Exercise 1 — Block Diagrams and Transfer Functions

Covers: Lessons 01–02 Difficulty: Foundation

Problem 1: Identify the Elements

Draw the block diagram for the following verbal description:

“A thermostat measures room temperature, compares it to the setpoint (22°C), and adjusts the heater power. The room’s temperature changes based on heater power minus heat loss through the walls.”

Identify: sensor, controller, plant, reference, error, feedback path.

Problem 2: Transfer Function from Block Diagram

Given:

R(s) ──→(+)──→ [ C(s) = 5 ] ──→ [ G(s) = 1/(s+2) ] ──→ Y(s)
          ↑ (-)                                           │
          └───────────────────────────────────────────────┘

a) Write the closed-loop transfer function $T(s) = Y(s)/R(s)$.

b) Find the poles. Is the system stable?

c) If $R(s) = 1/s$ (unit step), what is the steady-state value of $y(t)$?

Problem 3: DC Motor Transfer Function

A DC motor has: $R_a = 1.5 \Omega$, $L_a = 0.002$ H, $K_t = K_e = 0.05$ N·m/A, $J = 0.001$ kg·m², $B = 0.005$ N·m·s/rad.

a) Write the transfer function $G(s) = \Omega(s) / V(s)$.

b) What are the poles? Classify as overdamped, critically damped, or underdamped.

c) Can you neglect $L_a$? (Hint: compare electrical and mechanical time constants.)

Problem 4: Cascaded Blocks

Simplify:

R(s) → [G1 = 2/(s+1)] → [G2 = 3/(s+5)] → Y(s)

What is the overall transfer function? Find the DC gain (value at $s = 0$).

Problem 5: Feedback with Sensor Dynamics

R(s) ──→(+)──→ [ C(s) ] ──→ [ G(s) = 10/(s+1) ] ──→ Y(s)
          ↑ (-)                                        │
          └──────── [ H(s) = 1/(0.01s+1) ] ←──────────┘

The sensor $H(s)$ has a 10 ms time constant. At what frequency does the sensor start introducing significant phase lag (> 5°)?

Solutions