1). For full-wave: FF = 1.11, RF = 0.482. Memorise these two numbers to answer ripple questions without full calculation.
- Firing angle alpha and extinction angle beta: for resistive load, beta = pi (current stops at zero crossing). For inductive load, beta > pi due to stored energy; conduction angle = beta - alpha.