Power Electronics and Control

1

This section covers the fundamental distinction between open-loop and closed-loop control systems, and the concept and use of a system transfer function to describe system behavior.

2

Open Loop vs Closed Loop Systems System Type Description Open-loop system Output has no effect on the control action (no feedback); simple, cheap, but sensitive to disturbances and parameter variations, with no automatic error correction (e.g. a simple washing machine timer) Closed-loop (feedback) system Output is measured and fed back to compare with the reference input; the resulting error drives the controller, automatically correcting for disturbances/parameter variations, but more complex and can introduce stability issues (e.g. a thermostat-controlled heater) Key elements of a closed-loop system: reference input, error detector/summing junction, controller, actuator/plant (process), output, and a feedback element/sensor that returns a measured signal for comparison.

3

System Transfer Function The transfer function G(s) of a linear, time-invariant (LTI) system is defined as the ratio of the Laplace transform of the output to the Laplace transform of the input, assuming zero initial conditions:

4

The transfer function fully characterizes the input-output behavior of a linear system independent of the specific input applied, and is widely used for analysis (stability, time/frequency response) and design of control systems.

5

For a closed-loop system with forward path gain G(s) and feedback path gain H(s) (negative feedback), the overall (closed-loop) transfer function is C(s)/R(s) = G(s)/(1+G(s)H(s)).

1

This section covers the time-domain response of first-order and second-order control systems, methods for assessing control system stability (pole-zero plot, Root Locus), and the widely used PID controller.

2

First-Order & Second-Order Systems System Order Characteristics First-order system Described by a single time constant τ; step response is a simple exponential rise/decay, reaching ≈63.2% of final value in one time constant, no overshoot Second- order system Characterized by natural frequency ωₙ and damping ratio ζ; step response can be underdamped (ζ<1, oscillatory with overshoot), critically damped (ζ=1, fastest non-oscillatory response), or overdamped (ζ>1, slow, no oscillation) Key second-order time-response specifications: rise time (time to go from 10%/0% to 90%/100% of final value), peak time (time to reach first peak overshoot), percentage overshoot (amount by which response exceeds final value), and settling time (time to stay within a specified tolerance band, e.g. ±2% or ±5%, of final value).

3

Control System Stability & Pole-Zero Plot A linear control system is stable if and only if all poles of its closed-loop transfer function lie in the left half of the s-plane (negative real parts); a pole on the imaginary axis gives marginal (sustained oscillation) stability, and any pole in the right half s-plane makes the system unstable.

4

A pole-zero plot graphically shows the locations of a transfer function's poles (denominator roots, marked ×) and zeros (numerator roots, marked °) on the s-plane, giving quick insight into stability and transient response character.

5

Root Locus The Root Locus technique plots the paths traced by the closed-loop poles of a system as a chosen parameter (typically the loop/controller gain K) is varied from 0 to ∞, providing a graphical tool to study how gain affects stability and transient response, and to design a controller gain that achieves desired performance.

6

PID Controller Term Effect Proportional (P) Output ∝ current error; reduces rise time but leaves a steady-state (offset) error and can increase overshoot if gain is too high Integral (I) Output ∝ accumulated (integrated) past error; eliminates steady-state error but can increase overshoot and settling time, and risks instability if gain is too high Derivative (D) Output ∝ rate of change of error; improves transient response/damping (reduces overshoot, improves stability) but amplifies high-frequency noise

1

This section covers two classical frequency-domain graphical tools used to analyze control system stability and performance: the Bode plot and the Nyquist plot.

2

Bode Plot A Bode plot represents a system's frequency response as two separate plots against log-frequency: magnitude (in decibels, dB) and phase (in degrees), making it straightforward to sketch by hand (via asymptotic approximations) and to read off key stability margins.

3

Gain margin (GM): the amount of additional gain (in dB) that can be added before the system becomes unstable, measured at the frequency where phase = −180° (the phase crossover frequency).

4

Phase margin (PM): the additional phase lag (in degrees) that can be added before the system becomes unstable, measured at the frequency where the magnitude = 0 dB (the gain crossover frequency).

5

For a stable system, both gain margin and phase margin should be positive; larger margins generally indicate a more robustly stable system (though excessively large margins can mean a sluggish response).

6

Nyquist Plot A Nyquist plot is a polar plot of the open-loop frequency response G(jω)H(jω) as ω varies from −∞ to +∞, plotted on the complex plane (real vs imaginary parts).

7

The Nyquist stability criterion determines closed-loop stability by counting encirclements of the critical point (−1+j0) by the Nyquist plot: the system is closed-loop stable if the number of counter-clockwise encirclements of (−1,0) equals the number of open-loop right-half-plane poles (N=P, so Z=0 unstable closed-loop poles), a powerful method that can assess stability even for systems with open-loop unstable poles (unlike the Bode plot alone).

8

Gain margin and phase margin can also be read directly from a Nyquist plot, based on how closely the plot approaches the critical point (−1+j0).

1

This section covers the construction and characteristics of the major power semiconductor switching devices — power diode, thyristor, GTO, TRIAC, IGBT, and MOSFET — along with power loss and protection considerations.

2

Power Semiconductor Devices Device Key Characteristics Power diode Two-terminal, uncontrolled device; conducts when forward biased, blocks when reverse biased; simplest power switch, used in rectifiers and freewheeling applications Thyristor (SCR) Three-terminal (anode, cathode, gate) device; once triggered 'on' via a gate pulse (while forward biased), it latches on and conducts until current falls below a holding value (naturally commutated) or is forced off; cannot be turned off via the gate alone GTO (Gate Turn-Off thyristor) A thyristor variant that CAN be turned off via a (negative) gate pulse, without relying on natural commutation; offers full controllability but with more complex, higher-power gate drive requirements than a standard SCR TRIAC A bidirectional thyristor-like device (equivalent to two SCRs connected in anti-parallel with a common gate); can conduct and be triggered in both current directions, widely used for AC power control (e.g. light dimmers, fan speed controllers) IGBT (Insulated Gate Bipolar Transistor) A voltage-controlled device combining a MOSFET's high-impedance gate (easy drive, low gate power) with a bipolar transistor's low conduction loss at high current/voltage; widely used in medium-to-high power inverters and motor drives MOSFET (Power MOSFET) A voltage-controlled, unipolar device with very fast switching speed and low gate drive power; well suited to high-frequency switching but has relatively higher conduction loss at high voltage/current compared to IGBTs Power Loss in Power Switches Loss Type Description Conduction loss Power dissipated while the device is fully on, due to its on-state voltage drop/resistance and the current flowing through it Switching loss Power dissipated during the (non-instantaneous) turn-on and turn-off transitions, as voltage and current simultaneously have significant values; increases with switching frequency Off-state (leakage) loss Small power dissipated due to leakage current while the device is off; usually negligible compared to conduction and switching losses Protection of Power Switching Devices Snubber circuits (typically RC or RCD networks) limit the rate of change of voltage (dV/dt) and/or current (dI/dt) across a switching device during transitions, protecting it from voltage/current spikes and reducing switching stress.

3

Heat sinks and thermal management limit device junction temperature rise from conduction/switching losses, preventing thermal runaway/damage.

4

Fuses and overcurrent protection protect against excessive fault currents; freewheeling (flyback) diodes protect switching devices from inductive voltage spikes when switching inductive loads.

1

This section covers the three major categories of power converters: rectifiers (AC-DC), inverters (DC-AC), and DC-DC converters, including their operating principles, output waveforms, and key performance parameters.

2

Rectifiers (AC to DC) Rectifier Type Characteristics Half-wave rectifier Conducts (converts) only during one half-cycle of the AC input; simple (single diode) but produces a highly pulsating DC output with significant ripple and poor efficiency Full-wave rectifier Conducts during both half-cycles of the AC input (via a center-tapped transformer with two diodes, or a bridge of four diodes); much lower ripple and higher average DC output/efficiency than half-wave Key performance parameters of a rectifier include: average (DC) output voltage, ripple factor (ratio of AC ripple component to DC component in the output — lower is better), efficiency (DC output power / AC input power), and Total Harmonic Distortion (THD), which quantifies the distortion of the input current/output voltage waveform due to harmonics as a percentage of the fundamental component.

3

Power factor improvement techniques for rectifier/converter loads include passive filtering (LC filters to smooth current and reduce harmonics), and active power factor correction (PFC) circuits (which shape the input current to closely track the input voltage waveform, achieving near-unity power factor and low THD).

4

Inverters (DC to AC) An inverter converts DC input into an AC output of desired voltage/frequency, using power semiconductor switches (e.g.

5

IGBTs/MOSFETs) operated in a specific switching sequence; basic square-wave inverters produce a simple square output waveform, rich in harmonics (high THD).

6

Pulse Width Modulation (PWM) is the dominant modern technique for inverter control: switches are turned on/off rapidly with varying pulse widths (duty cycle) to synthesize an output waveform whose fundamental component closely approximates a sine wave, pushing harmonic content to much higher frequencies (easier to filter) and significantly reducing THD compared to simple square-wave switching.

7

DC-DC Converters Converter Type Principle & Output Buck (step-down) converter Output voltage is lower than input voltage; average output voltage V₀ = D × V₁₄, where D is the duty cycle (0<D<1) Boost (step-up) converter Output voltage is higher than input voltage; average output voltage V₀ = V₁₄/(1−D) Buck-boost converter Output voltage can be either higher or lower than (and of opposite polarity to) input voltage, depending on duty cycle D;

8

V₀ = −D×V₁₄/(1−D) All basic DC-DC converters operate by rapidly switching a semiconductor device on/off, storing/releasing energy in an inductor (and smoothing with a capacitor), with the average output voltage controlled by adjusting the duty cycle (fraction of switching period the switch is on).

1

This final section of the chapter covers two significant power electronics applications:

2

Uninterruptible Power Supply (UPS) systems and High Voltage Direct Current (HVDC) transmission systems.

3

Uninterruptible Power Supply (UPS) UPS Type Operating Principle Offline (standby) UPS Load is normally supplied directly from the AC mains; on mains failure, a static switch quickly transfers the load to a battery-inverter backup (small transfer time gap, typically a few milliseconds) Line-interactive UPS Similar to offline, but includes a voltage regulation stage (e.g. tap-changing autotransformer) to correct minor voltage sags/surges without switching to battery, improving power quality Online (double- conversion) UPS Load is continuously supplied via a rectifier-battery-inverter chain (mains → rectifier/charger → DC bus/battery → inverter → load) at all times, providing zero transfer time and the highest power quality/protection, at the cost of higher losses/cost Key UPS components: rectifier/battery charger (converts AC to DC, charges battery), battery bank (energy storage for backup), inverter (converts DC back to AC for the load), and (for offline/line-interactive types) a static transfer switch.

4

HVDC Transmission System HVDC (High Voltage Direct Current) transmission converts AC power to DC for transmission over long distances (or via submarine cable, or between asynchronous AC grids), then converts back to AC at the receiving end — offering lower transmission losses over long distances, no reactive power/stability limits associated with long AC lines, and the ability to interconnect grids of different frequencies.

5

Types & Components of HVDC Systems Type/Component Description Monopolar HVDC link Uses a single conductor (positive or negative polarity) with ground/sea return for the return current; simpler and cheaper but with a single path (less redundancy) Bipolar HVDC link Uses two conductors (one positive, one negative polarity) operating with respect to a common (often grounded) reference; under normal operation, if one pole fails the other can continue supplying at reduced capacity (improved reliability) Back-to-back HVDC link Rectifier and inverter stations are located at the same site (no DC transmission line), used specifically to interconnect two asynchronous AC systems (different frequency or unsynchronized) at a common point Converter station (rectifier/inverter) The core AC-DC-AC conversion equipment at each end (using thyristor or IGBT-based converters), the most critical and costly component of an HVDC link Smoothing reactor A large inductor on the DC side that reduces ripple in the DC current and limits the rate of rise of fault currents DC filters Filter harmonic components on the DC side to reduce interference with nearby communication lines and improve power quality