Voltage series negative feedback

Reproduces textbook Figure 4.4.1. The source $V_s$ is in series with the input; input current $I_i$ enters the basic amplifier of gain $A$ as $V_i$; the output appears as $V_o$ across the load resistor $R_L$; the 'Feedback network $\beta$' samples the output voltage across $R_L$ and returns $V_f = \beta V_o$ in series with the input. This figure matters because it is the refe …

$A_f = \dfrac{A}{1 + A\beta}$ — the voltage gain with negative feedback, reduced from the open-loop gain $A$ by the factor $(1+A\beta)$. Here $A$ is the open-loop gain and $\beta$ the feedback fraction; since $(1+A\beta) > 1$, the closed-loop gain $A_f$ is always smaller than $A$ — the price paid for t …

$A_f \approx \dfrac{1}{\beta}$ when $A\beta \gg 1$ — the gain then depends only on the stable feedback fraction $\beta$, making it independent of device parameters, supply voltage and ageing. Because $\beta$ is fixed by stable passive elements (resistors, capacitors, inductors) in the feedback network, the clos …

$\dfrac{dA_f}{A_f} = \dfrac{dA}{A} \times \dfrac{1}{1 + A\beta}$ — since $(1+A\beta) > 1$, a given percentage change in the internal gain $A$ produces a much smaller percentage cha …

Reproduces textbook Figure 4.4.2, the same voltage-series topology as Figure 4.4.1 but drawn to analyse input impedance. The input current $I_i$ flows from the source $V_s$ into $V_i$ of the basic amplifier; the output $V_o$ across $R_L$ is sampled by the 'Feedback network $\beta$', which returns $V_f = \beta V_o$ in series with the input. This figure matters because it defines the loop used to compute $Z_i …

$Z_{if} = Z_i(1 + A\beta)$ — voltage series negative feedback raises the input impedance by the factor $(1+A\beta)$, where $Z_i$ is the input impedance of the basic amplifier. A high input impedance is desirable so the amplifier does not load the preceding stage or the input source; this result fo …

Reproduces textbook Figure 4.4.3, an equivalent circuit for the output-impedance derivation. The input is short-circuited ($V_s = 0$); the basic amplifier is modelled as a dependent voltage source $A V_i$ in series with its internal output impedance $Z_o$; a hypothetical external source $V_o$ is applied at the output, driving current $I_o$ into the loop; the feedback network returns $V_f = \beta V_o$ to the shorted input. This figure matters because applying KVL around the o …

$Z_{of} = \dfrac{Z_o}{1 + A\beta}$ — voltage series negative feedback lowers the output impedance by the factor $(1+A\beta)$, where $Z_o$ is the output impedance of the basic amplifier. A low output impedance lets the amplifier deliver voltage or power to the load with little loss; it is derived with the …

Reproduces textbook Figure 4.4.4, a gain-versus-frequency graph. The 'Before FB' curve peaks at $A_{max}$ with 3 dB points at $f_1$ and $f_2$ (the 0.707 $A_{max}$ level) and bandwidth $BW$. The 'After FB' curve is lower (peak $A_{f\,max}$) but wider, with 3 dB points $f_1'$ (below $f_1$) and $f_2'$ (above $f_2$) and bandwidth $BW_f$. This figure matters because it shows visually that negative feedback reduces gain but increases …

$BW_f = BW(1 + A\beta)$ and $A \times BW = A_f \times BW_f$ — the bandwidth increases by the same factor $(1+A\beta)$ by which the gain decreases, so the gain-bandwidth product stays constant. The lower cut-off falls to $f_1' = f_1/(1+A\beta)$ and the upper rises to $f_2' = f_2(1+A\beta)$, so th …