Question

How to find RTH and VTH?

Answer 1

Finding \( R_{TH} \) (Thevenin Resistance) and \( V_{TH} \) (Thevenin Voltage) is essential for simplifying complex circuits into manageable equivalent circuits, particularly when analyzing circuits with multiple sources and components. The Thevenin theorem allows you to replace a portion of the circuit with a simple equivalent circuit composed of a single voltage source (\( V_{TH} \)) in series with a single resistor (\( R_{TH} \)). Here's how to find these values step-by-step.

### Steps to Find \( V_{TH} \)

1. **Identify the Terminals**:
   - Determine the two terminals across which you want to find the Thevenin equivalent.

2. **Remove the Load**:
   - If there’s a load connected across the terminals, remove it to avoid affecting the measurements.

3. **Calculate \( V_{TH} \)**:
   - **Method 1: Open Circuit Voltage**:
     - The Thevenin voltage \( V_{TH} \) is the voltage across the open terminals (where the load was connected).
     - You can use techniques such as voltage division, nodal analysis, or mesh analysis to find this voltage.
   - **Method 2: Test Source**:
     - Apply a known voltage or current source at the terminals and calculate the resulting voltage across the terminals, then remove the test source. The resulting voltage is \( V_{TH} \).

### Steps to Find \( R_{TH} \)

1. **Deactivate Independent Sources**:
   - **Voltage Sources**: Replace all independent voltage sources with short circuits (0 V).
   - **Current Sources**: Replace all independent current sources with open circuits (0 A).

2. **Calculate \( R_{TH} \)**:
   - **From the Terminals**:
     - Now look into the circuit from the terminals where the load was connected and calculate the equivalent resistance.
   - **Parallel and Series Combinations**: Use series and parallel resistance combinations to find the total resistance seen from the terminals.

### Example Problem

Let’s consider an example circuit to illustrate these steps.

#### Given Circuit:

- A 10 V battery is in series with a 2 kΩ resistor and a parallel combination of a 1 kΩ resistor and a 4 kΩ resistor.

#### Steps to Find \( V_{TH} \):

1. **Identify Terminals**: Assume we want to find \( V_{TH} \) across the parallel resistors.

2. **Remove Load**: There’s no load resistor in this case.

3. **Calculate \( V_{TH} \)**:
   - The voltage across the parallel combination can be calculated using voltage division.
   - First, calculate the equivalent resistance of the parallel resistors:
     \[
     R_{eq} = \frac{1}{\frac{1}{1k} + \frac{1}{4k}} = \frac{4k}{5} = 0.8kΩ
     \]
   - Then find the total resistance in the circuit:
     \[
     R_{total} = R_{series} + R_{eq} = 2k + 0.8k = 2.8kΩ
     \]
   - Use voltage division:
     \[
     V_{TH} = V_{source} \times \frac{R_{eq}}{R_{total}} = 10V \times \frac{0.8k}{2.8k} = 2.857V
     \]

#### Steps to Find \( R_{TH} \):

1. **Deactivate Sources**:
   - Short the 10 V battery.

2. **Calculate \( R_{TH} \)**:
   - Look into the circuit from the terminals:
     - The 1 kΩ and 4 kΩ resistors are still in parallel:
       \[
       R_{TH} = \frac{1}{\frac{1}{1k} + \frac{1}{4k}} = 0.8kΩ
       \]

### Summary

- The Thevenin voltage \( V_{TH} \) is found by calculating the open circuit voltage across the terminals.
- The Thevenin resistance \( R_{TH} \) is found by deactivating all sources and calculating the resistance seen from the terminals.

### Final Thevenin Equivalent Circuit

After calculating \( V_{TH} \) and \( R_{TH} \), you can represent the original circuit with a single voltage source \( V_{TH} = 2.857V \) in series with a resistance \( R_{TH} = 0.8kΩ \). This simplification makes analysis much easier, particularly for circuit simulation and load calculations.

Answer 2

To find **RTH** (Thevenin resistance) and **VTH** (Thevenin voltage) for a given electrical network, you follow the process of finding the Thevenin equivalent circuit. The Thevenin equivalent of a complex network simplifies that network to a single voltage source in series with a resistor. This is useful for analyzing the behavior of a circuit when connected to different loads.

Here’s a detailed step-by-step guide on how to find **RTH** and **VTH**:

### 1. Identify the portion of the circuit
You need to choose the part of the circuit where you want to find the Thevenin equivalent. This is typically done at two terminals (A and B), where you want to simplify the circuit.

### 2. Find the Thevenin Voltage (VTH)

The **Thevenin voltage (VTH)** is the open-circuit voltage between the two terminals of interest (where you’re going to place the load). Here’s how you find it:

#### Steps to calculate **VTH**:
- **Remove the load resistor** (if there is one) from the circuit at the terminals.
- **Calculate the open-circuit voltage** across the two terminals (A and B).
  - This is the voltage that appears across the terminals when no load is connected.
  - Use voltage division, nodal analysis, or mesh analysis to calculate this voltage if necessary.

#### Example:
Suppose you have a voltage source of 10 V in series with a 5Ω resistor and a 10Ω resistor connected to the terminals A and B. The open-circuit voltage (VTH) would be calculated using voltage division:

\[
V_{TH} = V_{source} \times \frac{R_{B}}{R_A + R_B}
\]

where \(R_A\) and \(R_B\) are the resistances in the circuit, and \(V_{source}\) is the supply voltage.

### 3. Find the Thevenin Resistance (RTH)

The **Thevenin resistance (RTH)** is the equivalent resistance seen from the terminals (A and B) with the voltage sources replaced by short circuits and current sources replaced by open circuits.

#### Steps to calculate **RTH**:
- **Turn off all independent sources**:
  - Replace all independent voltage sources with short circuits (i.e., just a wire).
  - Replace all independent current sources with open circuits (i.e., remove them from the circuit).
- **Calculate the equivalent resistance** seen from the terminals A and B.
  - Combine series and parallel resistors as needed to find the total resistance between the terminals.

#### Example:
If the circuit has a 5Ω resistor and a 10Ω resistor in series between terminals A and B, the Thevenin resistance is the sum of the two:

\[
R_{TH} = R_A + R_B = 5\Omega + 10\Omega = 15\Omega
\]

### 4. Reconnect the load (if any)

Once you've calculated **VTH** and **RTH**, you can reconnect the load resistor, and the entire circuit between the terminals A and B can be replaced with a simple Thevenin equivalent circuit: a voltage source of **VTH** in series with a resistance **RTH**.

---

### Example Problem

**Problem Statement**:
You have a circuit with a 12 V battery in series with a 10Ω resistor, and a 40Ω resistor connected to the terminals A and B. Find the Thevenin equivalent seen by the load across terminals A and B.

#### Step-by-step Solution:
1. **Identify VTH (Thevenin Voltage)**:
   - The circuit has a 12V source with a 10Ω resistor and a 40Ω resistor. The 40Ω resistor is the load.
   - Remove the load (40Ω resistor), and calculate the open-circuit voltage across the terminals A and B.
   - Apply voltage division:

   \[
   V_{TH} = V_{source} \times \frac{R_{B}}{R_A + R_B}
   \]
   
   \[
   V_{TH} = 12V \times \frac{40Ω}{10Ω + 40Ω} = 12V \times \frac{40Ω}{50Ω} = 9.6V
   \]

2. **Identify RTH (Thevenin Resistance)**:
   - Turn off the voltage source (replace it with a short circuit).
   - The 10Ω and 40Ω resistors are now in series between terminals A and B.
   - The Thevenin resistance is the sum of the resistances:

   \[
   R_{TH} = 10Ω + 40Ω = 50Ω
   \]

3. **Thevenin Equivalent Circuit**:
   - The Thevenin equivalent is a 9.6V voltage source in series with a 50Ω resistor.

You can now reconnect the load (40Ω resistor) to this Thevenin equivalent circuit and analyze the circuit's behavior.

---

### Summary of Steps:
1. **Remove the load** and find the open-circuit voltage at the terminals to get **VTH**.
2. **Turn off all independent sources** (voltage sources shorted, current sources opened) and find the equivalent resistance seen from the terminals to get **RTH**.
3. Replace the circuit with its **Thevenin equivalent**: a voltage source **VTH** in series with a resistance **RTH**.

This approach simplifies the analysis of complex circuits, especially when dealing with multiple loads.