AM-GM Inequality for Minimum Power Dissipation

Problem: AM-GM Inequality for Minimum Power Dissipation

An electrical source is connected to a circuit containing two variable resistors ##R_1## and ##R_2## in series. If the product of the two resistances is held constant such that ##R_1 R_2 = C##, where ##C## is a positive constant, determine the minimum value of the total series resistance ##R_s = R_1 + R_2## using the Arithmetic Mean-Geometric Mean (AM-GM) inequality. Provide a rigorous mathematical proof.

Worked Solution & Step-by-Step Explanation

1. Understanding the Physical and Mathematical Constraints

In passive electrical circuits, resistance is a physical quantity that represents the opposition to the flow of electric current. By physical laws, the resistance of standard passive components must be non-negative real numbers. Therefore, we establish our domain constraints as:

We are given that the product of these two positive resistances is constrained to a constant value ##C##:

The total equivalent resistance of the two resistors connected in series, denoted as ##R_s##, is given by the sum of their individual resistances:

Our objective is to find the minimum possible value of ##R_s## under the given product constraint using the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

2. Rigorous Proof of the AM-GM Inequality for Two Variables

To ensure mathematical rigor, we first prove the AM-GM inequality for two positive real variables ##x_1## and ##x_2##. Let ##x_1, x_2 \in \mathbb{R}^+##.

Consider the square of the difference between their square roots. Since the square of any real number is non-negative, we can write:

Expanding the left-hand side of this inequality using the algebraic identity ##(a - b)^2 = a^2 - 2ab + b^2##, we obtain:

Since ##x_1## and ##x_2## are positive, ##(\sqrt{x_1})^2 = x_1## and ##(\sqrt{x_2})^2 = x_2##. Substituting these back into the expression yields:

Adding ##2\sqrt{x_1 x_2}## to both sides of the inequality:

Dividing both sides by ##2##, we arrive at the standard formulation of the AM-GM inequality for two variables:

Analyzing the Condition for Equality

The equality holds if and only if the initial squared term is zero:

Taking the square root of both sides:

Squaring both sides yields the final equality condition:

3. Application of AM-GM to the Circuit Problem

We map our physical variables directly to the proven inequality. Let ##x_1 = R_1## and ##x_2 = R_2##. Since ##R_1 > 0## and ##R_2 > 0##, we apply the AM-GM inequality:

Multiplying both sides of the inequality by ##2##:

We substitute the expression for total series resistance ##R_s = R_1 + R_2## and the given constraint ##R_1 R_2 = C## into the inequality:

This inequality establishes that the total series resistance ##R_s## is bounded below by ##2\sqrt{C}##. Therefore, the minimum value of ##R_s## is:

4. Determining the Minimizing Resistance Values

The minimum value ##R_{s,\text{min}}## is achieved when the condition for equality in the AM-GM inequality is satisfied. As proven in Section 2, equality holds if and only if the terms are equal:

Using the constraint equation ##R_1 R_2 = C##, we substitute ##R_2## with ##R_1##:

Since resistance must be positive (##R_1 > 0##), we take the positive square root:

Consequently, because ##R_2 = R_1##, we find:

Thus, the minimum total series resistance is achieved when both variable resistors are set to equal values of ##\sqrt{C}##.

5. Physical Significance in Power Dissipation

In electrical engineering, if this series combination is connected to a constant voltage source ##V##, the total power dissipated by the circuit is given by:

To minimize the power dissipation of the circuit for a fixed voltage source, we must maximize the total resistance ##R_s##. Conversely, to maximize the power dissipation (or to design for maximum current flow), we minimize ##R_s##. The minimum resistance occurs when ##R_1 = R_2 = \sqrt{C}##, yielding the maximum power dissipation state for a constant voltage supply:

This demonstrates how algebraic optimization tools like the AM-GM inequality directly govern the limits of physical systems and energy efficiency calculations in circuit design.