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The image you uploaded contains some circuit design calculations. Here’s a breakdown of the key steps and equations involved:

1. **Equation 1:**
   \[
   I_2 = I_1 = I = \frac{V_T \ln(n)}{R} + \frac{V_{BE}}{R_B}
   \]
   This appears to be part of a current equation involving thermal voltage \(V_T\), the natural logarithm of \(n\), the base-emitter voltage \(V_{BE}\), and resistances \(R\) and \(R_B\).

2. **Equation 2:**
   \[
   V_{\text{Ref}} = R_{\text{Ref}} \cdot I
   \]
   A simple voltage equation derived from Ohm's law for the reference voltage.

3. **Design assumptions:**
   You have assumed:
   - \( I = 15 \, \mu A \)
   - \( n = 8 \)
   - From the simulator, \( V_{BE} = 730 \, mV \) and \( V_T = 26 \, mV \)

4. **First substitution:**
   After substituting values into equation 2:
   \[
   500 \, mV = 15 \mu A \cdot R_{\text{Ref}} \quad \Rightarrow \quad R_{\text{Ref}} = 33.33 \, k\Omega
   \]

5. **Second substitution (Equation 3):**
   Substituting into equation 1:
   \[
   15 \mu A = \frac{26 \, mV \cdot \ln(8)}{R} + \frac{730 \, mV}{R_B}
   \]

6. **Equation from Dr. Banba’s paper (Equation 4):**
   \[
   V_{\text{Ref}} = \frac{R_{\text{Ref}}}{R_B} V_{GO} = 1.25
   \]
   Where \( V_{GO} \) is the bandgap voltage of Silicon extrapolated at absolute zero Kelvin.

7. **Solving for \( R_B \):**
   From equation 4:
   \[
   R_B = \frac{R_{\text{Ref}}}{500 \, mV} \cdot 1.25 = 83.325 \, k\Omega
   \]

8. **Solving for \( R \):**
   Substituting into equation 3:
   \[
   15 \mu A = \frac{26 \, mV \cdot \ln(8)}{R} + \frac{730 \, mV}{83.325 \, k\Omega} \quad \Rightarrow \quad R = 8.66 \, k\Omega
   \]

### Summary of values:
- \( R_{\text{Ref}} = 33.33 \, k\Omega \)
- \( R_B = 83.325 \, k\Omega \)
- \( R = 8.66 \, k\Omega \)

The final note mentions that these values might slightly change or need tuning to meet specifications.

---

Would you like further details or clarifications? Here are 5 related questions you could explore:

1. How would different values of \( I \) and \( n \) affect the resistances?
2. Can you explain the significance of \( V_{GO} \) and how it influences the calculations?
3. What is the role of \( V_T \) (thermal voltage) in this design?
4. How could you optimize \( R \) for lower power consumption?
5. What simulator tool did you use to obtain \( V_{BE} \) and \( V_T \)?

**Tip:** When designing circuits, always double-check that your component tolerances allow for the slight variances expected in real-world conditions.

How to determine the resistances in a circuit design using equations involving thermal voltage, base-emitter voltage, and reference voltage?

Learn how to calculate the resistances in a circuit design using equations that involve thermal voltage (V_T), base-emitter voltage (V_BE), and reference voltage (V_Ref). This solution includes step-by-step calculations and key formulas based on Ohm's Law and Bandgap Voltage Theory, suitable for undergraduate engineering students.

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