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Chapter 3: Problem 19

Equivalent Circuits and Power Suppose we have an arbitrary circuit of resistors that we collapse into an equivalent resistor using the series and parallel rules. Is the power dissipated by the equivalent resistor equal to the sum of the powers dissipated by the actual resistors comprising the circuit? Let's start with simple cases and build up to a complete proof. a) Suppose resistors \(R_{1}\) and \(R_{2}\) are connected in parallel. Show that the power dissipated by \(R_{1} \| R_{1}\) equals the sum of the powers dissipated by the component resistors. b) Now suppose \(R_{1}\) and \(R_{2}\) are connected in series. Show the same result for this combination. c) Use these two results to prove the general result we seek.

Short Answer

Expert verified
Yes, the power dissipated by the equivalent resistor equals the sum of the powers dissipated by the actual resistors.

Step by step solution

01

Initial setup for parallel resistors

We have two resistors, \( R_1 \) and \( R_2 \), connected in parallel. The voltage across each resistor is the same. Let this voltage be \( V \). For resistors in parallel, the equivalent resistance \( R_{eq} \) is given by \( \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} \).
02

Power in parallel resistors

The power dissipated by the equivalent parallel resistor \( R_{eq} \) is \( P_{eq} = \frac{V^2}{R_{eq}} \). The power dissipated by each individual resistor is \( P_1 = \frac{V^2}{R_1} \) and \( P_2 = \frac{V^2}{R_2} \). Adding these, the total power is \( P_1 + P_2 = \frac{V^2}{R_1} + \frac{V^2}{R_2} = \frac{V^2}{R_{eq}} \). Thus, the power across the equivalent resistor is the sum of the power across \( R_1 \) and \( R_2 \).
03

Initial setup for series resistors

Two resistors, \( R_1 \) and \( R_2 \), are connected in series. The current \( I \) flowing through both resistors is the same. The total resistance for series resistors is \( R_{eq} = R_1 + R_2 \).
04

Power in series resistors

The power dissipated by the equivalent series resistor \( R_{eq} \) is \( P_{eq} = I^2 R_{eq} = I^2 (R_1 + R_2) \). For each resistor, the power is \( P_1 = I^2 R_1 \) and \( P_2 = I^2 R_2 \), thus \( P_1 + P_2 = I^2 R_1 + I^2 R_2 = I^2 (R_1 + R_2) = P_{eq} \). The power dissipated by the equivalent series resistor equals the sum of the powers of individual resistors.
05

Combining results for general proof

In both cases, whether resistors are in series or parallel, the power dissipated by the equivalent resistance is equal to the sum of the powers of all individual resistors. This property holds due to the conservation of energy, as energy dissipated as heat in resistors is additive. Hence, the general result that the power dissipated by the equivalent resistor is equal to the sum of the powers of all resistors in the circuit holds true for any combination of series and parallel resistors.

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Key Concepts

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

Power Dissipation
Power dissipation refers to the process where electrical energy is converted into heat within a component, such as a resistor. When a current flows through a resistor, it encounters resistance, and energy is lost as heat, which is known as power dissipation.
  • Power dissipated by a resistor is calculated using the formula: \( P = I^2 R \) or \( P = \frac{V^2}{R} \) where \( P \) is the power, \( I \) is the current, \( V \) is the voltage, and \( R \) is the resistance.
  • In essence, power dissipation is the product of the current squared and the resistance, or the square of the voltage divided by the resistance.
Understanding power dissipation is crucial because it helps determine the efficiency of a circuit and ensures that components do not overheat beyond their endurance.
Resistors in Series
When resistors are placed in a sequence or series within a circuit, they share the same current. This setup is called resistors in series. In a series configuration,
  • The total or equivalent resistance, \( R_{eq} \), is the sum of all individual resistances: \( R_{eq} = R_1 + R_2 + R_3 + ... \).
  • Since the current is uniform throughout the series, the equivalent resistance affects the entire current flow.
Power considerations in series circuits show that the overall power dissipation is the sum of power dissipated by each resistor individually. Thus, the series circuit's share of voltage and total power helps understand how energy conversions impact each resistive component.
Resistors in Parallel
Resistors in parallel are connected across the same two points, so each resistor experiences the same voltage.
  • The equivalent resistance for parallel resistors is given by: \( \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + ... \).
  • With a parallel combination, this configuration allows multiple paths for current, dividing the current inversely according to resistance.
Power dissipation in a parallel circuit means that the power dissipated by the equivalent resistor equals the sum of the powers dissipated by individual resistors. This happens since current spreads to ensure equal voltages, influencing energy use correspondingly across resistors.
Conservation of Energy
The principle of conservation of energy states that energy cannot be created or destroyed within an isolated system, only transformed from one form to another.
  • In electrical circuits, the total energy supplied by a source is equal to the total energy consumed across all circuit components.
  • This principle is vital because it governs that the amount of power (energy per unit time) supplied to a circuit must equal the sum of all power dissipations within it.
Whether resistors are in series or parallel, the total power dissipated remains constant due to this energy conservation. Hence, understanding how energy distributes in circuits helps engineers design efficient and safe electronic systems.

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Most popular questions from this chapter

Linear, Time-Invariant Systems For a system to be completely characterized by a transfer function, it needs not only be linear, but also to be time- invariant. A system is said to be time-invariant if delaying the input delays the output by the same amount. Mathematically, if \(S(x(t))=y(t),\) meaning \(y(t)\) is the output of a system \(S(\bullet)\) when \(x(t)\) is the input, \(S(\bullet)\) is the time-invariant if \(S(x(t-\tau))=y(t-\tau)\) for all delays \(\tau\) and all inputs \(x(t) .\) Note that both linear and nonlinear systems have this property. For example, a system that squares its input is time- invariant. a) Show that if a circuit has fixed circuit elements (their values don't change over time), its input-output relationship is time-invariant. Hint: Consider the differential equation that describes a circuit's inputoutput relationship. What is its general form? Examine the derivative(s) of delayed signals. b) Show that impedances cannot characterize time-varying circuit elements \((\mathrm{R}, \mathrm{L},\) and \(\mathrm{C}) .\) Consequently, show that linear, time-varying systems do not have a transfer function. c) Determine the linearity and time-invariance of the following. Find the transfer function of the linear, time-invariant (LTI) one(s). i) diode ii) \(y(t)=x(t) \sin \left(2 \pi f_{0} t\right)\) iii) \(y(t)=x\left(t-\tau_{0}\right)\) iv) \(y(t)=x(t)+N(t)\)

Big is Beautiful Sammy wants to choose speakers that produce very loud music. He has an amplifier and notices that the speaker terminals are labeled " \(8 \Omega\) source." a) What does this mean in terms of the amplifier's equivalent circuit? b) Any speaker Sammy attaches to the terminals can be well-modeled as a resistor. Choosing a speaker amounts to choosing the values for the resistor. What choice would maximize the voltage across the speakers? c) Sammy decides that maximizing the power delivered to the speaker might be a better choice. What values for the speaker resistor should be chosen to maximize the power delivered to the speaker?

Cartesian to Polar Conversion Convert the following expressions into polar form. Plot their location in the complex plane \(^{36}\). a) \((1+\sqrt{-3})^{2}\) b) \(3+j^{4}\) c) \(\frac{2-j \frac{6}{\sqrt{3}}}{2+j \frac{6}{\sqrt{3}}}\) d) \(\left(4-j^{3}\right)\left(1+j \frac{1}{2}\right)\) e) \(3 e^{j \pi}+4 e^{j \frac{\pi}{2}}\) f) \((\sqrt{3}+j) 2 \times \sqrt{2} e^{-\left(j \frac{\pi}{4}\right)}\) g) \(\frac{3}{1+j 3 \pi}\)

Circuit Detective Work In the lab, the open-circuit voltage measured across an unknown circuit's terminals equals \(\sin (t) .\) When a \(1 \Omega\) resistor is place across the terminals, a voltage of \(\frac{1}{\sqrt{2}} \sin \left(t+\frac{\pi}{4}\right)\) appears. a) What is the Th茅venin equivalent circuit? b) What voltage will appear if we place a \(1 \mathrm{~F}\) capacitor across the terminals?

Fun in the Lab You are given an unopenable box that has two terminals sticking out. You assume the box contains a circuit. You measure the voltage \(\sin \left(t+\frac{\pi}{4}\right)\) across the terminals when nothing is connected to them and the current \(\sqrt{2} \cos (t)\) when you place a wire across the terminals. a) Find a circuit that has these characteristics. b) You attach a \(1 \mathrm{H}\) inductor across the terminals. What voltage do you measure?
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