Block diagrams are a compact way to represent how signals move through an engineering system. They are especially important in control systems, where an output is measured and compared with a desired input. Transfer functions describe each block using algebra in the Laplace variable s, so complex systems can be analyzed without solving differential equations directly.
This makes it easier to predict stability, speed of response, and steady state accuracy.
In a closed-loop system, the reference input R(s) is compared with a feedback signal to form an error signal E(s). The controller and plant act on this error to produce the output C(s), while the feedback path sends a measured version of the output back to the summing junction. Series blocks multiply, parallel blocks add, and feedback loops reduce to standard formulas.
These rules let engineers simplify a full block diagram into one equivalent transfer function.
Key Facts
- A transfer function is the ratio of output to input with zero initial conditions: G(s) = C(s)/R(s).
- Blocks in series multiply: G_eq(s) = G1(s)G2(s).
- Blocks in parallel add or subtract depending on the summing junction: G_eq(s) = G1(s) + G2(s).
- For negative feedback, the closed-loop transfer function is T(s) = G(s)/(1 + G(s)H(s)).
- For positive feedback, the closed-loop transfer function is T(s) = G(s)/(1 - G(s)H(s)).
- The loop transfer function is L(s) = G(s)H(s), and it strongly affects stability and transient response.
Vocabulary
- Block Diagram
- A graphical model that shows system components as blocks connected by arrows representing signal flow.
- Transfer Function
- A function of s that relates the Laplace transform of a system output to the Laplace transform of its input.
- Summing Junction
- A point in a block diagram where signals are added or subtracted according to their signs.
- Feedback Path
- The route that sends a portion or measurement of the output back to be compared with the input.
- Closed-Loop Transfer Function
- The equivalent transfer function from reference input to output when feedback is included.
Common Mistakes to Avoid
- Using T(s) = G(s)/(1 + G(s)) for every feedback system is wrong because a non-unity feedback block H(s) must be included as T(s) = G(s)/(1 + G(s)H(s)).
- Forgetting the sign at the summing junction is wrong because negative feedback and positive feedback give different denominators and very different system behavior.
- Adding transfer functions in series is wrong because cascaded blocks multiply, so two blocks G1(s) and G2(s) in sequence become G1(s)G2(s).
- Treating internal signals as the same variable is wrong because R(s), E(s), C(s), and feedback signals represent different points in the diagram.
Practice Questions
- 1 Two blocks in series have G1(s) = 4/(s + 2) and G2(s) = 3/(s + 5). Find the equivalent transfer function G_eq(s).
- 2 A negative feedback system has forward transfer function G(s) = 10/(s + 4) and feedback transfer function H(s) = 2. Find the closed-loop transfer function T(s) = C(s)/R(s).
- 3 Explain why increasing feedback gain can reduce steady state error but may also make a system less stable.