Power Electronics: Thyristor-Controlled Rectifiers

This section develops the SCR structure, two-transistor equivalent and $V$–$I$ characteristic, derives controlled-rectifier average voltage, and compares switched-mode and linear supplies. Rectifier formulas depend on the load and on whether load current is continuous or discontinuous; those conditions are stated with each result.

Silicon-Controlled Rectifier (SCR)

A silicon-controlled rectifier is a unidirectional, four-layer $p_1n_1p_2n_2$, three-junction power device with anode (A), cathode (K) and gate (G) terminals. It is a bistable, latching switch: a gate pulse can turn it on while it is forward biased, but the gate cannot normally turn it off.

The anode is connected to the outer $p_1$ layer, the cathode to $n_2$, and the gate to the inner $p_2$ layer near the cathode. The central junction $J_2$ is the forward-blocking junction.

Native representation of SCR $p$-$n$-$p$-$n$ construction and symbol.

Operating states

1. Reverse blocking: A is negative with respect to K. Junctions $J_1$ and $J_3$ are reverse biased and only reverse leakage flows. Reverse breakdown is normally destructive.

2. Forward blocking: A is positive and $I_G=0$. Junctions $J_1,J_3$ are forward biased, but $J_2$ is reverse biased. The device blocks until the forward breakover voltage $V_{BO}$ is reached.

3. Forward conduction: A positive gate-current pulse injects carriers into $p_2$ and starts regeneration. The SCR snaps to a low on-state voltage, typically about $1$–$2\,\text{V}$, and anode current is then limited by the external circuit.

Note

Key Point — Latching Is Not Gate Control

Immediately after firing, $I_A$ must exceed the latching current $I_L$ before the gate pulse is removed. Once latched, the SCR remains on while $I_A>I_H$, where the holding current $I_H$ is normally smaller than $I_L$. To turn off, $I_A$ must remain below $I_H$ for at least the specified device turn-off time $t_q$ so stored charge can recombine.

Two-transistor regeneration

The four layers can be separated into a PNP transistor $T_1$ and an NPN transistor $T_2$. Each collector supplies the other transistor’s base current, creating positive feedback. The gate injects current into the base region of $T_2$.

Cross-coupled PNP–NPN equivalent of an SCR.

Let $\alpha_1,\alpha_2$ be the common-base current gains and $I_{CBO1},I_{CBO2}$ the collector leakage currents. Then

$$I_{C1}=\alpha_1 I_A+I_{CBO1}, \qquad I_{C2}=\alpha_2 I_K+I_{CBO2}.$$

Kirchhoff’s current law gives $I_K=I_A+I_G$ and $I_A=I_{C1}+I_{C2}$. Therefore

$$\begin{aligned} I_A&=\alpha_1I_A+I_{CBO1} +\alpha_2(I_A+I_G)+I_{CBO2},\\ I_A\{1-(\alpha_1+\alpha_2)\} &=I_{CBO1}+I_{CBO2}+\alpha_2I_G. \end{aligned}$$

Caution

Key Formula — SCR Two-Transistor Current

$$\boxed{I_A= \frac{I_{CBO1}+I_{CBO2}+\alpha_2 I_G} {1-(\alpha_1+\alpha_2)}}$$

As current rises, $\alpha_1+\alpha_2\rightarrow1$ and the denominator becomes very small. The equation does not imply infinite current: the load and transistor saturation limit current while both internal transistors latch on.

$V$–$I$ characteristic

SCR static characteristic; the gate shifts the forward turn-on point.

Triggering, commutation and protection

Method	Mechanism and significance
Gate triggering	A rated positive G–K current pulse injects carriers; this is the normal, controllable method.
Forward breakover	Raising $V_{AK}$ to $V_{BO}$ avalanches $J_2$; normally avoided because it stresses the device.
$dv/dt$ triggering	Junction capacitance carries $i=C_j\,dv/dt$; excessive displacement current can start regeneration.
Thermal triggering	Higher temperature increases leakage and gains; unintended turn-on can lead to thermal failure.
Light triggering	Optical carriers initiate regeneration in a light-activated SCR (LASCR), useful for high-voltage isolation.

SCR turn-on mechanisms.

An SCR is commutated only when anode current falls below $I_H$ for at least $t_q$:

• Natural or line commutation: AC load current crosses zero and the supply then reverse biases the SCR. This is the usual rectifier process.

• Forced commutation: in DC circuits, an auxiliary switch and a capacitor/inductor network impose reverse current or voltage so net SCR current becomes zero.

Stress	Why it is dangerous	Protection
High $dv/dt$	Capacitive current through $J_2$ can cause false turn-on	Series RC snubber across the SCR; controlled layout
High turn-on $di/dt$	Conduction initially occupies a small cathode area, causing a hot spot	Series inductance and a gate pulse with adequate amplitude and rise time
Overcurrent	Junction heating or surge destruction	Semiconductor fuse/MCB, current limiting and coordinated surge rating
Overvoltage	Exceeds $V_{DRM}$ or $V_{RRM}$	MOV/TVS clamp, snubber and adequate repetitive/non-repetitive voltage margin
Overtemperature	Leakage rises and ratings fall as $T_j$ rises	Heat sink, thermal interface, airflow and thermal shutdown

Principal SCR stresses and practical protection.

Important data-sheet quantities are $V_{DRM}$, $V_{RRM}$, average/RMS/surge on-state current, $I_{GT}$, $V_{GT}$, $I_L$, $I_H$, $t_q$, allowable $dv/dt$ and $di/dt$, and maximum junction temperature.

Uncontrolled Full-Wave Bridge Rectifier

A diode bridge reverses the negative input half-cycle at the load. It requires no centre-tapped transformer and uses the entire secondary on both half-cycles.

Full-wave bridge and ideal conduction/output waveforms.

For ideal diodes and $v_s=V_m\sin\theta$, $v_o=V_m|\sin\theta|$. Thus

Caution

Key Formula — Ideal Bridge Performance

$$\boxed{V_{dc}=\frac{2V_m}{\pi}},\qquad \boxed{V_{rms}=\frac{V_m}{\sqrt{2}}},\qquad \boxed{f_r=2f},\qquad \boxed{\mathrm{PIV}=V_m\ \text{per diode}}.$$$$\gamma=\sqrt{\left(\frac{V_{rms}}{V_{dc}}\right)^2-1} =\sqrt{\frac{\pi^2}{8}-1}=0.482, \qquad \eta_{\max}=\frac{V_{dc}^2}{V_{rms}^2}=\frac{8}{\pi^2}=81.2\%.$$

Here $\gamma$ is ripple factor and $\eta$ is ideal rectification efficiency for an $R$ load. In a practical silicon bridge, two diodes conduct in series, so $v_o\simeq |v_s|-2V_D$ while conducting. Transformer utilisation is better than in a half-wave circuit. A capacitor or LC filter can reduce ripple, but it also makes diode and transformer currents more sharply pulsed.

Quantity	Half-wave	Full-wave bridge
$V_{dc}$	$V_m/\pi=0.318V_m$	$2V_m/\pi=0.637V_m$
$V_{rms}$	$V_m/2$	$V_m/\sqrt{2}$
Ripple factor	$1.21$	$0.482$
Maximum efficiency	$40.6\%$	$81.2\%$
Ripple frequency	$f$	$2f$
PIV per device	$V_m$	$V_m$

Ideal half-wave and bridge-rectifier comparison.

For a centre-tapped full-wave rectifier, PIV is $2V_m$ per diode when $V_m$ is the peak of one half-secondary; this must not be confused with bridge PIV.

Single-Phase SCR-Controlled Rectifiers

The firing angle $\alpha$ is the delay, measured from the natural forward-voltage zero crossing, at which the gate pulse turns on a forward- biased SCR. Changing $\alpha$ changes the conduction interval and hence the average output.

Half-wave controlled rectifier with an $R$ load

Half-wave SCR phase controller and delayed $R$-load waveform (illustrated at $\alpha=60^\circ$).

During $0<\theta<\alpha$, the SCR is forward biased but off. The gate pulse at $\theta=\alpha$ turns it on; with a pure resistance, $i_o=v_o/R$ and conduction lasts only to $\pi$. At the supply zero crossing, current falls below $I_H$ and natural commutation turns the SCR off. It remains reverse biased throughout the negative half-cycle. Hence

$$v_o(\theta)= \begin{cases} V_m\sin\theta, & \alpha\leq\theta\leq\pi,\\ 0, & \text{otherwise over }0\leq\theta<2\pi. \end{cases}$$

The average over a complete supply cycle is

$$\begin{aligned} V_{dc} &=\frac{1}{2\pi}\int_{\alpha}^{\pi}V_m\sin\theta\,d\theta\\ &=\frac{V_m}{2\pi}[-\cos\theta]_{\alpha}^{\pi}. \end{aligned}$$

Caution

Key Formula — Half-Wave Controlled Rectifier, Pure $R$ Load

$$\boxed{V_{dc}=\frac{V_m}{2\pi}(1+\cos\alpha)}$$

At $\alpha=0$, $V_{dc}=V_m/\pi$; at $\alpha=\pi/2$ it is $V_m/(2\pi)$; and at $\alpha=\pi$ it is zero.

Full-wave circuits: the load distinction

For a full-wave centre-tapped circuit, alternate SCRs are fired at $\alpha$ and $\pi+\alpha$. A bridge may use two SCRs plus two diodes (half-controlled) or four SCRs (fully controlled).

For a pure $R$ load, load current reaches zero at every supply zero crossing. Two identical positive output pulses occur per cycle:

$$\begin{aligned} V_{dc,R} &=\frac{1}{2\pi}\left[ 2\int_{\alpha}^{\pi}V_m\sin\theta\,d\theta\right]\\ &=\boxed{\frac{V_m}{\pi}(1+\cos\alpha)}. \end{aligned}$$

For a fully controlled four-SCR bridge with a sufficiently inductive load, current is assumed continuous. Each diagonal pair conducts for $\pi$ radians, including an interval in which output voltage is negative:

$$\begin{aligned} V_{dc,\mathrm{cont}} &=\frac{1}{\pi}\int_{\alpha}^{\pi+\alpha} V_m\sin\theta\,d\theta\\ &=\boxed{\frac{2V_m}{\pi}\cos\alpha}. \end{aligned}$$

Note

Law — Do Not Conflate the Full-Wave Formulas

$$\begin{aligned} \text{Half-wave, $R$ load:}\quad &V_{dc}=\frac{V_m}{2\pi}(1+\cos\alpha),\\ \text{Full-wave, $R$ load:}\quad &V_{dc}=\frac{V_m}{\pi}(1+\cos\alpha),\\ \text{Full bridge, continuous current:}\quad &V_{dc}=\frac{2V_m}{\pi}\cos\alpha. \end{aligned}$$

The $R$-load expressions apply when current ends at each supply zero. In the continuous-current bridge each pair conducts for $\pi$ radians. Its average is negative for $\alpha>90^\circ$; with a suitable DC source in the load, this is inverter or regenerative operation.

Feature	Half-wave	Half-controlled bridge	Fully controlled bridge
Power devices	One SCR	Two SCRs and two diodes	Four SCRs
Output use	One half-cycle	Both half-cycles	Both half-cycles
Ripple frequency	$f$	$2f$	$2f$
Average-output sign	Non-negative	Non-negative	Positive or negative with continuous current and an active load
Regeneration	No	No	Possible
Control/complexity	Lowest	Moderate	Greatest

Single-phase controlled-rectifier comparison.

Increasing $\alpha$ reduces average output and usually worsens input power factor and harmonic distortion. Full-wave conversion gives higher output, higher ripple frequency and easier filtering than half-wave conversion.

Note

Example — Three Controlled-Rectifier Averages

A $230\,\text{V}_{\mathrm{rms}}$ source has $V_m=\sqrt{2}(230)=325.3\,\text{V}$. Find the average output at $\alpha=60^\circ$ for (i) a half-wave $R$ load, (ii) a full-wave $R$ load, and (iii) a fully controlled bridge with continuous current.

Since $\cos60^\circ=0.5$,

$$\begin{aligned} V_{dc,\mathrm{half},R} &=\frac{325.3}{2\pi}(1+0.5)=77.7\,\text{V},\\ V_{dc,\mathrm{full},R} &=\frac{325.3}{\pi}(1+0.5)=155.3\,\text{V},\\ V_{dc,\mathrm{full},\mathrm{cont}} &=\frac{2(325.3)}{\pi}(0.5)=103.5\,\text{V}. \end{aligned}$$

The different numerical answers come from different conduction intervals, not from alternative conventions for the same circuit.

For completeness, industrial high-power converters commonly use three phases. With $V_{ph,m}$ the phase-to-neutral peak and $V_{LL}$ the line-to-line RMS voltage, ideal continuous-current averages are

$$V_{dc,3\text{-pulse}} =\frac{3\sqrt{3}V_{ph,m}}{2\pi}\cos\alpha,$$ $$\boxed{V_{dc,6\text{-pulse}} =\frac{3\sqrt{3}V_{ph,m}}{\pi}\cos\alpha =\frac{3\sqrt{2}V_{LL}}{\pi}\cos\alpha \simeq1.35V_{LL}\cos\alpha.}$$

The three-pulse circuit uses three SCRs; the six-pulse bridge uses six SCRs and produces lower output ripple than a single-phase converter.

Linear and Switched-Mode Power Supplies

A linear supply transforms at mains frequency, rectifies and filters, then operates a series-pass device in its active region. An SMPS rectifies where necessary, switches energy at high frequency, transfers or stores it in magnetic components, rectifies/filters the output and closes a feedback loop around duty ratio.

Energy and control flow in linear and switched-mode supplies.

For a linear regulator, ignoring quiescent current,

Caution

Key Formula — Linear-Regulator Loss

$$P_{loss}\simeq(V_{in}-V_o)I_o, \qquad \eta\simeq\frac{V_o}{V_{in}}\times100\%.$$

The voltage difference is continuously converted to heat in the pass device.

For ideal continuous-conduction switching converters with duty ratio $D$,

Caution

Key Formula — Common Ideal SMPS Conversion Ratios

$$\frac{V_o}{V_{in}}=D\quad\text{(buck)},\qquad \frac{V_o}{V_{in}}=\frac{1}{1-D}\quad\text{(boost)},$$$$\frac{V_o}{V_{in}}=-\frac{D}{1-D}\quad\text{(inverting buck--boost)}, \qquad \left|\frac{V_o}{V_{in}}\right| \simeq\frac{N_s}{N_p}\frac{D}{1-D}\quad\text{(ideal flyback)}.$$

A forward converter transfers energy to the secondary while its switch is on; a flyback stores energy in magnetising inductance while on and delivers it to the secondary while off. Buck, boost and buck–boost stages are normally non-isolated; flyback, forward and bridge-derived converters can provide galvanic isolation.

Property	SMPS	Linear supply
Efficiency	Typically $80$–$95\%$ or higher	Often $30$–$60\%$ when the voltage drop is large
Size and mass	Small high-frequency magnetics; high power density	Bulky 50/60-Hz transformer and heat sink
Conversion range	Step-down, step-up, inversion, wide input and multiple isolated outputs	Mainly step-down; constrained by dropout and dissipation
Heat	Low when switching and magnetic losses are controlled	High pass-device loss for large $V_{in}-V_o$
Noise/EMI	Switching ripple and conducted/radiated EMI require filtering and careful layout	Very low switching noise and EMI
Complexity	Controller, compensation, gate drive, magnetics and protection	Simple regulator; easier design and fault finding
Best use	Computers, chargers, telecom and portable/high-power equipment	Precision analog, audio, laboratory and very low-noise rails

SMPS and linear power-supply comparison.

Note

Example — Linear Loss Versus SMPS Loss

A supply must deliver $5\,\text{V}$ at $2\,\text{A}$ from $12\,\text{V}$. The load power is $P_o=5(2)=10\,\text{W}$.

For a linear regulator,

$$P_{loss}=(12-5)(2)=14\,\text{W},\qquad \eta=\frac{10}{10+14}\times100\%=41.7\%.$$

For a $90\%$ efficient SMPS,

$$P_{in}=\frac{10}{0.90}=11.11\,\text{W},\qquad P_{loss}=11.11-10=1.11\,\text{W}.$$

The SMPS wastes about one thirteenth as much power, so its heat sink and energy cost are much smaller; the trade-off is switching noise and greater complexity.

Practical SMPS design adds an input fuse, MOV and inrush limiter, common- and differential-mode EMI filters, cycle-by-cycle current limiting, snubbers or clamps, overvoltage and thermal shutdown, safe bulk-capacitor discharge, and adequate isolation creepage and clearance. Feedback crossing an isolation barrier commonly uses an optocoupler or isolated controller.

Protection from Electrical Hazards

Electrical safety is the coordinated use of insulation, enclosures, earthing, automatic disconnection, isolation, energy limiting and safe work practices to prevent shock, burns, fire and equipment damage in normal use and under a credible single fault.

Layered Class-I protection: insulation, enclosure, PE, overcurrent protection and residual-current detection.

Measure	Protective action
Basic/reinforced insulation and enclosure	Prevent contact with live parts; use adequate creepage, clearance, flame resistance and an appropriate IP rating. Class II equipment uses double/reinforced insulation and no PE.
Protective earthing (Class I)	Bonds accessible metal to PE. A live-to-case fault follows a low-resistance path and operates the protective device rather than leaving the case hazardous.
Fuse or MCB	Interrupts overload and short-circuit current to protect wiring and prevent fire. It is not sensitive enough to replace shock protection.
RCD/ELCB	Compares live and neutral currents and disconnects on residual earth leakage. A $30\,\text{mA}$ RCD is commonly used as additional personal protection, but it does not replace a fuse/MCB, insulation or PE.
Isolation transformer/SELV	Provides galvanic separation and limits available voltage/current, removing a direct mains-to-earth return path for a single secondary contact.
Surge and thermal protection	MOV/TVS devices clamp transients; thermal cut-outs and current limiting prevent excessive energy and temperature.
Cable and mechanical design	Correct conductor size, strain relief, secure terminals and guarded connectors prevent overheating and exposed conductors.
Inspection and testing	Visual checks, earth continuity, insulation resistance, leakage and RCD tests reveal deterioration before service.

Electrical-hazard controls and the fault each addresses.

Before maintenance: de-energise, isolate, lock out/tag out, prove dead with a verified tester, discharge stored capacitors, and use appropriate tools and PPE. Relevant product standards include IEC 62368-1 for ICT/AV equipment and IEC 61010 for measurement and laboratory equipment; installation must also follow applicable national wiring rules.

Note

Key Point — Protection mechanism distinctions

Earthing provides a deliberate fault-current path; a fuse/MCB clears excessive current; an RCD detects live–neutral imbalance; double insulation avoids reliance on PE; and galvanic isolation separates hazardous mains from the user circuit. These protections are complementary, not interchangeable.