The Plate Current in a Triode Can Be Written as I P = K ( V G + V P μ ) 3 / 2 Show that the Mutual Conductance is Proportional to the Cube Root of the Plate Current. - Physics

Sum

The plate current in a triode can be written as $i_p = k \left( V_g + \frac{V_p}{\mu} \right)^{3/2}$ Show that the mutual conductance is proportional to the cube root of the plate current.

Solution

Given:- The plate current varies with plate and grid voltage as

$i_p = K \left( V_g + \frac{V_p}{\mu} \right)^{3/2}........... (1)$

Differentiating the equation w.r.t V_G, we get:-

$d i_p = K\frac{3}{2} \left( V_g + \frac{V_p}{\mu} \right)^{1/2} d V_g$

$\Rightarrow g_m = \frac{d i_p}{d V_g} = \frac{3}{2}K \left( V_g + \frac{V_p}{\mu} \right)^{1/2}$

From (1), plate current can be written in terms of transconductance as:-

$i_p = \left[ \frac{3}{2}K \left( V_g + \frac{V_p}{\mu} \right)^{1/2} \right]^3 \times K'$

Here, $K'\text{ is a constant } = \left(\frac{2}{3} \right)^3 \times \frac{1}{K^2}$

$i_p = K'( g_m )^3$

$\Rightarrow g_m \propto \sqrt[3]{i_p}$

Concept: Concept of Semiconductor Electronics: Materials, Devices and Simple Circuits
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HC Verma Class 11, Class 12 Concepts of Physics Vol. 2
Chapter 19 Electric Current through Gases
Q 20 | Page 353