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Chapter 1: Problem 56

Sketch the diagram of a circuit that contains a \(5-\Omega\) resistor, a \(10-V\) voltage source, and a voltage-controlled voltage source having a gain constant of \(0.5 \mathrm{V} / \mathrm{V}\), Assume that the voltage across the resistor is the control voltage for the controlled source. Place all three elements in series. Several answers are possible, depending on the polarities chosen for the sources and the control voltage.

Short Answer

Expert verified
The circuit has a 5-ohm resistor, a 10-V source, and a VCVS (0.5 * V_R) in series.

Step by step solution

01

Analyze Circuit Components

List the components required for the circuit: a 5-ohm resistor, a 10-V voltage source, and a voltage-controlled voltage source (VCVS) with a gain of 0.5 V/V. The control voltage for the VCVS is taken across the 5-ohm resistor.
02

Understand Series Configuration

In a series circuit, components are connected end-to-end so that there is only one path for current flow. This means the same current flows through all the components, but the voltage across each component can differ.
03

Sketch the Circuit

Start by placing the 10-V voltage source with a positive terminal on the left. Next, connect the 5-ohm resistor in series to the positive terminal of the voltage source. Lastly, connect the VCVS in series, leaving one terminal to connect to the negative terminal of the 10-V source, completing the loop.
04

Determine Polarity and Control Voltage

The control voltage ( trl) is the voltage across the 5-ohm resistor. Assume conventional current flow (from positive to negative). Connect the '+' terminal of the VCVS toward the 10-V source and '-' toward the resistor, so it adds or subtracts as per the gain constant of 0.5 * V_Ctrl.
05

Calculate VCVS Output

The output voltage of the VCVS will be 0.5 times the control voltage across the resistor. If the resistor has a voltage drop of V_R, then the VCVS output would be 0.5*V_R.
06

Finalize the Diagram

Double-check all connections: the 10-V source, 5-ohm resistor, and VCVS are all in series, ensuring the circuit is a closed loop and polarities correspond to the intended control voltage flow.

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Key Concepts

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

Series Circuit
A series circuit is a straightforward type of electrical circuit. All components are connected in a single, continuous loop. This connection pattern means that there is only one path for the current to travel. The same current flows sequentially through each component.

Key features of series circuits include:
  • Identical current throughout: In a series circuit, the current flowing through each component is the same.
  • Different voltages: Each component may have a different voltage across it, depending on its resistance and the total circuit voltage.
  • A single loop: All components are neatly connected end-to-end in one continuous loop, forming a complete circuit.
These characteristics are essential when designing and analyzing any series circuit, such as the one in the exercise that includes a resistor, a voltage source, and a voltage-controlled voltage source.
Circuit Diagram
A circuit diagram is a simplified drawing representing an electrical circuit. To understand and construct circuits, recognizing the symbols for various components in the diagram is crucial.

In the exercise, the circuit diagram includes the following components:
  • Resistor: Denoted typically by a zig-zag line. Our circuit uses a 5-ohm resistor.
  • Voltage Source: Usually represented by a circle with a '+/-' sign. Here, it鈥檚 a 10-V source.
  • Voltage-Controlled Voltage Source (VCVS): A specialized component shown by a diamond shape and often includes an arrow indicating the direction of the controlled output voltage.
A well-drawn circuit diagram aids in visualizing the connections and understanding how electricity flows, ensuring that our design functions as intended. For our example, the components align in series, confirming that all are interconnected correctly in sequence.
Resistor Control Voltage
In certain circuits, the voltage across a resistor is pivotal. This concept, known as the control voltage, plays a significant role in managing how other components function, especially in circuits with controlled sources.

For example:
  • In our exercise, the voltage across the 5-ohm resistor is the control voltage.
  • This control voltage drives the voltage-controlled voltage source (VCVS), which has a gain constant indicating how it amplifies or affects the voltage it controls.
  • The gain constant for our VCVS is 0.5 V/V, meaning the output voltage is half the control voltage across the resistor.
Understanding the resistor control voltage is crucial for correctly implementing and adjusting circuits with controlled sources, as it directly affects the behavior and performance of the circuit.

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Most popular questions from this chapter

Figure \(P 1.27\) shows an ammeter \((A M)\) and voltmeter \((\mathrm{VM})\) connected to measure the current and voltage, respectively, for circuit clement \(A\). When current actually enters the + terminal of the ammeter, the reading is positive, and when current leaves the \(+\) terminal, the reading is negative. If the actual voltage polarity is positive at the \(+\) terminal of the \(\mathrm{VM}\), the reading is positive; otherwise, it is negative. (Actually, for the connection shown, the ammeter reads the sum of the current in element \(A\) and the very small current taken by the voltmeter. For purposes of this problem, assume that the current taken by the voltmeter is negligible.) Find the power for element \(A\) and state whether energy is being delivered to element \(A\) or taken from it if a. the ammeter reading is \(+2 A\) and the voltmeter reading is \(-25 \mathrm{V} ;\) b. the ammeter reading is \(-2 \mathrm{A}\) and the voltmeter reading is \(+25 \mathrm{V} ; \mathrm{c}\), the ammeter reading is \(-2 \mathrm{A}\) and the voltmeter reading is \(-25 \mathrm{V}\).

In the fluid-flow analogy for electrical circuits, what is analogous to: a a conductor; b. an open switch; \(\mathbf{c}\), a resistance; da battery?

We have a circuit element with terminals \(a\) and \(b\). Furthermore, the element has \(v_{a b}=\) \(5 \mathrm{V}\) and \(i_{a b}=2 \mathrm{A}\). Over a period of \(10 \mathrm{sec}\)onds, how much charge moves through the element? If electrons carry the charge, which terminal do they enter? How much energy is transferred? Is it delivered to the clement or taken from it?

Write a few paragraphs describing an interesting application of electrical engineering in your field, Consult engineering journals and trade magazines such as the IEEE Spectrum, Automotive Engineering Chemical Engineering, or Civil Engineering for ideas.

A copper wire has a diameter of \(2.05 \mathrm{mm}\) and carries a current of 5 A duc solely to electrons. (These values are common in residential wiring, ) Each electron has a charge of \(1.60 \times 10^{-19} \mathrm{C}\). Given that the free electron (these are the electrons capable of moving through the copper) concentration in copper is \(10^{29}\) electrons/m \(^{3}\), find the average velocity of the electrons in the wire.
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