Equivalent Components

Series, Parallel, Neither, Both

Elements in a circuit can be connected in four different ways:

• Series

• Parallel

• Neither

• Both

Students often assume that two elements are either in parallel or series. Beware of this false dichotomy and avoid it by studying the definitions of series and parallel connections carefully.

In series or parallel, some components can be combined. When neither, they can not.

Series Resistors

Two elements connected in series share one node exclusively.

Fig. 19 \(R_1\) and \(R_2\) are in series.

When two elements are in series, the same current flows through each.

\[I_{R1}=I_{R2}\]

When two resistors are in series they can be redrawn as a single resistor.

Fig. 20 The equivalent resistance of \(R_1\) and \(R_2\) in series is \(R_S = R_1 + R_2\).

Parallel Resistors

Two elements are in parallel when they are connected to the same two nodes.

Fig. 21 \(R_1\) and \(R_2\) are in parallel.

When two resistors are in parallel they can be redrawn as a single resistor.

Fig. 22 The equivalent conductance of \(G_1\) and \(G_2\) in series is \(G_P = G_1 + G_2\).

The conductances add. Recall that

\[G=\frac{1}{R}\]

so

\[\frac{1}{R_P}=\frac{1}{R_1}+\frac{1}{R_2}\]

Solving for \(R_P\) and adding additional resistors

(1)\[R_P=\frac{1}{\frac{1}{R_1}+\frac{1}{R_2}+\dots+\frac{1}{R_N}}\]

The value of two resistors in parallel is commonly expressed as

\[R_P=\frac{R_1R_2}{R_1+R_2}\]

Careful:

• They may not be drawn geometrically parallel.

Fig. 23 \(R_1\) and \(R_2\) are still electrically in parallel.

• If they are drawn geometrically parallel they may not be connected in parallel.

Fig. 24 \(R_1\) and \(R_3\) are not electrically in parallel.

Question

\(R_1\) and \(R_3\) are not in parallel. How are they connected?

More Complex Circuits

Example

Find \(R_{AB}\)

Fig. 25 A more complex example.

Solution

\(R_{AB}=7.4\Omega\)

Example

Find \(R_{AB}\)

Fig. 26 Another complex example.

Solution

\(R_{AB}=11.2\Omega\)

Example

Find \(R_{AB}\)

Fig. 27 Yet another complex example.

Solution

\(R_{AB}=4k\Omega\)

Series Voltage Supplies

• Polarity matters when adding voltage supplies in series.

• Circuit components can effectively be in series if they are both in series with another component.

Fig. 28 Series voltage supplies, take one.

Fig. 29 Series voltage supplies, take two.

Fig. 30 Series voltage supplies, take three.

Fig. 31 Series voltage supplies, take four.

Fig. 32 Series voltage supplies, take five.

Parallel Current Supplies

• Polarity matters when adding current supplies in parallel.

• Circuit components can effectively be in parallel if they are both in parallel with another component.

Fig. 33 Parallel current supplies, take one.

Fig. 34 Parallel current supplies, take two.

Fig. 35 Parallel current supplies, take three.

Fig. 36 Parallel current supplies, take four.

Delta-Wye Conversions

(2)\[\begin{align} R_a &= \frac{R_1 R_2}{R_1 + R_2 + R_3} \\ R_b &= \frac{R_1 R_3}{R_1 + R_2 + R_3} \\ R_c &= \frac{R_2 R_3}{R_1 + R_2 + R_3} \end{align}\]

Fig. 37 The wye configuration.

(3)\[\begin{align} R_1 &= \frac{R_a R_b + R_b R_c + R_a R_c}{R_c} \\ R_2 &= \frac{R_a R_b + R_b R_c + R_a R_c}{R_b} \\ R_3 &= \frac{R_a R_b + R_b R_c + R_a R_c}{R_a} \end{align}\]

Fig. 38 The delta configuration.