Question

A.C. Fundamentals - Some Cases of Parallel Connected Elements

Answer 1

In the realm of electrical engineering and circuit theory, parallel-connected elements are components that are connected across the same two nodes, sharing the same voltage difference. Here are some common cases of parallel-connected elements:

Resistors in Parallel:
When multiple resistors are connected in parallel, they share the same voltage across their terminals. The total resistance

total
R
total
    ​

 of

n resistors (

1
,

2
,
…
,


R
1
    ​

,R
2
    ​

,…,R
n
    ​

) in parallel can be calculated using the formula:

1

total
=
1

1
+
1

2
+
…
+
1


R
total
    ​

1
    ​

=
R
1
    ​

1
    ​

+
R
2
    ​

1
    ​

+…+
R
n
    ​

1
    ​


The reciprocal of the total resistance is the sum of the reciprocals of individual resistances.

Capacitors in Parallel:
Parallel-connected capacitors have the same voltage across their terminals. The total capacitance

total
C
total
    ​

 of

n capacitors (

1
,

2
,
…
,


C
1
    ​

,C
2
    ​

,…,C
n
    ​

) in parallel is calculated using the formula:


total
=

1
+

2
+
…
+


C
total
    ​

=C
1
    ​

+C
2
    ​

+…+C
n
    ​


The total capacitance is the sum of individual capacitances.

Inductors in Parallel:
Inductors in parallel also share the same voltage across their terminals. The total inductance

total
L
total
    ​

 of

n inductors (

1
,

2
,
…
,


L
1
    ​

,L
2
    ​

,…,L
n
    ​

) in parallel is calculated using the formula:

1

total
=
1

1
+
1

2
+
…
+
1


L
total
    ​

1
    ​

=
L
1
    ​

1
    ​

+
L
2
    ​

1
    ​

+…+
L
n
    ​

1
    ​


Again, the reciprocal of the total inductance is the sum of the reciprocals of individual inductances.

Combination of Different Elements:
Circuits can have combinations of different elements in parallel. For instance, you might encounter circuits with both resistors and capacitors in parallel. The rules for calculating total resistance, capacitance, or inductance still apply, combining the respective formulas.

Mixed Parallel-Serial Combinations:
More complex circuits can have combinations of parallel and series elements. When solving such circuits, it's often helpful to simplify them step by step, converting parallel elements into equivalent single components, and then analyzing the resulting series combinations.

Remember that in parallel connections, voltage across elements is the same, while in series connections, current through elements is the same. Calculations involving parallel elements can become more complex in circuits with multiple branches, so it's important to apply the appropriate formulas and techniques to simplify the analysis.