Question

A control system shown in the figure below has a forward gain \(G = 100\). An input \(V_{in} = 5\) volts is supplied to the system. What is the absolute error between the output \(V_{out}\) and input \(V_{in}\) in volts?

• A. 0.00
• B. 0.05
• C. 0.50
• D. 1.00

Hint:

For unity feedback, \(V_{out} = \frac{G}{1+G}V_{in}\), so the error is \(\frac{V_{in}}{1+G}\).

Answer 1

The Correct Option is B

Step 1: Set up the closed-loop equations.
The block diagram shows unity negative feedback: the summing junction forms an error signal \(e = V_{in} - V_{out}\) (since \(V_{out}\) is fed back directly, with no extra scaling block in the feedback path), and this error is multiplied by the forward gain block to give the output: \(V_{out} = G\,e\).

Step 2: Combine the two equations.
Substitute \(e = V_{in} - V_{out}\) into \(V_{out} = G\,e\):
\[ V_{out} = G\,(V_{in} - V_{out}) \]
\[ V_{out} + G\,V_{out} = G\,V_{in} \]
\[ V_{out}(1 + G) = G\,V_{in} \]
\[ V_{out} = \frac{G}{1+G}\,V_{in} \]

Step 3: Plug in the numbers.
With \(G = 100\) and \(V_{in} = 5\) V:
\[ V_{out} = \frac{100}{101}\times 5 = \frac{500}{101} \approx 4.95 \text{ V} \]

Step 4: Find the absolute error.
The error asked for is the gap between input and output:
\[ |V_{in} - V_{out}| = |5 - 4.95| = 0.05 \text{ V} \]
This matches the general formula for steady-state error of a unity-feedback system with a high forward gain: \(e = \dfrac{V_{in}}{1+G} = \dfrac{5}{101} \approx 0.05\) V.

Step 5: Final Answer.
The absolute error between \(V_{out}\) and \(V_{in}\) is 0.05 V. \[ \boxed{0.05 \text{ V}} \]