The important aspects in the study of feedback systems are to control

1. Sensitivity

2. Effect of an internal disturbance

3. Distortion in a nonlinear system

Effects of feedback:

Consider a negative feedback network as shown:

The transfer function of the above negative feedback control system will be:

\(T = \frac{G}{{1 + GH}}\)    ---(1)

Overall Gain: The overall gain may increase or decrease depending on the value of (1 + GH).

Sensitivity: The sensitivity of the overall gain of negative feedback closed-loop control system (T) to the variation in the open-loop gain (G) is defined as the ratio of the percentage change in T to the percentage change in G, i.e.

\(S_G^T = \frac{{\frac{{\partial T}}{T}}}{{\frac{{\partial G}}{G}}}\)   ---(2)

∂T is the incremental change in T due to incremental change in G.

Equation (2) can be rewritten as:

\(S_G^T = \frac{{\partial T}}{{\partial G}} \times \frac{G}{T}\)    ---(3)

Applying partial differentiation with respect to G in Equation (1), we get:

\(\frac{{\partial T}}{{\partial G}} = \frac{\partial }{{\partial G}}\left( {\frac{G}{{1 + GH}}} \right)\)

\(\frac{{\partial T}}{{\partial G}} = \frac{{\left( {1 + GH} \right).1 - G\left( H \right)}}{{{{\left( {1 + GH} \right)}^2}}} = \frac{1}{{{{\left( {1 + GH} \right)}^2}}}\)   ---(4)

From equation 1, we can write:

\(\frac{G}{T} = 1 + GH\)  ---(5)

Substituting Equation (4) and Equation (5) in Equation (3), we get:

\(S_G^T = \frac{1}{{{{\left( {1 + GH} \right)}^2}}}\left( {1 + GH} \right)\)

\(S_G^T = \frac{1}{{1 + GH}}\)

∴ The sensitivity of the overall gain of the closed-loop control system is the reciprocal of (1 + GH).

So, Sensitivity may increase or decrease depending on the value of (1 + GH).

Noise or internal disturbance:

To know the effect of feedback on noise, let us compare the transfer function relations with and without feedback due to noise signal alone.

Consider an open-loop control system with a noise signal as shown below.

The open-loop transfer function due to a noise signal alone is:

\(\frac{{C\left( s \right)}}{{N\left( s \right)}} = {G_b}\)   ---(5)

Consider a closed-loop control system with a noise signal as shown below:

The closed-loop transfer function due to a noise signal alone is:

\(\frac{{C\left( s \right)}}{{N\left( s \right)}} = \frac{{{G_b}}}{{1 + {G_a}{G_b}H}}\)      ---(6)

It is obtained by making the other input R(s) equal to zero.

Comparing Equations (5) and (6), we can conclude that in the closed-loop control system, the gain due to noise signal is decreased by a factor of (1 + G_aG_bH) provided that the term (1 + G_aG_bH) is greater than one.

Other important effects of negative feedback include:

• Negative feedback reduces distortion
• Improves bandwidth
• Input and output impedance increases