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Chapter 2: Problem 82

List the steps in determining the Th茅venin and Norton equivalent circuits for a general two-terminal circuit. Try this as if it were a "closed-book" exam question.

Short Answer

Expert verified
Determine \( V_{oc} \), calculate \( I_{sc} \), find \( R_{th} \), and build equivalent circuits.

Step by step solution

01

Open-Circuit Voltage

Find the open-circuit voltage, denoted as \( V_{oc} \), across the two terminals where the Th茅venin or Norton equivalent is to be found. This is done by determining the voltage across the terminals when no external load is connected.
02

Short-Circuit Current

Calculate the short-circuit current, \( I_{sc} \), between the same two terminals. This is done by connecting a wire (short) across the terminals and measuring the current that flows through it.
03

Th茅venin Resistance

Determine the Th茅venin resistance, \( R_{th} \), by using the formula \( R_{th} = \frac{V_{oc}}{I_{sc}} \) if both open-circuit voltage and short-circuit current can be easily found. Alternatively, deactivate all independent voltage sources (replace them with a short circuit) and all independent current sources (replace them with an open circuit) and then calculate the resistance across the terminals.
04

Form Th茅venin Equivalent Circuit

Using the values obtained, the Th茅venin equivalent circuit is made of a voltage source \( V_{th} = V_{oc} \) in series with the resistance \( R_{th} \).
05

Convert to Norton Equivalent Circuit

To convert the Th茅venin equivalent circuit to a Norton equivalent, use the current \( I_{N} = I_{sc} \) and the same resistance \( R_{N} = R_{th} \). The Norton equivalent circuit consists of a current source \( I_{N} \) in parallel with the resistance \( R_{N} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Open-Circuit Voltage
The open-circuit voltage, often symbolized as \( V_{oc} \), is a fundamental concept when determining the Th茅venin equivalent circuit. Imagine you have two terminals of a circuit with nothing connected between them. The voltage across these terminals is what we call the open-circuit voltage.
The beauty of open-circuit voltage is that it reveals the potential difference when there is no external load, meaning no current flows through it. It's like assessing the pressure in a water hose with no water flowing.
To find \( V_{oc} \), simply make sure no load is connected to your terminals. Measure the voltage across them with a voltmeter. This reading is your \( V_{oc} \).
  • Disconnect all external loads.
  • Measure the voltage across the open terminals.
It鈥檚 essential to get this right, as it sets the stage for calculating Th茅venin resistance and other parameters.
Short-Circuit Current
Short-circuit current, noted as \( I_{sc} \), is another crucial piece of the puzzle when working with equivalent circuits. Essentially, it is the current that would flow if the terminals in question were connected directly by a conductor鈥攍ike pulling a water plug and letting the water rush out.
To determine \( I_{sc} \), you connect a wire directly across the two terminals, essentially shorting the circuit. A current meter will help you read the current flowing through the wire鈥攁 direct measure of \( I_{sc} \). Remember:
  • Use a low-resistance conductor to connect the terminals.
  • Measure the current flowing through using an ammeter.
This step not only helps ascertain the current flow under direct short-circuit conditions but also assists in evaluating the circuit's response in extreme situations.
Th茅venin Resistance
Th茅venin resistance is key to simplifying and analyzing circuits, represented as \( R_{th} \). It is the equivalent resistance seen from the terminals when all the independent sources are turned off.
You have two primary methods to find \( R_{th} \):
The first method involves the formula:
  • Use \( R_{th} = \frac{V_{oc}}{I_{sc}} \).
If both \( V_{oc} \) and \( I_{sc} \) are known, this provides an effortless way to compute \( R_{th} \).
The second approach deactivates all sources:
  • For voltage sources鈥攔eplace them with a short circuit.
  • For current sources鈥攔eplace them with an open circuit.
  • Measure the resistance across the terminals.
Th茅venin resistance helps you build a more straightforward representation of complex networks, making circuit analysis and design much more intuitive.

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Most popular questions from this chapter

A parallel circuit (i.e., all elements are in parallel with one another) has a \(60-\Omega\) resistance, a \(20-\Omega\) resistance, an unknown resistance \(R_{x}\), and \(30 \mathrm{~mA}\) current source. The current through the unknown resistance is \(10 \mathrm{~mA}\). Determine the value of \(R_{x}\).

If we measure the voltage at the terminals of a two-terminal network with two known (and different) resistive loads attached, we can determine the Th茅venin and Norton equivalent circuits. When a 1-k \(\Omega\) load is attached to a two- terminal circuit, the load voltage is \(8 \mathrm{~V}\). When the load is increased to \(2 \mathrm{k} \Omega\), the load voltage becomes 10 V. Find the Th茅venin voltage and resistance for the circuit.

The heating element of an electric cook top has two resistive elements, \(R_{1}=40 \Omega\) and \(R_{2}=100 \Omega\), which can be operated separately, in series, or in parallel from voltages of either \(120 \mathrm{~V}\) or \(240 \mathrm{~V}\). For the lowest power, \(R_{1}\) is in series with \(R_{2}\), and the combination is operated from \(120 \mathrm{~V}\). What is the lowest power? For the highest power, how should the elements be operated? What power results? List three more modes of operation and the resulting power for each.

We have a \(15-\mathrm{V}\) source and a load that absorbs power and requires a current varying between 0 and \(100 \mathrm{~mA}\). The voltage across the load must remain between \(4.7\) and \(5.0 \mathrm{~V}\) for all values of load current. Design a voltagedivider network to supply the load. You may assume that resistors of any value desired are available. Also, give the minimum power rating for each resistor.

The open circuit voltage of a certain twoterminal circuit is \(15 \mathrm{~V}\). When a \(150-\Omega\) load is connected, the voltage across the load is \(12 \mathrm{~V}\). Determine the Th茅venin resistance for the circuit.
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