https://file.aiquickdraw.com/mathbot/file/1726407372625m7rnylkp.jpg

To obtain the mathematical model for the circuits provided in the image, let's analyze each one in detail.

### Circuit on the Left: Inverting Operational Amplifier with Resistor \( R \) and Capacitor \( C \)

1. **Configuration**: This is an inverting amplifier circuit with a resistor \( R \) connected to the input and a capacitor \( C \) in the feedback loop.
   
2. **Kirchhoff's Current Law (KCL)**: At the inverting input node, the sum of currents should be zero:
   \[
   \frac{V_{in} - V_{-}}{R} + C \frac{d(V_{-} - V_o)}{dt} = 0
   \]
   Since the inverting input is connected to a virtual ground (ideal op-amp assumption, where \( V_{-} = 0 \)), this simplifies to:
   \[
   \frac{V_{in}}{R} + C \frac{d(V_o)}{dt} = 0
   \]
   
3. **Differential Equation**:
   \[
   V_{in} = -RC \frac{dV_o}{dt}
   \]
   This equation describes the relationship between the input voltage \( V_{in} \) and the output voltage \( V_o \).

### Circuit on the Right: Capacitor and Resistor with Operational Amplifier

1. **Configuration**: In this configuration, the input voltage \( V_{in} \) is passed through a capacitor \( C \), followed by a resistor \( R \) at the output of the operational amplifier.

2. **Low-Pass Filter Behavior**: The capacitor \( C \) blocks any DC component, allowing only AC signals to pass. The resistor \( R \) forms a low-pass filter with the capacitor.

3. **Voltage Relationship**:
   At the output node, considering the capacitor \( C \) and resistor \( R \), the transfer function for a simple low-pass filter is:
   \[
   \frac{V_{out}}{V_{in}} = \frac{1}{1 + j \omega RC}
   \]
   This transfer function shows that at higher frequencies, the output will be attenuated, and at lower frequencies, the output will approximate the input voltage.

In summary:
1. **First Circuit (Inverting Op-Amp with Capacitor)**: The equation is \( V_{in} = -RC \frac{dV_o}{dt} \).
2. **Second Circuit (Low-Pass Filter with Capacitor and Resistor)**: The transfer function is \( \frac{V_{out}}{V_{in}} = \frac{1}{1 + j \omega RC} \).

Would you like more details on any of the components or their functions?

Here are some related questions you might explore:
1. How does the frequency response of the second circuit change with \( R \) and \( C \)?
2. What would happen if the positions of the capacitor and resistor were swapped in either circuit?
3. How do the time-domain and frequency-domain representations relate for each circuit?
4. Can the first circuit be considered a high-pass filter? Why or why not?
5. What are the practical applications of each of these circuits in signal processing?

**Tip**: For both circuits, the capacitor's role is crucial in controlling how the circuit behaves with changing frequencies, especially in the context of signal filtering.

Obtain the mathematical model of the following circuits.

Discover how to derive the mathematical models for an inverting operational amplifier with a resistor and capacitor, as well as a low-pass filter circuit. This involves using Kirchhoff’s laws and differential equations, suitable for undergraduate students in electrical engineering.

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