Question

An ideal p-n junction diode (ideality factor \(\eta = 1\)) is operating in forward bias at room temperature (thermal energy = 26 meV). If the diode current is 26 mA for an applied bias of 1.0 V, the dynamic resistance (\(r_{ac}\)) is \rule{1cm{0.15mm} \(\Omega\). (up to two decimal places)}

Hint:

The formula \(r_{ac} \approx \frac{\eta V_T}{I}\) is a very useful approximation for the dynamic resistance of a forward-biased diode. At room temperature, \(V_T\) is approximately 25-26 mV. For an ideal diode (\(\eta=1\)), this simplifies to \(r_{ac} \approx \frac{26 \text{ mV}}{I}\). This is a great shortcut for quick calculations.

Answer 1

Correct Answer: 1

Step 1: Understanding the Concept:
The dynamic resistance (also called differential or AC resistance) of a diode is the resistance it offers to a small AC signal superimposed on a DC bias. It is defined as the inverse of the slope of the diode's I-V characteristic curve at the operating point (Q-point).
Step 2: Key Formula or Approach:
The ideal diode equation gives the current \(I\) as a function of the applied voltage \(V\): \[ I = I_0 \left( e^{\frac{V}{\eta V_T}} - 1 \right) \] where \(I_0\) is the reverse saturation current, \(\eta\) is the ideality factor, and \(V_T\) is the thermal voltage. The thermal voltage is given by \(V_T = \frac{k_B T}{e}\). The problem gives the thermal energy \(k_B T = 26\) meV, so \(V_T = 26\) mV. The dynamic resistance \(r_{ac}\) is defined as: \[ r_{ac} = \frac{dV}{dI} = \left( \frac{dI}{dV} \right)^{-1} \] Step 3: Detailed Explanation:
First, we find the derivative of the current \(I\) with respect to the voltage \(V\): \[ \frac{dI}{dV} = \frac{d}{dV} \left[ I_0 \left( e^{\frac{V}{\eta V_T}} - 1 \right) \right] = I_0 \cdot e^{\frac{V}{\eta V_T}} \cdot \frac{1}{\eta V_T} \] For a forward-biased diode, especially with \(V = 1.0\) V which is much larger than \(V_T = 26\) mV, the exponential term is much greater than 1. So, we can approximate the diode current as: \[ I \approx I_0 e^{\frac{V}{\eta V_T}} \] Substituting this approximation back into the expression for the derivative: \[ \frac{dI}{dV} \approx \frac{I}{\eta V_T} \] Now, we can find the dynamic resistance by taking the reciprocal: \[ r_{ac} = \left( \frac{dI}{dV} \right)^{-1} \approx \frac{\eta V_T}{I} \] We are given the following values: Ideality factor, \(\eta = 1\) Thermal voltage, \(V_T = 26 \text{ mV} = 0.026 \text{ V}\) Diode current at the operating point, \(I = 26 \text{ mA} = 0.026 \text{ A}\) The applied bias of 1.0 V simply sets the operating point where the current is 26 mA. Substituting the values into the formula for \(r_{ac}\): \[ r_{ac} \approx \frac{(1) \times (0.026 \text{ V})}{0.026 \text{ A}} = 1 \, \Omega \] Step 4: Final Answer:
The dynamic resistance (\(r_{ac}\)) is 1.00 \(\Omega\).