Active and Passive Elements Explained

Recall that a source is any device that delivers energy into to a system. A source is a device that actively generates electrical energy, such as a battery — whereas, an electrical load is considered to be a device that passively extracts energy from a system, such as a speaker.

There are two types of elements we want to consider in circuits — those two types of elements are active and passive.

• Active Elements: An active element is the circuit component that supplies energy to other circuit components. Active elements control the flow of charge (electrons) within a circuit — this means that every circuit requires at least one active element to create the flow of current. In short, active elements deliver energy or power to a circuit.
• Passive Elements: Unlike an active element, which is a component that creates energy for a circuit, a passive element is a component that can only receive energy within the circuit. A passive element cannot supply energy into a circuit and they don’t provide any gain in energy or current to the system. They require an active element to function in the circuit. In short, passive elements absorb energy or power within a circuit.

A Review of The Conservation of Power

Recall the definition of power in purely resistive circuits is the measure of how much work or energy transfer the circuit can do over an amount of time. The formula for power is given as the product of the voltage (V) across a purely resistive element and the current (I) through that element — or what I like to call “piv”:

\begin{equation}\label{eq:NL}
P = IV
\end{equation}

Recall Ohm’s Law (V = IR). Power (P) can also be written as:

\begin{equation}
P = \frac{V^2}{R}
\end{equation}

or written as:

\begin{equation}
P = {I^2}{R}
\end{equation}

The conservation of power states that the total power absorbed in a system is equivalent to the total power delivered by the system:

Total Power Absorbed in The System = Total Power Delivered by The System

Σ (Power Absorbed) = Σ (Power Delivered)

where Σ is the capital Greek letter “sigma” that is used in mathematics as a symbol to represent the summing up of values or to add all values — in this case we want to separately add all the values for the power absorbed and add all the values for the power delivered — and the conservation of power tells us that these two sums should equal each other for the statement of conservation of power to be true.

A Circuit of Unknown Types of Elements, Their Given Voltage and Current Values, and Labeled Orientations:

Let’s look at the following generic circuit example to further understand the idea of passive and active elements of a circuit.

Note the labeled voltage and current values in our circuit above. Also, note the direction of the arrows for our labeled currents in the circuit and the directions of the positive (+) and negative (-) labels that are across each element in the circuit.

Passive Convention of Elements

Passive and active elements are identified by looking at the direction of the positive (+) and negative (-) labels in association with the arrow direction of the current. The images below describe what’s known as passive convention — where a passive element is identified as a component that shows the direction of current going from positive (+) to negative (-) through the element.

An active element is identified as a component that shows the direction of current going from negative (-) to positive (+) through the element.

Passive Convention in Power Calculations

When referring to an element in a circuit that’s unknown to whether it’s passive or active, the outcome of the power calculation let’s us know which type of element we have:

• If the value of power P is positive (+), then the element is passive.
• If the value of power P is negative (-), then the element is active.

Note that labeled current arrow directions in our circuit affects the outcome of the type of element in our circuit. We use the help of passive convention we discussed above.

• If the arrow direction of current is labeled in a way that makes it passive, then we write our power calculation as: P = +IV
• If the arrow direction of current is labeled in a way that makes it active, then we write our power calculation as: P = -IV
• So, for our power calculations we’ll use the general formula P = ±IV, where the plus sign over the minus sign just tells us that it’ll either be positive or negative — based on convention.

A Table of Values for Our Given Circuit

The following table provides a concise description of our circuit example above — observing the table from left-to-right:

Element Label	Type of Element (based on passive convention)	Power Calculation P = ±IV	Type of Element (based on computation)
a	passive P = +IV	Pa = +IaVa Pa = +(2mA)(5V) = +10mW	+10mW (positive) absorbs power (passive)
b	active P = -IV	Pb = +IbVb Pb = -(3mA)(1V) = -3mW	-3mW (negative) delivers power (active)
c	active P = -IV	Pc = -IcVc Pc = -(-2mA)(7V) = +14mW	+14mW (positive) absorbs power (passive)
d	active P = -IV	Pd = IdVd Pd = -(1mA)(-9V) = +9mW	+9mW (positive) absorbs power (passive)
e	passive P = +IV	Pe = IeVe Pe = +(5mA)(-20V) = -100mW	-100mW (negative) delivers power (active)
f	passive P = +IV	Pf = IfVf Pf = +(2mA)(20V) = +40mW	+40mW (positive) absorbs power (passive)
g	active P = -IV	Pg = IgVg Pg = -(-2mA)(-3V) = -6mW	-6mW (negative) delivers power (active)
h	passive P = +IV	Ph = IhVh Ph = +(-3mA)(-12V) = +36mW	+36mW (positive) absorbs power (passive)

Is The Power Conserved?

Now, the moment of truth. Is the power conserved in our example circuit above? Let’s go through the steps:

1. First, we see in the table above, that we performed a passive sign convention on the unknown elements in the given circuit. This was done by observation of the given arrow directions of current and their association with the given positive (+) and negative (-) orientations across each element in the circuit. These observations do not necessarily tell us that an observed unknown element is actually either passive or active — this will be determined after our power calculation.
2. We then performed a power calculation based on passive convention and on the given voltages and currents for each element in the circuit.
3. Then, we determined the type of element for each element in the circuit based on the computational outcome of their power — whether the power calculation of the element turned out to be positive or negative. Doing so told us whether each element was truly passive or active.
4. Now, to find out if power has been conserved we add (sum up) all the passive outcomes on one side of the conservation of power equality formula, and add all the active outcomes on the other side of the equality — these separate summations should equal each other.

Σ (Power Absorbed) = Σ (Power Delivered)

Σ P_Absorb = Σ -P_Deliv

Σ P_Absorb = -Σ P_Deliv

P_a + P_c + P_d + P_f + P_h = -[P_b + P_e + P_g]

10mW + 14mW + 9mW + 40mW + 36mW = -[-3mW + (-100mW) + (-6mW)]

109mW = -(-109mW)

109mW = 109mW

The power is conserved!

If you’re wondering where the negative (-) sign came from for Σ P_Absorb = Σ -P_deliv, remember we are using passive convention for power calculations, so an active element delivers power — therefore, we use a negative sign for its calculation. Here’s the gist of it:

• Think of it as an active element letting go of power or releasing it — or think of power as being taken away from the active element by the passive elements in the circuit.
• Remember, active elements are delivering power (P = -IV) and the passive elements are absorbing that power (P = +IV).

Conclusion

We discussed what is means for an element in a circuit is to be either active or passive. Active elements deliver energy to a circuit, whereas passive elements absorb energy from a circuit.

You were briefly given a review on the conservation of power and later learned the concept of passive convention of circuits to understand how to make power calculations of circuits. Each topic learned taught you how to calculate the important circuit value concept of the conservation of power of given unknown circuit elements.

It is highly recommended that from here, that you learn the concepts of circuit nodes, loops and meshes on our page here, as well as the concepts of Kirchoff’s Voltage Laws here and Kirchoff’s Current Laws here — to further your understanding of circuit theory and its applications to real world circuit problems and design.