Easy Block Diagram - Transfer Function
We wish to find the transfer function \(H(s) = \frac{Y(s)}{U(s)}\) from the block diagram.
In general, for a loop with negative feedback the closed loop transfer function can be found with the following: $$G_{cl}(s) = \frac{G_{forward}(s)}{1+G_{ol}(s)} $$ Where \(G_{forward}(s)\) is the every block in the forward path multiplied together, and \(G_{ol}\) is every block in the loop multiplied together (also known as the open loop transfer function .
In this case, there are two forward paths; A->B->C or E->C. So the total forward path is $$G_{forward}(s) =ABC + EC = (AB+E)\cdot C $$ The open loop transfer function is obvious. $$G_{ol}(s) = BCF $$ The closed loop transfer function finally becomes $$G_{cl}(s) = \frac{Y(s)}{U(s)} = \frac{G_{forward}(s)}{1+G_{ol}(s)} = \frac{(AB+E)C}{1+BCF}$$ And we are done.
In general, for a loop with negative feedback the closed loop transfer function can be found with the following: $$G_{cl}(s) = \frac{G_{forward}(s)}{1+G_{ol}(s)} $$ Where \(G_{forward}(s)\) is the every block in the forward path multiplied together, and \(G_{ol}\) is every block in the loop multiplied together (also known as the open loop transfer function .
In this case, there are two forward paths; A->B->C or E->C. So the total forward path is $$G_{forward}(s) =ABC + EC = (AB+E)\cdot C $$ The open loop transfer function is obvious. $$G_{ol}(s) = BCF $$ The closed loop transfer function finally becomes $$G_{cl}(s) = \frac{Y(s)}{U(s)} = \frac{G_{forward}(s)}{1+G_{ol}(s)} = \frac{(AB+E)C}{1+BCF}$$ And we are done.