Parallel circuits Series and parallel circuits
1 parallel circuits
1.1 voltage
1.2 current
1.3 resistors
1.4 inductors
1.5 capacitors
1.6 switches
1.7 cells , batteries
parallel circuits
if 2 or more components connected in parallel have same potential difference (voltage) across ends. potential differences across components same in magnitude, , have identical polarities. same voltage applicable circuit components connected in parallel. total current sum of currents through individual components, in accordance kirchhoff’s current law.
voltage
in parallel circuit voltage same elements.
v
=
v
1
=
v
2
=
…
=
v
n
{\displaystyle v=v_{1}=v_{2}=\ldots =v_{n}}
current
the current in each individual resistor found ohm s law. factoring out voltage gives
i
t
o
t
a
l
=
v
(
1
r
1
+
1
r
2
+
⋯
+
1
r
n
)
{\displaystyle i_{\mathrm {total} }=v\left({\frac {1}{r_{1}}}+{\frac {1}{r_{2}}}+\cdots +{\frac {1}{r_{n}}}\right)}
.
resistors
to find total resistance of components, add reciprocals of resistances
r
i
{\displaystyle r_{i}}
of each component , take reciprocal of sum. total resistance less value of smallest resistance:
1
r
t
o
t
a
l
=
1
r
1
+
1
r
2
+
⋯
+
1
r
n
{\displaystyle {\frac {1}{r_{\mathrm {total} }}}={\frac {1}{r_{1}}}+{\frac {1}{r_{2}}}+\cdots +{\frac {1}{r_{n}}}}
.
for 2 resistors, unreciprocated expression reasonably simple:
r
t
o
t
a
l
=
r
1
r
2
r
1
+
r
2
.
{\displaystyle r_{\mathrm {total} }={\frac {r_{1}r_{2}}{r_{1}+r_{2}}}.}
this goes mnemonic product on sum .
for n equal resistors in parallel, reciprocal sum expression simplifies to:
1
r
t
o
t
a
l
=
1
r
×
n
{\displaystyle {\frac {1}{r_{\mathrm {total} }}}={\frac {1}{r}}\times n}
.
and therefore to:
r
t
o
t
a
l
=
r
n
{\displaystyle {r_{\mathrm {total} }}={\frac {r}{n}}}
.
to find current in component resistance
r
i
{\displaystyle r_{i}}
, use ohm s law again:
i
i
=
v
r
i
{\displaystyle i_{i}={\frac {v}{r_{i}}}\,}
.
the components divide current according reciprocal resistances, so, in case of 2 resistors,
i
1
i
2
=
r
2
r
1
{\displaystyle {\frac {i_{1}}{i_{2}}}={\frac {r_{2}}{r_{1}}}}
.
an old term devices connected in parallel multiple, such multiple connection arc lamps.
since electrical conductance
g
{\displaystyle g}
reciprocal resistance, expression total conductance of parallel circuit of resistors reads:
g
t
o
t
a
l
=
g
1
+
g
2
+
⋯
+
g
n
{\displaystyle {g_{\mathrm {total} }}={g_{1}}+{g_{2}}+\cdots +{g_{n}}}
.
the relations total conductance , resistance stand in complementary relationship: expression series connection of resistances same parallel connection of conductances, , vice versa.
inductors
inductors follow same law, in total inductance of non-coupled inductors in parallel equal reciprocal of sum of reciprocals of individual inductances:
1
l
t
o
t
a
l
=
1
l
1
+
1
l
2
+
⋯
+
1
l
n
{\displaystyle {\frac {1}{l_{\mathrm {total} }}}={\frac {1}{l_{1}}}+{\frac {1}{l_{2}}}+\cdots +{\frac {1}{l_{n}}}}
.
if inductors situated in each other s magnetic fields, approach invalid due mutual inductance. if mutual inductance between 2 coils in parallel m, equivalent inductor is:
1
l
t
o
t
a
l
=
l
1
+
l
2
−
2
m
l
1
l
2
−
m
2
{\displaystyle {\frac {1}{l_{\mathrm {total} }}}={\frac {l_{1}+l_{2}-2m}{l_{1}l_{2}-m^{2}}}}
if
l
1
=
l
2
{\displaystyle l_{1}=l_{2}}
l
total
=
l
+
m
2
{\displaystyle l_{\text{total}}={\frac {l+m}{2}}}
the sign of
m
{\displaystyle m}
depends on how magnetic fields influence each other. 2 equal tightly coupled coils total inductance close of each single coil. if polarity of 1 coil reversed m negative, parallel inductance 0 or combination non-inductive. assumed in tightly coupled case m equal l. however, if inductances not equal , coils tightly coupled there can near short circuit conditions , high circulating currents both positive , negative values of m, can cause problems.
more 3 inductors becomes more complex , mutual inductance of each inductor on each other inductor , influence on each other must considered. 3 coils, there 3 mutual inductances
m
12
{\displaystyle m_{12}}
,
m
13
{\displaystyle m_{13}}
,
m
23
{\displaystyle m_{23}}
. best handled matrix methods , summing terms of inverse of
l
{\displaystyle l}
matrix (3 3 in case).
the pertinent equations of form:
v
i
=
∑
j
l
i
,
j
d
i
j
d
t
{\displaystyle v_{i}=\sum _{j}l_{i,j}{\frac {di_{j}}{dt}}}
capacitors
the total capacitance of capacitors in parallel equal sum of individual capacitances:
c
t
o
t
a
l
=
c
1
+
c
2
+
⋯
+
c
n
{\displaystyle c_{\mathrm {total} }=c_{1}+c_{2}+\cdots +c_{n}}
.
the working voltage of parallel combination of capacitors limited smallest working voltage of individual capacitor.
switches
two or more switches in parallel form logical or; circuit carries current if @ least 1 switch closed. see or gate.
cells , batteries
if cells of battery connected in parallel, battery voltage same cell voltage current supplied each cell fraction of total current. example, if battery comprises 4 identical cells connected in parallel , delivers current of 1 ampere, current supplied each cell 0.25 ampere. parallel-connected batteries used power valve filaments in portable radios rare. solar electric systems have batteries in parallel increase storage capacity; close approximation of total amp-hours sum of batteries in parallel.
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