Follow the signal, then reduce the blocks
Predict series, parallel and feedback outputs before switching topology. Then change only H from 1 to 2. Which error does feedback reduce?
First, picture it
A block maps an input signal to an output. In this bench every block is a constant dimensionless gain, so the result is instantaneous. Set r=1, G₁=2 and G₂=3. In series the intermediate signal is 2 and the output is 6. In parallel both blocks see the same input and their outputs add to 5. Changing the wiring changes the system even when the numbers stay fixed.
What the model says
For negative feedback, e=r−Hy and y=G₁G₂e. Substitute y into the first equation: e(1+G₁G₂H)=r. Hence y/r=G₁G₂/(1+G₁G₂H). With the default gains and H=1, e=1/7 and y=6/7. The loop product L=G₁G₂H is 6; it is not the closed-loop gain 6/7.
Series: G₁G₂ · Parallel: G₁+G₂ · Negative feedback: G₁G₂/(1+G₁G₂H)The same algebra applies to compatible SISO transfer functions under zero initial conditions: multiply in series, add parallel paths, and solve the feedback equation. Dynamic blocks can have poles and phase lag; this static bench makes no stability prediction. Moving a summing point or pickoff requires compensating gains. Never cancel an unstable hidden mode and call the physical realization stable.
Write the signals before memorizing a rule
A block maps an input signal to an output. In this bench every block is a constant dimensionless gain, so the result is instantaneous. Set r=1, G₁=2 and G₂=3. In series the intermediate signal is 2 and the output is 6. In parallel both blocks see the same input and their outputs add to 5. Changing the wiring changes the system even when the numbers stay fixed.
Close the loop by solving simultaneous equations
For negative feedback, e=r−Hy and y=G₁G₂e. Substitute y into the first equation: e(1+G₁G₂H)=r. Hence y/r=G₁G₂/(1+G₁G₂H). With the default gains and H=1, e=1/7 and y=6/7. The loop product L=G₁G₂H is 6; it is not the closed-loop gain 6/7.
A sensor changes what the loop tracks
Increase H to 2. The loop reduces r−Hy, not necessarily r−y. At high forward gain, Hy approaches r, so y approaches r/H. A calibrated reference scale can compensate a known sensor gain. Blindly interpreting the summing-junction signal as physical tracking error is a common mistake. The bench displays both errors.
From gains to dynamics
The same algebra applies to compatible SISO transfer functions under zero initial conditions: multiply in series, add parallel paths, and solve the feedback equation. Dynamic blocks can have poles and phase lag; this static bench makes no stability prediction. Moving a summing point or pickoff requires compensating gains. Never cancel an unstable hidden mode and call the physical realization stable.
Make it concrete
Predict series, parallel and feedback outputs before switching topology. Then change only H from 1 to 2. Which error does feedback reduce?
Open the concept benchThe same algebra applies to compatible SISO transfer functions under zero initial conditions: multiply in series, add parallel paths, and solve the feedback equation. Dynamic blocks can have poles and phase lag; this static bench makes no stability prediction. Moving a summing point or pickoff requires compensating gains. Never cancel an unstable hidden mode and call the physical realization stable.
Check your understanding+
Predict series, parallel and feedback outputs before switching topology. Then change only H from 1 to 2. Which error does feedback reduce?
Increase H to 2. The loop reduces r−Hy, not necessarily r−y. At high forward gain, Hy approaches r, so y approaches r/H. A calibrated reference scale can compensate a known sensor gain. Blindly interpreting the summing-junction signal as physical tracking error is a common mistake. The bench displays both errors.