Control Systems and Power Electronics — ESE Electrical

Weightage: Control systems and power electronics are two compact, high-scoring blocks in the Electrical papers. Both reward a clean toolkit: for control, the second-order formulas and Routh table, and for power electronics, the output-voltage relation of each converter.

1. Models and block diagrams

The transfer function is the ratio of the Laplace transform of the output to that of the input, with zero initial conditions. A closed loop with forward path $G(s)$ and feedback $H(s)$ has:

$\frac{C(s)}{R(s)}=\frac{G(s)}{1+G(s)H(s)}$

for negative feedback. Feedback reduces the sensitivity to parameter change, improves disturbance rejection and widens bandwidth, at the cost of gain and the risk of instability. Mason's gain formula finds the overall gain of a signal-flow graph from the forward paths, loops and non-touching loops.

2. Time response

A first-order system $1/(\tau s+1)$ has a step response $1-e^{-t/\tau }$, reaching 63.2 percent at $t=\tau$.

A standard second-order system is:

$\frac{{\omega }_n^2}{s^2+2\zeta {\omega }_{n}s+{\omega }_n^2}$

The damping ratio $\zeta$ sets the shape: undamped at 0, underdamped for $0<\zeta <1$, critically damped at 1 and overdamped above 1. For the underdamped case:

Worked example. For $\zeta =0.5$ and ${\omega }_n=10$ rad/s, $M_p=e^{-\pi \times 0.5/0.866}=e^{-1.81}=16.3\%$ and $t_s=4/5=0.8$ s.

Overshoot depends only on $\zeta$, so it is fixed by the pole angle, while settling time depends on the real part of the poles.

3. Steady-state error

For a unity-feedback system the type is the number of poles at the origin of $G(s)$. The error constants are $K_p={\lim }_{s\to 0}G$, $K_v={\lim }_{s\to 0}sG$ and $K_a={\lim }_{s\to 0}{s}^{2}G$.

For $G=K/(s(s+2))$ (type 1), $K_v=K/2$ and the ramp error is $2/K$. Raising the type improves accuracy, usually at the cost of stability.

4. Stability and the Routh criterion

A system is stable if all poles of the closed loop lie in the left half of the $s$-plane. The Routh array counts right-half-plane roots by the sign changes in its first column. A necessary condition is that all coefficients of the characteristic polynomial are present and positive.

Worked example. For $s^3+6s^2+11s+K=0$ the first column is $1,6,(66-K)/6,K$. The system is stable for $0<K<66$, and at $K=66$ it is marginally stable with an oscillation at $\omega =\sqrt{11}$.

5. Root locus

The root locus plots the closed-loop poles as the gain $K$ varies from 0 to infinity. Rules:

• It starts at open-loop poles ($K=0$) and ends at open-loop zeros or at infinity.
• Real-axis segments lie to the left of an odd number of real poles and zeros.
• The number of asymptotes is $P-Z$, with angles $\frac{180^{\circ }(2k+1)}{P-Z}$ and a centroid $\frac{\sum \text{poles}-\sum \text{zeros}}{P-Z}$.
• Adding a pole pushes the locus toward the right, which reduces stability, while adding a zero pulls it to the left.

6. Frequency response: Bode and Nyquist

A Bode plot shows magnitude (in dB) and phase against log frequency. Each pole gives a slope of $-20$ dB/decade after its corner frequency, and each zero $+20$ dB/decade.

• Gain margin: how much the gain can rise at the phase crossover frequency before instability.
• Phase margin: how much extra phase lag is tolerated at the gain crossover frequency.

A stable system needs positive margins, and a phase margin of about 45 degrees is a common target. The Nyquist criterion says $Z=N+P$, with $Z$ the closed-loop right-half-plane poles, $P$ the open-loop right-half-plane poles and $N$ the clockwise encirclements of $-1$.

Compensators. A lead network adds phase and speeds up response. A lag network improves steady-state accuracy and lowers bandwidth. A PID controller combines proportional, integral (removes steady-state error) and derivative (adds damping) action.

7. Power semiconductor devices

An SCR (thyristor) is a four-layer device that turns on by a gate pulse once forward biased and remains on until its current falls below the holding current. The latching current is the minimum anode current to keep it on after the gate pulse ends, and it is larger than the holding current. Turn-off needs commutation: natural (AC line), forced or load commutation.

Compare the controllable devices:

8. Controlled rectifiers

For a single-phase full-wave fully controlled bridge with continuous current, the mean output voltage is:

$V_{dc}=\frac{2V_m}{\pi }\cos \alpha$

with firing angle $\alpha$. For a half-wave controlled rectifier with a resistive load, $V_{dc}=\frac{V_m}{2\pi }(1+\cos \alpha )$.

Worked example. At $V_m=325$ V and $\alpha =60^{\circ }$, $V_{dc}=(2\times 325/\pi )\times 0.5=103.5$ V.

When $\alpha >90^{\circ }$ the average voltage is negative, so with a source such as a DC motor the converter can invert, returning energy to the AC supply. A three-phase fully controlled bridge gives $V_{dc}=\frac{3V_{m,LL}}{\pi }\cos \alpha$, with $V_{m,LL}$ the peak line voltage. A freewheeling diode keeps load current flowing and raises the output of an inductive load circuit.

9. DC-DC converters and inverters

With duty ratio $D$ and input voltage $V_s$:

So with $D=0.4$ and $V_s=100$ V a buck gives 40 V, and a boost with $D=0.5$ and $V_s=50$ V gives 100 V.

An inverter turns DC into AC. A single-phase full-bridge square-wave inverter gives an RMS output equal to the DC input, and PWM control shifts the harmonic content to higher frequencies so filtering is easier and the fundamental can be varied. A cycloconverter converts AC at one frequency to a lower frequency AC directly, and an AC voltage controller varies RMS voltage by phase angle. Snubbers protect devices against $dv/dt$, and series inductors against $di/dt$.

Common traps

• Reading settling time as dependent on $\zeta$ only. It depends on $\zeta {\omega }_n$.
• Missing the sign-change count in the Routh array when a row of zeros arises.
• Confusing holding and latching currents. Latching is larger.
• Using the half-wave formula for a full bridge.
• Boost converter output below input. It is always at least $V_s$.

Memory aids

• "Overshoot from zeta, settling from zeta omega": second-order response.
• "Left half is stable": pole location.
• "2Vm over pi cos alpha": full converter.

Summary

Control systems describe linear systems by transfer functions, time response, steady-state error and stability, checked by Routh, root locus, Bode and Nyquist methods. Compensators reshape the response.

Power electronics converts electrical energy using SCRs and controllable switches, in controlled rectifiers, choppers and inverters whose output voltages follow compact formulas.

Exam protocol

• Identify the system type and order before choosing a formula.
• Build the Routh array row by row.
• Name the converter before quoting its output voltage.
• Check the sign of the average voltage when $\alpha$ exceeds 90 degrees.