Derive it from the loop equations. With input R, output C, error E and negative feedback:

\(E=R-H(s)C, \qquad C=G(s)E\)

Substituting the first into the second,

\(C=G(s)\left[R-H(s)C\right]=G(s)R-G(s)H(s)C\)

Collecting the C terms,

\(C\left[1+G(s)H(s)\right]=G(s)R\)

\(T(s)=\dfrac{C}{R}=\dfrac{G(s)}{1+G(s)H(s)}\)

which is option 1.

Two structural checks that identify it without any algebra.

The numerator must be G, not H. The forward path carries the signal from input to output, so the closed-loop gain must reduce to G when the feedback is removed. Setting H = 0 in option 1 gives exactly \(T=G\) ✓. Options 3 and 4 would give H, which is absurd — with no feedback the answer would depend only on the feedback element.

The sign must be plus. Negative feedback subtracts at the summing junction, which puts \(+GH\) in the denominator; positive feedback gives \(1-GH\). That eliminates option 2.

The general form worth memorising.

\(T(s)=\dfrac{\text{forward path gain}}{1\mp(\text{loop gain})}\)

with the minus sign for positive feedback and the plus sign for negative feedback. For unity feedback, H = 1 and the result reduces to \(G/(1+G)\).

What the denominator means. Setting it to zero,

\(1+G(s)H(s)=0\)

gives the characteristic equation, whose roots are the closed-loop poles. Every stability tool in control theory is an examination of this one expression: the root locus traces its roots as gain varies, the Nyquist criterion counts encirclements of the point \(-1\) by GH, and Routh–Hurwitz tests its coefficients.

Why negative feedback is used. When the loop gain is large, \(|GH|\gg1\), the expression collapses to \(T\approx1/H\) — the closed-loop behaviour is set almost entirely by the feedback element and becomes nearly independent of G. Since G is the part that drifts with temperature, ageing and device spread, this is precisely what buys accuracy and repeatability.

Hence, the equivalent single block is \(\dfrac{G(S)}{1+G(S)H(S)}\).