**Definition of Norton's Theorem**
Norton’s theorem states that any linear, active, single-port network containing independent sources, linear resistors, and linear controlled sources can be simplified into an equivalent circuit consisting of a current source in parallel with a resistor. This current source is equal to the short-circuit current at the port, and the resistor represents the equivalent resistance seen from the port when all independent sources are turned off (voltage sources are replaced by short circuits, and current sources are replaced by open circuits).
This equivalent circuit is known as the Norton equivalent, and it allows for easier analysis of the external behavior of the original network without needing to consider its internal structure.
**How to Prove Norton’s Theorem**
**(1) Content of Norton’s Theorem**
In a linear, active, single-port network, the external circuit can be represented as a current source in parallel with a resistor. The value of the current source equals the short-circuit current of the network, while the resistor corresponds to the equivalent resistance obtained after removing all independent sources.
For example, if we have a network labeled as N, which contains both independent and dependent sources, we can replace it with a Norton equivalent circuit: a current source I_sc (short-circuit current) in parallel with a resistor R_eq (equivalent resistance).
**(2) Proof of Norton’s Theorem**
To prove this, we can use the principle of superposition. Consider a linear active network connected to an external load. By applying the substitution theorem, we can replace the external load with a voltage source and analyze the resulting currents.
The total current at the port is the sum of the current due to the internal sources and the external source. When the external source is removed, the remaining current is the short-circuit current I_sc. When the internal sources are turned off, the network behaves like a passive resistor R_eq.
Combining these two results gives the Norton equivalent circuit: a current source in parallel with a resistor. This proves that any linear, active, single-port network can be simplified using Norton’s theorem.
**Verification of Norton’s Theorem**
To verify Norton’s theorem, we can conduct experiments using different methods to measure the equivalent parameters of an active two-terminal network. These include:
- **Open Circuit Voltage and Short Circuit Current Method**: Measure the open-circuit voltage (Uoc) and short-circuit current (Isc), then calculate the equivalent resistance R0 = Uoc / Isc.
- **Voltammetry**: Use a voltmeter and ammeter to plot the external characteristic curve and determine R0 from the slope.
- **Half Voltage Method**: Adjust the load until the voltage is half of the open-circuit voltage; the load resistance is equal to R0.
- **Zero Indication Method**: Used for high-resistance networks to avoid measurement errors caused by the voltmeter’s internal resistance.
**Experimental Setup and Procedure**
The experiment involves building a simple active two-terminal network using a DC power supply and a current source. Then, various measurements are taken to determine the Norton equivalent parameters. The steps include:
1. Measuring the open-circuit voltage (Uoc) and short-circuit current (Isc) to calculate R0.
2. Testing the network under different load conditions to observe its external characteristics.
3. Verifying Thevenin’s and Norton’s theorems by replacing the original network with its equivalent models and comparing the results.
4. Directly measuring the equivalent resistance using an ohmmeter after zeroing all independent sources.
**Important Notes on Applying Norton’s Theorem**
- Norton’s theorem is only valid for the external circuit and does not represent the internal structure of the original network.
- It can be applied recursively to complex circuits, simplifying them step by step.
- The theorem is applicable only to linear networks. If the network contains nonlinear components, Norton’s theorem cannot be used.
By understanding and applying Norton’s theorem, engineers and students can simplify the analysis of complex electrical networks, making it easier to predict their behavior under different conditions.
Diamond Saw Blade Labeling Machine
Diamond Saw Blade Labeling Machine,Saw Blade Labeling Machine,Labeling Machine For Saw Blade,Customized Labeling Machine
Suzhou Mountain Industrial Control Equipment Co., Ltd , https://www.szmountain.com