Series and Parallel Resistors (Voltage and Current Division)

Review: Spotting Series and Parallel combinations

Series Resistors ⇒ same current flowing through them.

⇒ 1 path

The resistors in series could look like a V (a). They could look like an L (b). They could look like a stepwise function (c). It doesn’t matter what it looks like; if the resistors are connected at one end, it is in series.

Series Resistors & Voltage Division

• v1= iR1 & v2 = iR2

KVL:

• v-v1-v2=0
• v= i(R1+R2)
• i = v/(R1+R2 ) =v/Req
• or v= i(R1+R2 ) =iReq
• iReq = R1+R2

VOLTAGE DIVISION FORMULA

Before:

v1 = iR1   &   v2 = iR2
i = v/(R1+R2 )
Thus:

v1=vR1/(R1+R2)

v2=vR2/(R1+R2)

To solve for Req:
Req = R1+R2 ...

---

---

Parallel Resistors & Current Division

Parallel Combinations
Now it gets a bit trickier. Are the resistor leads connected at two ends? Sometimes it’s very easy to see, like these nice block-y circuits.

Sometimes, though, you get things that look like this.

When this is the case, shorten the wires.

Do the resistors end up side-by-side and connected at both ends? Then they are connected in parallel, with the current being split between the junctions that are in parallel.
For more info. visit: https://craftsandcircuits.wordpress.com/2014/06/19/visualizing-circuit-analysis-with-gifs/

---

 

Parallel Resistors ⇒ common voltage across it.

⇒ Still equal

• v = i1R1 = i2R2
• i   = i1+ i2

= v/R1+ v/R2
= v(1/R1+1/R2)
=v/Req

• v  =iReq
• 1/Req = 1/R1+1/R2
• Req = R1R2 / (R1+R2 )

CURRENT DIVISION FORMULA

Before:

v = i1R1 = i2R2
v=iReq = iR1R2 / (R1+R2 )
and i1 = v /R1  &  i2 =v/ R2

Thus:

i1= iR2/(R1+R2)

i2= iR1/(R1+R2 )

To solve for Req:
Req = 1/R1+  1/R2 ...  1/Rn

---

---

Conductance (G)

Series conductance:

1/Geq = 1/G1 +1/G2+…

Parallel conductance:

Geq = G1 +G2+…

---

---

Now putting all this together…
Remember, orientation does not matter. We’ve been conditioned by nice, predictable circuit diagrams to think of series and parallel circuits in very limited forms, be it as a straight line or as a block-y succession of resistors. However, this will clearly not be the case all the time.
What we should appreciate is the basics. Connected on one end, series. Two ends, parallel. Resistor leads touching, short circuit. Because no matter what the complexity of the circuit is, these simple rules work every time.
Going back to the nasty circuit in the beginning…
(Assume that all unmarked resistors are 1 ohm.)

Source: https://craftsandcircuits.wordpress.com/2014/06/19/visualizing-circuit-analysis-with-gifs/

---

---

LEARNING/SUMMARY

• In series circuits, you SHOULD use voltage division to find the desired unknowns.

• In parallel circuits, you SHOULD use current division to find the desired unknown.

• If the circuit is complicated, always be conscious and observe your steps when you are minimizing your circuit without disobeying the rules of parallel and series circuits.

• ALWAYS BE AWARE AND DO NOT HASSLE YOURSELF.