**Definition of Norton's Theorem**
Norton's theorem states that any linear active one-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 conductance (or resistance). The value of the current source is equal to the short-circuit current at the port, while the conductance or resistance corresponds to the equivalent input conductance or resistance seen from the port after all independent sources are turned off (voltage sources are replaced by short circuits, and current sources are replaced by open circuits).
This equivalence allows engineers to simplify complex networks for easier analysis. A visual representation of this concept can be found in Figure 1.
**How to Prove Norton’s Theorem**
**(1) Content of Norton’s Theorem**
In general, any linear active single-port network can be represented as a current source connected in parallel with a resistor. The current of the source equals the short-circuit current (Isc), and the resistor represents the equivalent resistance (R0) of the network when all independent sources are removed. This model is known as the Norton equivalent circuit.
As shown in Figure 2, the Norton equivalent circuit can be derived from the Thevenin equivalent by converting the voltage source and series resistor into a current source and parallel resistor.
**(2) Proof of Norton’s Theorem**
To prove Norton’s theorem, consider a linear active one-port network N connected to an external circuit. Let the voltage at the port be U and the current be I. Using the substitution theorem, we replace the external circuit with a voltage source US = U, as shown in Figure 3.
According to the superposition principle, the current I at the port is the sum of two components: one due to the internal sources of the network, and the other due to the external source. When the external source is removed, the current becomes the short-circuit current Isc. When the internal sources are zeroed, the network behaves like a passive resistor R0.
Thus, the resulting circuit consists of a current source Isc in parallel with a resistor R0. This proves Norton’s theorem.
**Verification of Norton’s Theorem**
**First, Experimental Principle**
1) Any linear network containing sources can be treated as an active two-terminal network when analyzing a specific branch.
2) Thevenin’s theorem says such a network can be replaced by a voltage source in series with a resistor. Norton’s theorem, on the other hand, replaces it with a current source in parallel with a resistor.
**Second, Measurement Methods for Equivalent Parameters**
- **Open Circuit Voltage and Short Circuit Current Method**: Measure the open-circuit voltage (Uoc) and short-circuit current (Isc) to calculate the equivalent resistance R0. However, this method should not be used if the internal resistance is very low, as it may damage components.
- **Voltammetry**: Use a voltmeter and ammeter to measure the external characteristic curve. The slope of the curve gives the internal resistance.
- **Half-Voltage Method**: Adjust the load resistance until the output voltage is half of the open-circuit voltage. This load resistance equals the equivalent internal resistance.
- **Zero-Display Method**: Used for high-impedance networks to avoid measurement errors caused by the voltmeter’s own resistance.
**Third, Experimental Equipment**
A typical setup includes a DC power supply, a constant current source, multimeters, resistors, and a potentiometer for adjusting resistance values.
**Fourth, Experimental Procedure**
1. Measure the open-circuit voltage (Uoc) and short-circuit current (Isc) to calculate the equivalent resistance (R0).
2. Change the load resistance (RL) and record the corresponding voltage and current values to plot the external characteristic curve.
3. Verify Thevenin’s theorem by replacing the original network with its Thevenin equivalent and comparing results.
4. Similarly, verify Norton’s theorem using the Norton equivalent circuit.
5. Directly measure the equivalent resistance (R0) by turning off all independent sources and using an ohmmeter.
6. Use the half-voltage and zero-display methods to measure Uoc and R0.
**Fifth, Experimental Notes**
- Ensure proper range settings on meters before measurements.
- Avoid shorting voltage sources when they are set to zero.
- Always turn off the power before changing connections.
- Calibrate the multimeter before measuring resistance.
**Sixth, Experimental Report**
1. Plot the external characteristics and compare them with theoretical predictions to verify both Thevenin’s and Norton’s theorems.
2. Analyze discrepancies between measured and calculated values to identify possible sources of error.
**Notes on Norton’s Theorem**
1. Norton’s theorem applies only to the external behavior of the network, not to the internal structure. It cannot be used to determine internal voltages or currents.
2. If the network remains complex after applying Norton’s theorem, the process can be repeated iteratively.
3. Norton’s theorem is valid only for linear networks. Nonlinear components make the theorem inapplicable.
By understanding and applying Norton’s theorem, engineers can simplify the analysis of complex electrical networks, making it a powerful tool in circuit design and troubleshooting.
Diamond Tool Production And Processing Equipment
Diamond Tool Production And Processing Equipment,Automatic Brazing Machine For Diamond Sawblade,Big Diamond Segment Grinding Machine,Efficiency Diamond Segment Grinding Machine
Suzhou Mountain Industrial Control Equipment Co., Ltd , https://www.szmountain.com