Skip to content Skip to navigation

Connexions

You are here: Home » Content » Parallel Combination Rules

Navigation

Recently Viewed

This feature requires Javascript to be enabled.

Parallel Combination Rules

Module by: Don Johnson. E-mail the author

User rating (How does the rating system work?)
Ratings

Ratings allow you to judge the quality of modules. If other users have ranked the module then its average rating is displayed below. Ratings are calculated on a scale from one star (Poor) to five stars (Excellent).

How to rate a module

Hover over the star that corresponds to the rating you wish to assign. Click on the star to add your rating. Your rating should be based on the quality of the content. You must have an account and be logged in to rate content.

:
(0 ratings)

Summary: A discussion of combination rules for circuit elements connected in series and of the voltage divider rule.

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

One interesting simple circuit (Figure 1) has two resistors connected side-by-side, what we will term a parallel connection, rather than in series.

Figure 1: A simple parallel circuit.
Figure 1 (circuit25.png)

Here, applying KVL reveals that all the voltages are identical: v 1 =v v 1 v and v 2 =v v 2 v . This result typifies parallel connections. To write the KCL equation, note that the top node consists of the entire upper interconnection section. The KCL equation is i in − i 1 − i 2 =0 i in i 1 i 2 0 . Using the v-i relations, we find that

i out = R 1 R 1 + R 2 i in i out R 1 R 1 R 2 i in (1)

Exercise 1

Suppose that you replaced the current source in Figure 1 by a voltage source. How would i out i out be related to the source voltage? Based on this result, what purpose does this revised circuit have?

Solution

Replacing the current source by a voltage source does not change the fact that the voltages are identical. Consequently, v in = R 2 i out v in R 2 i out or i out = v in R 2 i out v in R 2 . This result does not depend on the resistor R 1 R 1 , which means that we simply have a resistor ( R 2 R 2 )across a voltage source. The two-resistor circuit has no apparent use.

This circuit highlights some important properties of parallel circuits. You can easily show that the parallel combination of R 1 R 1 and R 2 R 2 has the v-i relation of a resistor having resistance 1 R 1 +1 R 2 -1= R 1 R 2 R 1 + R 2 1 R 1 1 R 2 -1 R 1 R 2 R 1 R 2 . A shorthand notation for this quantity is R 1 ∥ R 2 ∥ R 1 R 2 . As the reciprocal of resistance is conductance, we can say that for a parallel combination of resistors, the equivalent conductance is the sum of the conductances.

Figure 2
Figure 2 (parallelR.png)

Similar to voltage divider for series resistances, we have current divider for parallel resistances. The current through a resistor in parallel with another is the ratio of the conductance of the first to the sum of the conductances. Thus, for the depicted circuit, i 2 = G 2 G 1 + G 2 i i 2 G 2 G 1 G 2 i . Expressed in terms of resistances, current divider takes the form of the resistance of the other resistor divided by the sum of resistances: i 2 = R 1 R 1 + R 2 i i 2 R 1 R 1 R 2 i .

Figure 3
Figure 3 (circuit14.png)

Content actions

Give Feedback:

E-mail the module author | Rate module ( How does the rating system work?)

Rating system

Ratings

Ratings allow you to judge the quality of modules. If other users have ranked the module then its average rating is displayed below. Ratings are calculated on a scale from one star (Poor) to five stars (Excellent).

How to rate a module

Hover over the star that corresponds to the rating you wish to assign. Click on the star to add your rating. Your rating should be based on the quality of the content. You must have an account and be logged in to rate content.

(0 ratings)

Download:

Module PDF

Add module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections directly in Connexions. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need a Connexions account to use 'My Favorites'.

| A lens (?)

Definition of a lens

Lenses

A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to Connexions materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual Connexions member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks