Dividers#

Equivalent resistances and Ohm’s law allows us to analyze to small, but important, circuits

Voltage Divider#

The voltage divider relates voltage across multiple resistors connected in series to the voltage across an individual resistor.

Example

_images/voltage-divider-1.svg

Fig. 39 The first voltage divider example.#

Find VR1 and VR2.

Can I Apply the Voltage Divider?#

Before you commit to using the voltage divider formula, ask yourself these questions:

  • Are the two resistors really in series? Do they share a common node, just with each other?

  • Do I really know the voltage across the two series resistors?

Current Divider#

The current divider relates current into multiple branches connected in parallel to the current through an individual branch.

Example

_images/current-divider-1.svg

Fig. 41 The first current divider example.#

Find IR1 and IR2.

Can I Apply the Current Divider?#

Before you commit to using the current divider formula, ask yourself these questions:

  • Are the two resistors really in parallel? Do they connect the same two nodes electrically?

  • Do I really know the current flowing into the nodes where they connect?

  • Am I really dividing a current, not a voltage?

A Look at Power#

Power in Voltage Dividers#

Let’s look at the power dissipated in the case of a voltage divider.

Example

Find PR1+2, PR1, and PR2 and show that

PR1+2=PR1+PR2.
_images/voltage-divider-power-1.svg

Fig. 43 The two circuits used above in the voltage divider example.#

Power in Current Dividers#

Let’s look at the power dissipated in the case of a current divider.

Example

Find PR12, PR1, and PR2 and show that

PR12=PR1+PR2
_images/current-divider-power-1.svg

Fig. 44 The two circuits used above in the current divider example.#

Kirchhoff’s Laws#

Kirchhoff’s Voltage Law#

The algebraic sum of voltages around a loop in a circuit is zero. Pay attention to the polarities.

_images/kvl-1.svg

Fig. 45 Kirchhoff’s Voltage Law circuit.#

If we consider the voltages around the loop starting and ending at node E we can write a KVL equation for the loop:

VAEVABVBCVCDVDE=0

We can also consider voltages across two distant nodes in the circuit (A and D for this example) and write a KVL such as:

VADVABVBCVCD=0

and rearrange it to find VAD:

VAD=VAB+VBC+VCD
_images/kvl-2.svg

Fig. 46 Showing how VAD is composed.#

Kirchhoff’s Current Law#

The algebraic sum of currents into a node in a circuit is zero. Pay attention to the direction of current flow.

_images/kcl-1.svg

Fig. 47 Kirchhoff’s Current Law, illustrated.#

If we consider the currents flowing into the central node we can write a KCL equation for that node:

I1+I2+I3+I4I5=0

Note that I5 is flowing out of the node, so its sign is negative.