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Q.Derive an expression for input impedance of an amplifier with negative feedback.

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[!TLDR]

Zif=Zi(1+Aβ)Z_{if} = Z_i(1+A\beta): voltage series negative feedback increases the input impedance by the factor (1+Aβ)(1+A\beta).

A high input impedance is desirable so that the amplifier does not load the preceding stage or source. Voltage series negative feedback achieves this (block diagram, Fig. 4.4.2). Let IiI_i be the input current, ViV_i the net input to the basic amplifier, VsV_s the source voltage, VfV_f the feedback voltage and VoV_o the output.

Step 1 — impedance without feedback. Looking into the basic amplifier,

Zi=ViIi.(1)Z_i = \frac{V_i}{I_i}. \qquad(1)

Step 2 — impedance with feedback. Looking in from the source,

Zif=VsIi.(2)Z_{if} = \frac{V_s}{I_i}. \qquad(2)

Step 3 — relate VsV_s and ViV_i. For negative feedback,

Vi=Vs−Vf=Vs−βVo.V_i = V_s - V_f = V_s - \beta V_o.

Using Vo=AViV_o = A V_i,

Vi=Vs−βAVi⇒Vs=Vi(1+Aβ).(3)V_i = V_s - \beta A V_i \quad\Rightarrow\quad V_s = V_i(1 + A\beta). \qquad(3)

Step 4 — combine. Substitute (3) into (2): …

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