01
Circuit foundations
The course begins with the real behavior of components. Ideal laws are useful models, but every design must respect thresholds, frequency response, and transient behavior.
Models have operating limits
An I–V curve maps an applied voltage to the expected current. A resistor is approximately linear only inside its useful region; real devices depart from the ideal line near their limits.
Resistance with memory
A memristor changes resistance according to previously applied voltage or current and retains that state. This makes it relevant to neuromorphic circuits that imitate learning and neural spike patterns.
Silicon diode: the 0.7 V threshold
VD ≈ 0.7 V when forward conductingBelow the knee voltage, the ideal model treats current as zero. In a 10 V series diode–resistor circuit, the resistor receives approximately 10 − 0.7 = 9.3 V.
Charge, hold, release
The capacitor initially draws charging current, then behaves as an open circuit at steady state. After the source is removed, it can temporarily act as a voltage source while discharging through the resistor.
Current follows impedance
Capacitive impedance falls as frequency rises, so high-frequency current favors the capacitor branch. At low frequency, more current favors the resistive path.
ZC = 1 / (jωC) · ZL = jωLCapacitors oppose low frequency and pass higher frequency more readily; inductors do the opposite. Impedance is frequency-dependent resistance to AC.
Series RLC and resonance
An RLC circuit stores energy alternately in the capacitor’s electric field and the inductor’s magnetic field. Its response depends on damping and on how closely the applied frequency matches its natural frequency.
ω0 = 1 / √(LC) · f0 = 1 / (2π√(LC))Near resonance, current can become large. Away from resonance, the forced response weakens; after excitation is removed, the natural response decays according to damping.
02
Control systems
A control system manages a plant so its output follows a desired target with acceptable accuracy, speed, stability, and resistance to disturbance.
Reference
r(t)Comparator
e(t)Controller
u(t)Plant
y(t)Sensor
b(t)e(t) = r(t) − b(t)Negative feedback compares the setpoint with the measured output. The controller acts on the error, while disturbances may enter the plant or sensor path.
| Architecture | What it measures | Strength | Limitation |
|---|---|---|---|
| Open loop | No measurement of actual output | Simple, low-cost, predictable when conditions do not change | Cannot automatically correct load changes, drift, or disturbance |
| Closed loop | Sensor returns actual output | Reduces error and rejects disturbance | Needs a trustworthy sensor and careful stability design |
PWM as an actuator technique
Pulse-width modulation changes the duty cycle of a fixed-amplitude switching waveform. A slow physical load—such as a motor, pump, heater, or lamp—responds mainly to the average drive.
Vavg ≈ D · Vsupply
D is the duty cycle from 0 to 1. PWM is not itself proof of closed-loop control; a system is closed loop only when actual output is measured and fed back.03
Response, damping & stability
How the system gets there
Rise time, overshoot, oscillation, and settling time describe what happens immediately after a command or disturbance.
Where the system ends
Steady-state error is the remaining difference between the setpoint and output after transients decay.
| Damping ratio ζ | Response | Interpretation |
|---|---|---|
ζ = 0 | Undamped | Oscillation persists indefinitely in the ideal model. |
0 < ζ < 1 | Underdamped | Fast initial motion with overshoot; oscillation decays over time. |
ζ = 1 | Critically damped | Fastest ideal return without overshoot. |
ζ > 1 | Overdamped | No overshoot, but slower response. |
Design lesson: the earliest first crossing is not necessarily the best response. Very low damping may reach the target quickly by overshooting, yet take much longer to settle. A practical target is often around
0.7 ≤ ζ ≤ 1, depending on safety and performance needs.Stability comes first. A useful system keeps its output bounded and eventually settles. Negative feedback often improves accuracy and disturbance rejection, but excessive gain, delay, or incorrect sign can make the loop unstable.
04
Block diagrams & transfer functions
Block
Applies an operation or transfer function to a signal.
Arrow
Shows the one-way direction of signal flow.
Summing point
Combines incoming signals with the signs shown.
Takeoff point
Copies a signal into another path without changing it.
Reduction rules
| Structure | Equivalent | Reminder |
|---|---|---|
| Series / cascade | Geq = G₁G₂… | Multiply blocks along the same path. |
| Parallel paths | Geq = G₁ ± G₂ | Use the signs at the summing point. |
| Negative feedback | G / (1 + GH) | The plus sign in the denominator comes from negative feedback under the standard convention. |
| Positive feedback | G / (1 − GH) | Check the summing-junction sign before reducing. |
E(s) = R(s) − H(s)C(s)
C(s) = G(s)E(s)
C(s) / R(s) = G(s) / [1 + G(s)H(s)]With unity feedback,
C(s) = G(s)E(s)
C(s) / R(s) = G(s) / [1 + G(s)H(s)]With unity feedback,
H(s)=1. Start there for a simpler first analysis, then restore the real sensor dynamics when needed.Moving a summing point, takeoff point, or block is allowed only if compensating gains preserve the original input–output relationship.
05
Transforms & controller action
The Laplace transform converts differential equations in time into algebraic equations in the s-domain, making dynamic models easier to combine and reduce.
F(s) = ∫₀∞ f(t)e−st dt where s = σ + jωThe real part
σ describes exponential growth or decay; jω carries oscillation frequency. With zero initial conditions, differentiation becomes multiplication by s.Proportional
Acts on present error. Faster correction is possible, but high gain can increase overshoot or oscillation.
Proportional + integral
Integral action accumulates error and helps remove steady-state offset, often with a settling-time trade-off.
Combined action
Adds derivative prediction to proportional and integral action to shape speed, damping, and accuracy together.
The Z-transform plays a similar role for discrete-time systems. Stability is interpreted relative to the unit circle rather than the continuous-time s-plane.
06
Case studies
Pressure regulation
A setpoint enters an MCU, PWM drives a blower, and pressure or flow sensors return actual output. With feedback it adapts to mask fit, posture, and breathing disturbance.
Potentiometer command
An Arduino maps a potentiometer voltage to a servo pulse-width command. Without an external measurement of actual shaft angle, the presented arrangement is open loop with respect to angle.
Smart water pump
A soil-moisture sensor closes the loop; an MCU uses switching, relay control, or PWM to keep moisture inside a target band while a solar subsystem manages energy.
Hysteresis control
Separate on and off thresholds prevent rapid relay chatter. Sensor feedback makes the watering decision responsive to actual soil condition.
Signal processing distinction
PWM can efficiently reproduce an audio signal, but transforming a waveform is not automatically control. Regulation requires feedback that drives a measured output toward a reference.
Performance in context
An elevator must balance speed, comfort, load disturbance, and final position. An air conditioner must reject heat disturbances while reducing temperature error without excessive cycling.
07
Six-step design workflow
- Define the physical system. State the desired output, operating range, performance metrics, disturbances, and environmental limits.
- Draw the block diagram. Identify references, signal directions, blocks, summing points, takeoff points, and feedback paths.
- Map blocks to hardware. Connect the abstract controller, plant, actuator, and sensor to real physical components.
- Build the mathematical model. Use circuit and physical laws, then linearize around an operating point when appropriate.
- Reduce to an overall transfer function. Combine series and parallel paths and reduce feedback loops without changing input–output behavior.
- Simulate and optimize. Use MATLAB, Simulink, or Python to inspect poles, stability, overshoot, settling, steady-state error, and disturbance response.
08
Recorded assignments
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