For two non-interacting first-order systems connected in series in process control, the overall transfer function of the combined system is obtained by what operation on the individual transfer functions? Provide the most appropriate relationship and assume standard linear, time-invariant blocks.

Introduction / Context:
In process control and instrumentation, complex plants are frequently modeled by connecting simpler dynamic blocks in series. Each block has a transfer function that relates its output to its input in the Laplace domain. This question tests your understanding of how to combine transfer functions for non-interacting first-order elements connected in series, which is a foundational concept for control-loop design and stability analysis.

Given Data / Assumptions:

• Two first-order, linear, time-invariant systems are connected in series.
• Systems are non-interacting (the output of the first is the input to the second with no side coupling or feedback between them).
• Standard Laplace-transform transfer function representation is used.

Concept / Approach:
For linear time-invariant blocks connected in series, the overall transfer function G_total(s) equals the product of the individual transfer functions. If G1(s) and G2(s) are the two blocks, then G_total(s) = G2(s) * G1(s). This follows directly from block-diagram algebra and the cascade property of Laplace-domain models.

Step-by-Step Solution:
Let the input be U(s) and the intermediate variable be X(s), with output Y(s).X(s) = G1(s) * U(s)Y(s) = G2(s) * X(s) = G2(s) * G1(s) * U(s)Therefore, G_total(s) = Y(s)/U(s) = G2(s) * G1(s)

Verification / Alternative check:
If each element is a standard first-order lag, G1(s) = K1/(1 + τ1s) and G2(s) = K2/(1 + τ2s). Then G_total(s) = (K1K2)/[(1 + τ1s)(1 + τ2s)], which shows the expected second-order denominator formed by multiplication—consistent with series connection behavior.

Why Other Options Are Wrong:
Ratio: Ratios arise in feedback loops, not in simple series cascades.Sum: Summation corresponds to parallel paths, not series.Difference: Also corresponds to summing junctions with a minus sign, not series.

Common Pitfalls:

• Confusing series combination (product) with parallel combination (sum).
• Assuming interaction or hidden feedback between stages, which would invalidate simple multiplication.
• Mixing time-domain convolution with Laplace-domain multiplication; in the Laplace domain, cascade implies multiplication.

Final Answer:
product