Feedback

Unity Feedback

In unity feedback, the difference between the output and input is feed back (get it?) into the input of the transfer function directly with no intermediate transfer function.

In this system, the overall transfer function becomes:

$$T(s)=\frac{G(s)}{1+G(s)}$$

The sign of $G(s)$ depends on whether it is positive or negative in the summing point (the circle with a cross). If the error is subtracted, (as is typical for stable systems), the sign in the denominator is positive.

Working

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Given:

• $R(s)$: System Input.
• $E(s)$: System Error.
• $C(s)$: System output.
• $G(s)$: Forward Transfer function.

And:

• $C(s)=E(s)\cdot G(s)$
• $E(s)=R(s)-C(s)$

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We are looking for $T(s)=\frac{C(s)}{R(s)}$

$$C(s)=G(s)\cdot [R(s)-C(s)]$$ $$C(s)=G(s)\cdot R(s)-G(s)\cdot C(s)$$ $$G(s)\cdot R(s)=C(s)+G(s)\cdot C(s)$$

Factor $C(s)$

$$G(s)\cdot R(s)=C(s)\cdot [1+G(s)]$$

Solve for $T(s)$:

$$T(s)=\frac{C(s)}{R(s)}=\frac{G(s)}{1+G(s)}$$

Non-Unity feedback

In non-unity feedback, a transfer function is applied between the output and the summing point.

The overall transfer function for this system becomes

$$T(s)=\frac{G(s)}{1+G(s)H(s)}$$

Working

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Given:

• $R(s)$: System Input.
• $E(s)$: System Error.
• $C(s)$: System output.
• $G(s)$: Forward Transfer function.
• $H(s)$: Feedback transfer function.
• $B(s)$: Feedback signal.

And:

• $C(s)=G(s)\cdot E(s)$
• $E(s)=R(s)-B(s)$
• $B(s)=C(s)\cdot H(s)$

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We need to find $T(s)=\frac{C(s)}{R(s)}$:
Perform substitutions:

$$C(s)=G(s)\cdot (R(s)-(C(s)\cdot H(s)))$$

Expand:

$$C(s)=G(s)\cdot R(s)-G(s)\cdot C(s)\cdot H(s)$$

Solve for $C(s)$

$$C(s)+G(s)\cdot C(s)\cdot H(s)=G(s)\cdot R(s)$$ $$C(s)\cdot (1+G(s)\cdot H(s))=G(s)\cdot R(s)$$

Solve for $T(s)$:

$$\frac{C(s)}{R(s)}=\frac{G(s)}{1+G(s)\cdot H(s)}$$