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Resistance in Parallel

 Determining the Equivalent Resistance

Two methods of combining parallel resistances to find REQ. (a) Using the reciprocal resistance formula to calculate REQ as 4 Ω. (b) Using the total line current method with an assumed line voltage of 20 V gives the same 4 Ω for REQ.

Resistance in Parallel

 Special Case: Two Unequal Resistors

 When there are only two branches in a parallel circuit and

their resistances are unequal, use the formula:

R 1 × R 2

R 1 + R 2

REQ =

For the special case of only two branch resistances, of any values, REQ equals their product divided by the sum. Here, REQ = 2400 / 100 = 24Ω.

Kirchhoff’s Current Law (KCL)

Kirchhoff’s Voltage Law (KVL)

The total voltage is equal

to the sum of the drops.

VT = V 1 + V 2 + V 3 + V 4 + V 5

V 1 V 2

V 3

V 5 V 4

VT

This is known as

Kirchhoff’s voltage law (KVL).

The total voltage is equal

to the sum of the drops.

VT = V 1 + V 2 + V 3 + V 4 + V 5

V 1 V 2

V 3

V 5 V 4

VT

V 1 V 2

V 3

V 5 V 4

VT

This is known as

Kirchhoff’s voltage law (KVL).

Series-Aiding and Series-Opposing Voltages

Example of voltage sources V 1 and V 2 in series.

( a ) Note the connections for series-aiding polarities. Here 8 V + 6 V = 14 V for the total VT.

( b ) Connections for series-opposing polarities. Now 8 V – 6 V = 2 V for VT.

Series-Aiding Voltages

Series-aiding voltages are connected with

polarities that allow current in the same

direction:

 The positive terminal of one is connected to

the negative terminal of the next.

They can be added for the total voltage.

Opens in Series Circuits

 The Effect of an Open in a Series Circuit

Effect of an open in a series circuit. ( b ) Open path between points P1 and P2 results in zero current in all parts of the circuit.

Shorts in Series Circuits

 The Effect of a Short in a Series Circuit

 When part of a series circuit is shorted, the current flow increases.  When part of a series circuit is shorted, the voltage drops across the non- shorted elements increase.  The voltage drop across the shorted component drops to 0 V.

Series Voltage Dividers

 The Largest Series R Has the Most V.

V 2 =

R 2

RT

× VT

999 k 1000 k

= × 1000 V = 999 V

V 1 =

R 1

RT

× VT

1 k 1000 k

× 1000 V = 1 V

KVL check: 1 V + 999 V = 1000 V

Fig. Example of a very small R 1 in series with a large R 2 ; V 2 is almost equal to the whole VT.

The analysis of a DC circuit involves determining either the voltage, current, or resistance between any two points. As an example, for the circuit

DC Circuit Analysis