VIPSolutionsGive step-by-step solution with explanation and final answer: 4.26 Two diodes with saturation currents Is₁ and Is2 are connected in parallel with their cathodes joined together and connected to grounds. The two anodes are joined together and fed with a constant current 7. Find the currents In and In2 that flow through the two diodes, and the voltage Vn that appears across their parallel combination.
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Give step-by-step solution with explanation and final answer: 4.26 Two diodes with saturation currents Is₁ and Is2 are connected in parallel with their cathodes joined together and connected to grounds. The two anodes are joined together and fed with a constant current 7. Find the currents In and In2 that flow through the two diodes, and the voltage Vn that appears across their parallel combination.
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Asked by: A j mahant
Created at: 2025-09-13 05:07:23
Asked by: A j mahant
Created at: 2025-09-13 05:07:23
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# Step-by-Step Solution
**Given:**
- Two diodes in parallel, with:
- Saturation currents: \(I_{S1}\) and \(I_{S2}\)
- Cathodes tied to ground
- Anodes tied together, fed by a constant total current \(I\)
- Need to find:
- Currents through each diode (\(I_1\) and \(I_2\))
- Voltage across the parallel combination (\(V_n\))
---
## 1. **Diode Equation**
For each diode, the current-voltage relationship is:
\[
I = I_S (e^{V/(\eta V_T)} - 1)
\]
Where:
- \(I_S\): Saturation current for the diode
- \(V\): Voltage across the diode
- \(\eta\): Ideality factor (assume \(\eta = 1\) unless stated)
- \(V_T\): Thermal voltage (\( \approx 26 \text{ mV} \) at room temperature)
Since both diodes are in parallel, both have the same voltage, \(V_n\), across them.
---
## 2. **Write Current Equations**
For diode 1:
\[
I_1 = I_{S1} \left(e^{V_n / V_T} - 1\right)
\]
For diode 2:
\[
I_2 = I_{S2} \left(e^{V_n / V_T} - 1\right)
\]
Total current:
\[
I = I_1 + I_2
\]
---
## 3. **Express Currents in Terms of \(V_n\)**
Add the two currents:
\[
I = I_{S1} \left(e^{V_n / V_T} - 1\right) + I_{S2} \left(e^{V_n / V_T} - 1\right)
\]
\[
I = (I_{S1} + I_{S2}) \left(e^{V_n / V_T} - 1\right)
\]
---
## 4. **Solve for \(V_n\)**
\[
\frac{I}{I_{S1} + I_{S2}} = e^{V_n / V_T} - 1
\]
\[
e^{V_n / V_T} = 1 + \frac{I}{I_{S1} + I_{S2}}
\]
\[
V_n = V_T \ln \left(1 + \frac{I}{I_{S1} + I_{S2}}\right)
\]
---
## 5. **Find \(I_1\) and \(I_2\)**
Use the value of \(V_n\):
\[
I_1 = I_{S1} \left(e^{V_n / V_T} - 1\right)
\]
But \(e^{V_n / V_T} - 1 = \frac{I}{I_{S1} + I_{S2}}\), so:
\[
I_1 = I_{S1} \cdot \frac{I}{I_{S1} + I_{S2}}
\]
Similarly,
\[
I_2 = I_{S2} \cdot \frac{I}{I_{S1} + I_{S2}}
\]
---
## 6. **Final Answers**
### Voltage across the parallel combination:
\[
\boxed{
V_n = V_T \ln \left(1 + \frac{I}{I_{S1} + I_{S2}}\right)
}
\]
### Currents through the two diodes:
\[
\boxed{
I_1 = \frac{I_{S1}}{I_{S1} + I_{S2}} I
}
\]
\[
\boxed{
I_2 = \frac{I_{S2}}{I_{S1} + I_{S2}} I
}
\]
---
## **Explanation**
- Both diodes share the same voltage in parallel.
- The current through each diode is proportional to its saturation current.
- The sum of their currents is the total current supplied.
- The voltage across the combination is determined by the total current and the sum of saturation currents.
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