(DC) Circuits
1.  Resistors in a circuit (serial and parallel
connections).
2.  The Direct Current (DC) and the
Alternating Current (AC) circuits.
3.  How to model a battery in a circuit.
4.  Circuits with 2 or more batteries –
Kirchhoff’s Rules
5.  The RC circuit (your low- or high- pass
filter in EE, and the drive in MOS
transistors): when we have R and C
together.
Review
The current is defined as: I =
dQ
dt
Its unit is ampere (A), a base unit in the SI system.
! !
Its relationship with the
I = ∫ J ⋅ dA or: J ≡ dI dA
current density J is:
!
!
Ohm’s Law: J = σ E
Here σ is the conductivity of the material.
The resistivity is defined as ρ ≡ 1/σ, and is a more commonly used
material parameter which linearly depends on temperature:
ρ = ρ0 "#1+ α (T − T0 )$%
The definition of
ΔV
resistance
R≡
I
(Ohm’s Law):
Its relationship
with material and
shape:
l
R≡ρ
A
Resistors in Series and in Parallel
Resistor connections
In series.
Condition:
Result:
I = I1 = I 2
ΔV = ΔV1 + ΔV2
ΔV2
ΔV1
ΔV ΔV1 + ΔV2 ΔV1 ΔV2
Req =
=
=
+
= R1 + R2
I
I
I1
I2
In parallel.
Condition:
Result:
I = I1 + I 2
ΔV = ΔV1 = ΔV2
I1 + I 2
I1
I2
1
I
1 1
=
=
=
+
= +
Req ΔV
ΔV
ΔV1 ΔV2 R1 R2
ΔV1
ΔV2
Resistor connections
In series,
∵ I = I1 = I 2
voltage sharing
ΔV1 R1
=
ΔV2 R2
power sharing
P1 R1
=
P2 R2
In parallel,
ΔV2
ΔV1
∵ ΔV = ΔV1 = ΔV2
current sharing
I 2 R1
=
I1 R2
ΔV1
power sharing
P2 R1
=
P1 R2
ΔV2
Resistors connections, summary
l
In series
I = I1 = I 2 = I 3 = ...
Req = R1 + R2 + R3 + ...
ΔV1:ΔV2 :ΔV3: ...=R1:R2 :R3: ...
P1:P2 :P3: ...=R1:R2 :R3: ...
l
In parallel
ΔV = ΔV1 = ΔV2 = ΔV3 = ...
1
1 1 1
= + + + ...
Req R1 R2 R3
I1R1 = I 2 R2 = I 3 R3 = ...
P1R1 = P2 R2 = P3 R3 = ...
Combinations of Resistors
l
l
l
The 8.0-Ω and 4.0-Ω
resistors are in series and
can be replaced with their
equivalent, 12.0 Ω
The 6.0-Ω and 3.0-Ω
resistors are in parallel and
can be replaced with their
equivalent, 2.0 Ω
These equivalent
resistances are in series
and can be replaced with
their equivalent resistance,
14.0 Ω
More examples
R1
R3
R2
R5
R4
Direct Current and Alternating Current
l
When the current direction (not magnitude) in a
circuit does not change with time, the current is
called a direct current (DC).
l
l
constant current magnitude, like the one powered through a
battery, is a common, but special case of DC.
When the current direction (often also the
magnitude) in a circuit changes with time, the current
is called an alternating current (AC).
l
l
The current from the wall outlet is AC.
The current from your car’s alternator is AC. Question: how
to charge the battery with the alternator? Question: are you
even interested in the first question?
Model of a battery
l
l
l
l
Two parameters, electromotive
force (emf), ε, and the internal
resistance r, are used to model
a battery.
When a battery is connected in
a circuit, the electric potential
measured at its + and –
terminals are called The
terminal voltage ΔV, with
ΔV = ε – Ir
If the internal resistance is zero
(an ideal battery), the terminal
voltage equals the emf ε.
The internal resistance, r, does
not change with external load
resistance R, and this provides
the way to measure the internal
resistance.
ΔV
battery
ΔV
load
Battery power figure
The power a battery generates
(ex. thrgh chemical reactions):
P =ε⋅ I = R+r ⋅ I2
The power the battery delivers to
the load, hence efficiency:
Pload = ΔV ⋅ I = R ⋅ I 2
(
)
battery
ΔV
load
Pload
R
efficiency=
=
P
R+r
The maximum power the battery
can deliver to a load
R 2
2
P
=
ε
We
have
load
From Pload = R ⋅ I and ε = ( R + r ) ⋅ I
R+r
Where the emf ε is a constant once the battery is chosen.
!
dP
From
# 1
load
=#
dR # R + r
"
(
2
$
2R & 2
−
ε =0
3&
R + r &%
) (
)
We get R = r to be the condition for
maximum Pload , or power delivered to
the load.
Battery power figure
One can also obtain this result
from the plot of
Pload =
R 2
ε
R+r
battery
ΔV
Where when R = r
Pload
reaches the maximum value
The efficiency of the battery at this
point is 50% because
Pload
R
efficiency=
=
P
R+r
load
circuits with 2+ batteries: Kirchhoff’s Rules
A typical circuit that
goes beyond
simplifications with the
parallel and series
formulas: ask for the
current in the diagram.
l  Kirchhoff’s rules can
be used to solve
problems like this.
l
Rule 1: Kirchhoff’s Junction Rule
l  Junction
Rule, from charge
conservation:
l
l
The sum of the currents at any
junction must equal zero
Mathematically:
∑
I =0
junction
l
The example on the left figure:
I1 − I 2 − I 3 = 0
Rule 2: Kirchhoff’s Loop Rule
l
l
Choose your loop
Loop Rule, from energy
conservation:
l  The sum of the potential
differences across all elements
around any closed circuit loop
must be zero
l  Mathematically:
∑
ΔV = 0
ΔV1
ΔV2
Loop direction
Remember two things:
1.  A battery supplies power.
l  One needs to pay attention the
Potential rises from the “–”
sign (+ or -) of these potential
terminal to “+” terminal.
changes, following the chosen loop 2.  Current follows the direction of
direction.
electric field, hence the
decrease of potential.
closed loop
Kirchhoff’s rules
Strict steps in solving a problem
Step 1: choose and mark
the loop.
Step 2: choose and mark
current directions. Mark the
potential change on resistors.
Step 3: apply junction rule:
I1
I1 + I2 − I3 = 0
Step 4: apply loop rule:
L1: +2.00I3 − 12.0 + 4.00I2 = 0
L2: − 8.00 − 2.00I3 − 6.00I1 = 0
Step 5: solve the three equations
for the three variables.
+
–
I2
L1
I3
+
–
L2
–
+
One more example
Step 1: choose and mark
the loop.
Step 2: choose and mark
current directions. Mark the
potential change on resistors.
Step 3: apply junction rule:
I1 + I2 − I3 = 0
Step 4: apply loop rule:
–
L1
+
+
–
L1: +6.0I1 − 10.0 − 4.0I2 − 14.0 = 0
L2
L2: − 2.0I3 + 10.0 − 6.0I1 = 0
Step 5: solve the three equations
for the three variables.
–
+
RC Circuits, solve with Kirchhoff’s rules
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When a circuit contains a
resistor and a capacitor
connected in series, the circuit
is called a RC circuit.
Current in RC circuit is DC,
but the current magnitude
changes with time.
There are two cases: charging
(b) and discharging (c).
Not a
circuit
charging
Discharging
Charging a Capacitor
When the switch turns to position
a, current starts to flow and the
capacitor starts to charge.
Kirchhoff’s rule says:
ε − ΔVc − ΔVR = 0
Re-write the equation in terms of the
charge q in C and the current I, and then
only the variable q:
q
q
dq
ε − − RI = 0 and then ε − − R
=0
C
C
dt
Solve for q:
The current I is
−t
−t
⎛
⎞
dq
ε
= e RC
q (t ) = Cε ⎜ 1− e RC ⎟ I (t ) =
dt R
⎝
⎠
+
–
+
Loop
–
Here RC has the unit
of time t, and is called
the time constant.
Charging a Capacitor, graphic presentation
l
The charge on the
capacitor varies with time
l
l
q(t) = Cε (1 – e-t/RC)
= Q(1 – e-t/RC)
The current decrease with
time
ε −t RC
I( t ) = e
R
l
τ is the time constant
l  τ = RC
Discharging a Capacitor
When the switch turns to position b,
after the capacitor is fully charged
to Q, current starts to flow and the
capacitor starts to discharge.
Kirchhoff’s rule says:
ΔVc − ΔVR = 0
Re-write the equation in terms of the
charge q in C and the current I, and then
only the variable q:
q
q
−dq
− RI = 0 and then − R
=0
C
C
dt
Solve for q:
The current I is
−t
−t
dq Q RC
I (t ) = −
=
e
q (t ) = Qe RC
dt RC
+
–
–
Loop
+
Ii =
Q
RC
Connections to EE
A low-pass filter
A high-pass filter
A MOSFET:
Reading material and Homework assignment
Please watch this video (about 50 minutes each):
http://videolectures.net/mit802s02_lewin_lec10/
Please check wileyplus webpage for homework
assignment.