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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


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.

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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
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