Handout 1: Review
• Basic Circuit Variables
– Voltage:
Voltage is an ”across variable” as it measures the difference between two electrical
potentials at two points. When a reference point is used (e.g., the ”ground”), the
voltage at a certain point is defined as the potential difference between this point
and the reference point.
– Current:
Current is a ”through variable” as it measures the amount of electrical charge
that flow through a pass per unit time.
• Basic Elements
– Resistor:
V = RI, I =
V
,
R
V
I
R=
– Capacitor:
V =
Q
,
C
i(t) = C
d
v(t),
dt
v(t) =
1
C
! t
−∞
i(τ )dτ
– Inductor:
W
,
V =
Q
d
d
v(t) = ψ(t) = L i(t),
dt
dt
1
i(t) =
L
• Series and Parallel Connections
– Series Connections:
Rs = R1 + · · · + Rn =
1
n
"
i=1
Ri
! t
−∞
v(τ )dτ
Ls = L1 + · · · + Ln =
n
"
1
1
1
1
=
+···+
=
,
Cs
C1
Cn i=1 Ci
If n = 2,
Cs =
1
C1
1
+
1
C2
=
n
"
Li
i=1
or Cs = #n
1
1
i=1 Ci
C1 C2
C1 + C2
– Parallel Connections:
n
"
1
1
1
1
=
+···+
=
,
Rp
R1
Rn i=1 Ri
If n = 2,
Rp =
1
R1
1
+
1
R2
=
Lp =
1
L1
1
+
1
L2
=
1
1
i=1 Ri
R1 R2
R1 + R2
n
"
1
1
1
1
=
+···+
=
,
Lp
L1
Ln i=1 Li
If n = 2,
or Rp = #n
or Lp = #n
1
1
i=1 Li
L1 L2
L1 + L2
Cp = C1 + C2
If n = 2, we have:
v-i relation
Series
Parallel
Resistor
v=Ri
Rs = R1 + R2
Rp = R1 R2 /(R1 + R2 )
Inductor
v = L di
dt
Ls = L1 + L2
Lp = L1 L2 /(L1 + L2 )
Capacitor
$
v = C1 i dt
Cs = C1 C2 /(C1 + C2 )
Cp = C1 + C2
Q: Why are R and L similar to each other while C is different?
A: Observe how R, L and C are related to voltage v and current i differently.
2
• Energy and Power
V =
W
, W = V Q,
Q
P =
dW
dQ
V2
=V
=VI =
= RI 2
dt
dt
R
• Voltage/Current Dividers
– Voltage Divider
I=
V
,
R1 + R2
V1 = IR1 = V
R1
,
R1 + R2
V2 = IR2 = V
R2
R1 + R2
Voltage across R is proportional to its own resistance.
When there are more than two resistors in series, we simply have:
Ri
Vi = IRi = V #
j Rj
– Current Divider
V =I
R1 R2
,
R1 + R2
I1 =
V
R2
=I
,
R1
R1 + R2
I2 =
V
R1
=I
R2
R1 + R2
Current through R is proportional to the other resistance.
When there are more than two resistors in parallel, we have:
1
V
1/Ri
Gi
, Ii =
= I#
= I#
Ri
j 1/Rj
j 1/Rj
j Gj
V = I#
where Gi = 1/Ri is the conductance of the ith resistor.
3
• Kirchoff’s Laws
– KCL: Due to conservation of electric charge, Kirchoff current Law (KCL) states:
#
The algebraic sum of all currents into a node is zero k Ik = 0 (Currents leaving
the node take negative values.)
– KVL: Due to conservation of electric energy, Kirchoff voltage Law (KCL) states:
#
The algebraic sum of voltage around a loop is zero k Vk = 0 (Voltages with
opposite polarity take negative values.)
• Reference Point or Ground
The voltage between any two nodes in a circuit is the difference between the potentials
at the two nodes. However, a particular node in the circuit is usually chosen as the
reference point, called the ground. The voltage at any node in the circuit is therefore
defined as the voltage between this node and the common ground.
4