In this first post to "Fun with Fundamentals," Bob Witte explains how phase affects the measurement of electrical power as it relates to voltage and current.

Welcome to my first blog post that explores fundamental principles of electrical engineering and attempts to tease out some new insights. For many of you, this will be a refresh of concepts you already know (or used to know) but I'll also try to add some new insights into the mix.

**Jacobi's Law**

Most engineers are familiar with the Maximum Power Transfer Theorem (also known as Jacobi's Law). **Figure 1** shows a resistive source and resistive load connected with the aim of transferring power from the source to the load. This principle can be stated as: “Maximum power is transferred when the internal resistance of the source equals the resistance of the load, when the external resistance can be varied, and the internal resistance is constant," (**Figure 1**).

**Figure 1. Circuit diagram showing a resistive source connected to a resistive load.**

Maximum power transfer to the load will occur when R_{L} = R_{S}. An often-overlooked constraint is that we are assuming the source resistance (R_{S}) is fixed and not under our control. Otherwise, we might want to choose R_{S} = 0 as a good value for getting maximum power transferred out of the source. (More on that in a future post).

**Figure 2** shows how the power to the load varies as a function of *R _{L}*/

*R*. Power delivered to R

_{S}_{L}depends on both the current through the load and the voltage across the load. Large values of

*R*increase the voltage (

_{L}*V*) but starve the current (

_{L}*I*). Similarly, small values of

_{L}*R*increase the load current but diminish the load voltage. A bit of calculus can show that maximum power occurs when

_{L}*R*=

_{L}*R*. (Ref. 1 provides a derivation without the use of calculus.)

_{S}**Figure 2. Plot of**

*P*vs._{L}*R*/_{L}*R*shows maximum power to the load when_{S}*R*/_{L}*R*=1._{S}**Complex impedance**

Now consider the AC case where the impedances are complex, as shown in **Figure 3**. The source impedance is *Z _{S}* =

*R*+

_{S}*jX*and the load impedance is

_{S}*Z*=

_{L }*R*+

_{L}*jX*. Maximum power transfer occurs when

_{L}*Z*is the complex conjugate of

_{L}*Z*. (References 1 and 2 provide a proof of this result.) In other words,

_{S}*R*=

_{L}*R*and

_{S}*X*= -

_{L}*X*. This is sometimes referred to as complex conjugate matching. As expected, if X

_{S}_{S}=0, the situation degenerates back to the resistive case.

**Figure 3. Circuit diagram showing a source connected to a load where both have complex impedances.**

[Continue reading on EDN US: It's all about the phase]

*Bob Witte is president of Signal Blue LLC, a technology consulting company.*

**References**

- [1] “Non-Calculus Derivation of the Maximum Power Transfer Theorem,” Kenneth V. Cartwright, Ph.D.
- [2] “Maximum Power Transfer Theorem in AC Circuit”
- [3] “True, Reactive, and Apparent Power”

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