# Shockley's diode equation

We owe a debt to Bell Labs with Shockley, Bardeen and Brattain for inventing the semiconductor. All have now passed on. I remember seeing my first transistor radio in the early 60's. Shockley developed the very important diode equation. I was reading through one of my good old engineering books where the dynamic resistance of a diode is derived. I'm trying to understand this because it's basic to many applications for transistors, current mirrors, long tailed pairs, etc. The dynamic resistance is used to get the volt drop over the small signal ac resistance. They give the equation $$I =I_s \left( e^{eV/kT}-1 \right) \tag{1}$$ where $$I$$ is the diode current, $$V$$ is the diode voltage, $$k$$ is Boltzmann's constant, $$T$$ is the temperature in Kelvin, and $$I_s$$ is the reverse leakage current.

They then derive the dynamic resistance as $$dV/dI$$ and get an answer as roughly $$1/40I$$.

I can't get the same answer that they do. Instead, I have to use this equation from Wikipedia to get the same answer: $$I=I_s \left( e^{V/V_t}-1 \right) \tag{2}$$ where $$V_t$$ is the thermal volts, which is 25mV at room temperature. Are they using the wrong equation in (1), or have I worked it out wrong?

• They're the same thing.... en.wikipedia.org/wiki/… – Chris Gerig Jun 10 '12 at 8:56
• As Chris says, if you calculate kT/e at 298K it's 25mV i.e. the same as $V_t$ in your second equation, so your two equations are exactly the same. If you get different answers with the two equations it must be a slip of the pen. Can you expand your post to show your working - we might be able to spot the problem. – John Rennie Jun 10 '12 at 19:10
• Hi John, Vt=kT/q =1.38x10^-23x298/1.6x10^-19volts =25mV – johann Jun 11 '12 at 23:53
• e here is not 2.718..., but the electron charge: 1.602E-19 coulomb. – Art Brown Jun 16 '12 at 6:46
• OK just to be clear, there are two e's: the one in the exponent is the magnitude of the electron charge (1.602E-19 coulombs) and the one being raised to a power is indeed the familiar 2.718... I can see that could be confusing... Often the charge is written as q instead. – Art Brown Jun 16 '12 at 17:32

The dynamic resistance is defined as $r_d = \frac{du}{di}$
from the Shockley equation $I_F = I_R(e^\frac{U_F}{m U_{Th}} -1)$
• $I_F$ : foreward current
• $U_F$ : foreward voltage
• $U_{Th}$ : the thermal voltage $U_{th} = \frac{kT} {e}$
$$\frac{1} {r_d} = \frac{dI_F} {dU_F} = \frac{dI_R(e^\frac{U_F}{m U_{Th}} -1)}{dU_F}=I_R e^\frac{U_F}{m U_{Th}}\frac{1}{m U_{Th}}$$ \If the forward current $I_F$ is much greater than the reverse saturation current $I_R$, then it does not mater whether we add or substract $I_{R}$ from $I_{F}$ therefore $$I_R e^\frac{U_F}{m U_{Th}}\approx I_R e^\frac{U_F}{m U_{Th}}-I_R = I_R (e^\frac{U_F}{m U_{Th}}-1)=I_F$$ $$\frac{1}{r_d} = \frac{I_F} {m U_{Th}}$$ and we finaly get $$r_d = \frac{m U_{Th}} {I_F}$$