To find the transfer function for the given block diagrams, we need to analyze each system based on the given structure and relationships between the blocks.

For Figure (a):

1. System Description:

   • The input is $R(s)$, and the output is $Y(s)$.
   • $G_1(s)$ and $G_2(s)$ are connected in such a way that there is both forward and feedback control.

2. Key Equations:

   • The forward path involves $G_1(s)$ and $G_2(s)$, while there is a feedback loop.
   • Let the output after $G_1(s)$ be $V(s)$. Then: $V(s) = G_1(s) \cdot E(s),$ where $E(s)$ is the error signal.
   • The error signal is: $E(s) = R(s) - G_2(s) \cdot Y(s).$
   • The output $Y(s)$ is: $Y(s) = G_1(s) \cdot E(s).$

3. Derivation of Transfer Function: Substitute $E(s)$ into the equation for $Y(s)$: $Y(s) = G_1(s) \cdot \left( R(s) - G_2(s) \cdot Y(s) \right).$ Rearrange: $Y(s) \cdot \left( 1 + G_1(s) \cdot G_2(s) \right) = G_1(s) \cdot R(s).$ Therefore, the transfer function is: $\frac{Y(s)}{R(s)} = \frac{G_1(s)}{1 + G_1(s) \cdot G_2(s)}.$

For Figure (b):

1. System Description:

   • The input is $R(s)$, and the output is $Y(s)$.
   • The blocks $A(s), B(s), C(s), G(s), H(s)$ are interconnected in a more complex structure with multiple feedback loops.

2. Key Equations:

   • The first summing point gives: $E(s) = R(s) - H(s) \cdot Y(s).$
   • The output of $A(s)$ is: $U(s) = A(s) \cdot E(s).$
   • The output of $G(s)$ is: $V(s) = G(s) \cdot U(s).$
   • The output of $B(s)$ is added to $C(s) \cdot Y(s)$ to produce $Y(s)$: $Y(s) = B(s) \cdot V(s) + C(s) \cdot Y(s).$

3. Derivation of Transfer Function: Substitute each block equation iteratively:

   • Substitute $V(s)$ and $U(s)$: $Y(s) = B(s) \cdot G(s) \cdot A(s) \cdot \left( R(s) - H(s) \cdot Y(s) \right) + C(s) \cdot Y(s).$
   • Expand terms: $Y(s) = B(s) \cdot G(s) \cdot A(s) \cdot R(s) - B(s) \cdot G(s) \cdot A(s) \cdot H(s) \cdot Y(s) + C(s) \cdot Y(s).$
   • Group $Y(s)$-terms: $Y(s) \cdot \left( 1 + B(s) \cdot G(s) \cdot A(s) \cdot H(s) - C(s) \right) = B(s) \cdot G(s) \cdot A(s) \cdot R(s).$
   • Solve for $\frac{Y(s)}{R(s)}$: $\frac{Y(s)}{R(s)} = \frac{B(s) \cdot G(s) \cdot A(s)}{1 + B(s) \cdot G(s) \cdot A(s) \cdot H(s) - C(s)}.$

If you need detailed steps or clarification, let me know. Here are related questions to explore:

1. How do feedback systems affect stability in transfer functions?
2. Can the transfer function include time delay? If yes, how is it represented?
3. How would you calculate the steady-state error for these systems?
4. What is the impact of $C(s)$ in Figure (b) on the overall system?
5. How can you derive the transfer function using Mason's Gain Formula?

Tip: Always simplify block diagrams step-by-step to avoid missing intermediate relationships.