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This problem involves a MOSFET transistor operating in the saturation region. We are asked to find the input voltage \( U_{IN} \) given the output voltage \( U_{OUT} = 5 \, \text{V} \). 

### Given Data:
- \( K = 1 \, \text{mA/V}^2 \)
- \( U_{TH} = 1 \, \text{V} \) (threshold voltage)
- \( R_D = 5 \, \text{k}\Omega \)
- \( R_S = 2 \, \text{k}\Omega \)
- \( U_{DD} = 10 \, \text{V} \)
- \( U_{OUT} = 5 \, \text{V} \)

### Step-by-Step Solution:

1. **Identify the MOSFET operating point:**

   Since the transistor is in saturation:
   \[
   I_D = \frac{K}{2} \left( U_{GS} - U_{TH} \right)^2
   \]
   where \( U_{GS} \) is the gate-source voltage.

2. **Determine the drain current \( I_D \):**

   The drain current \( I_D \) can be expressed in terms of the output voltage \( U_{OUT} \) as:
   \[
   I_D = \frac{U_{DD} - U_{OUT}}{R_D} = \frac{10 \, \text{V} - 5 \, \text{V}}{5 \, \text{k}\Omega} = \frac{5 \, \text{V}}{5000 \, \Omega} = 1 \, \text{mA}
   \]

3. **Use the relationship between \( U_{GS} \) and \( I_D \):**

   From the saturation region equation:
   \[
   I_D = \frac{K}{2} \left( U_{GS} - U_{TH} \right)^2
   \]
   Plugging in the known values:
   \[
   1 \, \text{mA} = \frac{1 \, \text{mA/V}^2}{2} \left( U_{GS} - 1 \, \text{V} \right)^2
   \]
   Simplifying:
   \[
   2 = \left( U_{GS} - 1 \, \text{V} \right)^2
   \]
   Taking the square root:
   \[
   U_{GS} - 1 \, \text{V} = \sqrt{2} \approx 1.414 \, \text{V}
   \]
   So,
   \[
   U_{GS} \approx 1.414 \, \text{V} + 1 \, \text{V} = 2.414 \, \text{V}
   \]

4. **Find \( U_{IN} \) using \( U_{GS} \):**

   The relationship between \( U_{IN} \) and \( U_{GS} \) is given by:
   \[
   U_{GS} = U_{IN} - I_D \cdot R_S
   \]
   Substituting the values:
   \[
   2.414 \, \text{V} = U_{IN} - (1 \, \text{mA} \cdot 2 \, \text{k}\Omega)
   \]
   \[
   2.414 \, \text{V} = U_{IN} - 2 \, \text{V}
   \]
   Therefore,
   \[
   U_{IN} = 2.414 \, \text{V} + 2 \, \text{V} = 4.414 \, \text{V}
   \]

### Final Answer:
\[
U_{IN} = 4.414 \, \text{V}
\]

Would you like further details on any of these steps? Here are five related questions that could extend this problem:

1. How would the input voltage \( U_{IN} \) change if the resistor \( R_S \) is doubled?
2. What happens to the output voltage \( U_{OUT} \) if \( U_{IN} \) is increased by 1 V?
3. How would the solution change if the transistor was operating in the linear region instead of saturation?
4. How do temperature changes potentially affect the threshold voltage \( U_{TH} \) and the operation of the transistor?
5. What is the significance of the transconductance parameter \( K \) in MOSFET operation?

**Tip:** Always verify the operating region of the transistor before proceeding with calculations, as it determines the appropriate equations to use.

Odredite vrijednost ulaznog napona UIN, ako je izlazni napon 5 V. Tranzistor radi u području zasićenja. K = 1 mA/V^2, UTH = 1 V, RD = 5 kΩ, RS = 2 kΩ, UDD = 10 V.

Solve a university-level electronics engineering problem involving a MOSFET transistor in the saturation region. Learn how to calculate the input voltage UIN given the output voltage and other circuit parameters. This problem covers key concepts in electronic circuit analysis and transistor theory.

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