A non-ideal diode is biased with a voltage of –0.03 V and a diode current of 1I is measured. The thermal voltage is 26 mV and the ideality factor for the diode is 15 /13. The voltage, in V, at which the measured current increases 1 to 1.5 I is closest to

Correct Answer: The correct option is –0.09 (or approximately –0.09 V).

Step-by-Step Explanation:

1. Understanding the Diode Equation:
The current-voltage relationship for a non-ideal diode is given by the Shockley diode equation:
$I=I_s\left(e^{\frac{V}{\eta V_T}}-1\right)$
where:
• $I$ is the diode current,
• $I_s$ is the reverse saturation current,
• $V$ is the applied voltage across the diode,
• $\eta$ is the ideality factor ($\eta =\frac{15}{13}$),
• $V_T$ is the thermal voltage ($V_T=26\text{mV}=0.026\text{V}$).

2. Calculate the Thermal-Ideality Product ($\eta V_T$):

$\eta V_T=\left(\frac{15}{13}\right)\times 26\text{mV}=30\text{mV}=0.03\text{V}$

3. Analyze Case 1 (Initial State):
Given initial voltage $V_1=-0.03\text{V}$ and measured current $I_1=1I$.
Substituting $V_1$ into the diode equation:

$I_1=I_s\left(e^{\frac{-0.03}{0.03}}-1\right)=I_s\left(e^{-1}-1\right)$

Since $e^{-1}\approx 0.3679$, we have:

$I_1=I_s\left(0.3679-1\right)=-0.6321I_s$

Thus, the magnitude of the measured current is $\left|I_1\right|=0.6321I_s$.

4. Analyze Case 2 (Final State):
We want to find the new voltage $V_2$ such that the current magnitude increases from $1I$ to $1.5I$ in reverse bias direction:
$I_2=1.5I_1=1.5\times \left(-0.6321I_s\right)=-0.94815I_s$

Substituting $I_2$ back into the diode equation:

$-0.94815I_{\mathrm{s}}=I_s\left(e^{\frac{V_2}{0.03}}-1\right)$

Dividing both sides by $I_s$:

$-0.94815=e^{\frac{V_2}{0.03}}-1$

$e^{\frac{V_2}{0.03}}=1-0.94815=0.05185$

5. Solving for $V_2$:
Taking the natural logarithm ($\ln$) on both sides:

$\frac{V_2}{0.03}=\ln \left(0.05185\right)\approx -2.959$

$V_2=-2.959\times 0.03\approx -0.0888\text{V}\approx -0.09\text{V}$

Therefore, the voltage at which the measured current increases to 1.5 I is closest to –0.09 V.