1 Answer
Tellegen's theorem is a fundamental theorem in electrical engineering, particularly in circuit theory. It states that for any linear network (containing both active and passive components), the total power entering the network equals the total power leaving the network, provided the voltage and current in each branch are related by a linear relationship.
To verify Tellegen’s theorem, we typically proceed by showing that the sum of powers entering and leaving a network is equal. Here’s a step-by-step approach to verify it:
### 1. **State the Theorem Mathematically:**
Tellegen’s theorem is usually written as:
\[
\sum_{k=1}^{N} v_k i_k = 0
\]
Where:
- \( v_k \) is the voltage across the \( k \)-th branch.
- \( i_k \) is the current through the \( k \)-th branch.
- \( N \) is the total number of branches in the network.
The left-hand side represents the power of each branch, where power \( P = V \times I \) (voltage times current). The sum of the powers over all branches must be zero.
### 2. **Choose a Linear Network:**
To verify this, you can work with a simple example of a linear network, like a resistor network or an RLC circuit. The network must be linear, meaning the elements should behave according to Ohm’s law (i.e., voltage and current are linearly related for each element).
### 3. **Apply Kirchhoff’s Laws (KVL & KCL):**
Use Kirchhoff’s Voltage Law (KVL) and Kirchhoff’s Current Law (KCL) to analyze the network. These laws ensure that the network follows the basic principles of conservation of energy and charge.
- **KVL**: The sum of voltages around any closed loop is zero.
- **KCL**: The sum of currents at any node is zero.
### 4. **Calculate the Total Power:**
For each branch, calculate the instantaneous power \( P_k = v_k i_k \), where:
- \( v_k \) is the voltage at the \( k \)-th branch.
- \( i_k \) is the current at the \( k \)-th branch.
Then, compute the total power of the network, which is the sum of the individual powers in each branch:
\[
P_{\text{total}} = \sum_{k=1}^{N} v_k i_k
\]
### 5. **Verify the Power Balance:**
Tellegen’s theorem asserts that the sum of powers entering and leaving the network must be zero. This means that:
\[
\sum_{k=1}^{N} v_k i_k = 0
\]
If this equality holds true, the network satisfies Tellegen’s theorem.
### 6. **Consider a Simple Example:**
Let’s take a simple network with resistors. Suppose there are two resistors connected in series with a voltage source. You can:
- Calculate the power delivered by the voltage source.
- Calculate the power dissipated by each resistor.
According to Tellegen’s theorem, the total power delivered by the source should equal the total power dissipated by the resistors (ignoring any energy storage elements, like capacitors or inductors).
### 7. **Conclusion:**
After performing the calculations, if the total incoming power is equal to the total outgoing power, you have verified Tellegen's theorem for that particular network.
### Key Notes:
- The theorem holds for any linear network, regardless of whether it contains active or passive components.
- It is a statement of conservation of energy, applied to electrical networks.
- The theorem is particularly useful in complex circuits, where direct calculation of powers is not always straightforward.
In summary, to verify Tellegen’s theorem, you analyze a network, calculate the power in each branch, and confirm that the sum of all powers is zero.
To verify Tellegen’s theorem, we typically proceed by showing that the sum of powers entering and leaving a network is equal. Here’s a step-by-step approach to verify it:
### 1. **State the Theorem Mathematically:**
Tellegen’s theorem is usually written as:
\[
\sum_{k=1}^{N} v_k i_k = 0
\]
Where:
- \( v_k \) is the voltage across the \( k \)-th branch.
- \( i_k \) is the current through the \( k \)-th branch.
- \( N \) is the total number of branches in the network.
The left-hand side represents the power of each branch, where power \( P = V \times I \) (voltage times current). The sum of the powers over all branches must be zero.
### 2. **Choose a Linear Network:**
To verify this, you can work with a simple example of a linear network, like a resistor network or an RLC circuit. The network must be linear, meaning the elements should behave according to Ohm’s law (i.e., voltage and current are linearly related for each element).
### 3. **Apply Kirchhoff’s Laws (KVL & KCL):**
Use Kirchhoff’s Voltage Law (KVL) and Kirchhoff’s Current Law (KCL) to analyze the network. These laws ensure that the network follows the basic principles of conservation of energy and charge.
- **KVL**: The sum of voltages around any closed loop is zero.
- **KCL**: The sum of currents at any node is zero.
### 4. **Calculate the Total Power:**
For each branch, calculate the instantaneous power \( P_k = v_k i_k \), where:
- \( v_k \) is the voltage at the \( k \)-th branch.
- \( i_k \) is the current at the \( k \)-th branch.
Then, compute the total power of the network, which is the sum of the individual powers in each branch:
\[
P_{\text{total}} = \sum_{k=1}^{N} v_k i_k
\]
### 5. **Verify the Power Balance:**
Tellegen’s theorem asserts that the sum of powers entering and leaving the network must be zero. This means that:
\[
\sum_{k=1}^{N} v_k i_k = 0
\]
If this equality holds true, the network satisfies Tellegen’s theorem.
### 6. **Consider a Simple Example:**
Let’s take a simple network with resistors. Suppose there are two resistors connected in series with a voltage source. You can:
- Calculate the power delivered by the voltage source.
- Calculate the power dissipated by each resistor.
According to Tellegen’s theorem, the total power delivered by the source should equal the total power dissipated by the resistors (ignoring any energy storage elements, like capacitors or inductors).
### 7. **Conclusion:**
After performing the calculations, if the total incoming power is equal to the total outgoing power, you have verified Tellegen's theorem for that particular network.
### Key Notes:
- The theorem holds for any linear network, regardless of whether it contains active or passive components.
- It is a statement of conservation of energy, applied to electrical networks.
- The theorem is particularly useful in complex circuits, where direct calculation of powers is not always straightforward.
In summary, to verify Tellegen’s theorem, you analyze a network, calculate the power in each branch, and confirm that the sum of all powers is zero.