ECE 222
Electric Circuit Analysis II
Chapter 2
Physics Rules for Circuits
Herbert G. Mayer, PSU
Status 3/15/2016
For use at CCUT Spring 2016
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Syllabus
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SI Units
Change of Current in L
Change of Voltage in C
Definitions
Passive Sign Convention
Displacement Current in Capacitor
Unit of Farad
Unit of Henry
Equations
Bibliography
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Only 7 SI Base Units
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Only 7 SI Base Units
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Units Derived from 7 SI
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Units Derived from 7 SI
 For example, take Hertz, Hz
 Second row, rightmost column
 Unit is s-1 AKA Hz
 Is derived from SI unit second, for time: s
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SI Units
 m: a meter is the length of light traveled in
1/299,792,458th of a second
 kg: kilogram is equal to the reference prototype,
i.e. a defined cube; will likely change
 s: second – duration of 9,192,631,770 periods of
radiation corresponding to the transition between
the two hyperfine levels of the ground state of
cesium 133 atom
 A: One ampere is the current which in 2 parallel
conductors 1 meter apart in a vacuum produces a
force of 2 * 10-7 newton per meter of conductor
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SI Units
 K: Kelvin – thermodynamic temperature unit of the
1/273.16 fraction of water temperature at triple point
 mol: mole – is amount of substance of a system
which contains as many elementary entities as
there are atoms in 0.012 kilogram of carbon 12;
entities can be atoms, molecules, electrons
 Old definition: the mole is the amount of substance
that contains 6.022,141,79 x 1023 specified
elementary entities
 cd: candela – is luminous intensity of a source that
emits monochromatic radiation of frequency 540 *
1012 hertz, plus some further constraints
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Change of Current in L
 Current i cannot change instantaneously within an
inductor
 If i would suddenly change, the voltage would grow
toward infinity; physically not possible
 Or infinite voltage would be required to accomplish
that; physically not possible
 See formula for v(t), with L being inductivity in Henry
v(t) = L * di / dt
 But yes, voltage can change instantaneously in an
inductor; i.e. an inductor can raise sudden voltage
across its terminals
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Change of Current in L
 Current instead has to grow gradually, from i = 0 A
in a quiet system
 Or from i(t=0) = i0 with i0 ≠ 0 A in a system with past
electric history
 Once current through L no longer changes, and i
flows through the inductor, the field no longer
changes, and there is no resistance for a DC current
 Initial resistance, even when a DC source is
connected to an inductor, is caused by the field
being built up, or being changed
 Thus one best not place an inductor directly parallel
to a constant voltage source; instead have a
resistor in series
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Change of Voltage in C
 Voltage v cannot change instantaneously in a
capacitor
 To change v in 0 time, an infinitely strong current
would be required; physically not possible
 See formula for i(t), with C being Capacity in Farad
i(t) = C * dv / dt
 But the current can change instantaneously in a
capacitor, i.e. have sudden displacement current
 Voltage grows gradually, from v = 0 in an initial
system
 Or voltage grows from vt0 = v0 , with v0 > 0 V in a
system with past electric history
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Def: Ampere
 Ampere is the unit of current; we saw one of the
base units of the SI
 Named after André Marie Ampère, French physicist
1775 – 1836
 When 6.2415093 * 1018 electrons stream though a
conductor in a second, the amount of charge moved
is 1 C and the current 1 A; AKA “amp”.
i = dq / dt
1A=1C/s
C here: Coulomb! Not capacitance
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Def: Capacitance
 Electrical capacitance represented by letter C,
measured in Farad F. Capacitor does not directly
conduct current; insulator separates its 2 plates
 But a charge placed onto one plate repels similarly
charged particles on the other plate, and so can
cause a charge to move; known as displacement
current. The current so created is proportional to the
rate at which the voltage across the plates varies
over time. Note: Farad is a very large unit; thus in
diagrams we see smaller units, such as μF or nF.
i ~ dv / dt
i = C * dv / dt
i
the resulting current in A, caused by the changing voltage
C the capacitor’s capacitance, measured in Farad
dv the change in voltage across the 2 plates
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Def: Capacitor
A capacitor’s power p and energy w?
p=v*i
p = C * v * dv / dt
w = C * v2 / 2
w
p
i
C
dv
energy in Joule
power v measured in Watt
the displacement current, in A
is the capacitor’s capacitance, measured in Farad
the change in voltage across the 2 plates
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Def: Capacitor Hint for Unit
 Farad is not one of the 7 elementary SI units
 To remember, one of the many ways that [F] is
expressed by simpler SI units, remember the
secret  and fictitious equation:
Q=Q
AKA: Q = CU
 Where Q is charge in Coulomb, SI units: Ampere
times seconds [A s]
 And C U is really capacity times Volt, i.e. [F V]
 With [A s] = [F V] derive Farad:
[F] = [A s V-1]
 Then Volt in turn can be substituted by other SI
units
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Def: Coulomb
 A coulomb? Is a fundamental unit of electrical
charge, and is also the SI derived unit of electric
charge; the symbol for Coulomb is C; the symbol
for charge flowing, creating a current, is: Q or q
 A coulomb is equal to a charge of approximately
6.241… ×1018 electrons
 Now what a charge really is, we don’t understand,
but we do know some key properties, and we can
measure it quite accurately
 Similar to gravity: we can measure and use it, but
we don’t fundamentally understand what it is; we
only observe that and how it works
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Def: Electron
 An electron? Subatomic particle with electric
charge; we call that charge negative; part of lepton
family
 Called an elementary particle, since it seems to have
no sub-particles
 Has mass of approx. 1/1836 of a proton
 Yet electrons have properties of particles AND
waves
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Def: Inductance
Electrical inductance and related power and energy?
p=i*v
p = i * L * di / dt
w = ( L / 2 ) * i2
w
p
L
i
di
the energy in Joule
the power measured in Watt
the inductance in Henry H
the current in A
the change of current over time, in A
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Def: Inductance
 Electrical inductance? A charge in motion (e.g. some
current) creates a magnetic field around its conductor
 If the current remains constant, so does the field
 If current i varies over time, the magnetic field also
changes as a direct function. A time-varying magnetic
field induces a voltage in any conductor linked to the
field; linked meaning it is close-by
v ~ di / dt
v = L * di / dt
v
L
di
measured in Volt V
inductance in Henry H
the change in current A
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Def: Magnetic Coupling
 Assumes circuit with 2 inductors, in circuit c1 and in
circuit c2
 C1 and c2 are not touching and are not connected, but
close to one another
 When voltage in c2 is induced by a current in c1, we
say, c1 and c2 are magnetically coupled
 AKA mutual inductance
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Def: Resistance
 Electrical resistance? A material’s opposition to the
free flow of electrons
 In an insulator, such as vacuum or porcelain,
resistivity is very large, typically >> 1 MΩ (Mega
Ohm)
R ~ ki * length / Area
A
I
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Def: Resistance
 Resistance Continued: In a conductor, such as
silver, carbon (graphene) or copper or gold,
resistivity is very small
 Resistance is expressed in units of Ohm, symbol: Ω
 Resistance grows proportional to the length l of
conducting material, and decreases inversely
proportional to the diameter A of the conductor; ki
being a material constant!
R~l/A
R = ki * l / A
ki
l
A
being a constant depending on material
being the length
being the diameter of the conducting material --not ampere!
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Def: Volt
 A Volt is the SI unit of electrical force to push one
ampere of current against a one Ω resistance
 Or the electric potential difference between 2 points
of a conductor when a current dissipates one watt
 A Volt is AKA the potential difference between 2
planes that are 1 m apart with an electric field of 1
newton / coulomb
 It is NOT one of the 7 base units on page 3 or 4!
 In the mks system the dimension is a derived unit:
[V] = [kg m2 A-1 s-3]
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Def: Volt
 A Volt is named named in honor of the Italian
physicist Alessandro Volta (1745-1827), inventor of
the first voltaic pile (chemical battery)
 A Volt is Amperes times Ohm, Watts per Ampere, or
Joules per Coulomb:
V=A*Ω
V=W/A
V=J/C
V = dw / dq
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Passive Element
 Capacitors, Inductors, Resistors are passive
elements
 They cannot generate energy
 Inductors and Capacitors can store energy
 Capacitor stores electric energy
 Inductor stores magnetic energy
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Passive Sign Convention
 Assigning a reference direction for current or
voltage in a circuit is arbitrary
 Used consistently, any method works out fine
 The most widely used method is the Passive Sign
Convention:
 When the reference direction for the current in a
passive element is in the direction of the voltage
drop across that element, use a + sign in any
expression that relates current to voltage
 Else use the – sign
 That we call the Passive Sign Convention
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Displacement Current in Capacitor
 Graphic symbol for capacitor is dual vertical bar, or
one side rounded; alludes to two plates
 Capacitor cannot directly transport current, due to
the separating dielectric, AKA insulator
 Yet one plate may repel charged particles on the
opposing side, creating impression that for brief
period a current is flowing
 Applying alternating voltage to the plates of a
capacitor, charge displacement continues with the
frequency of the AC, creating the impression of
conductivity
 Thus for AC the capacitor acts like a conductor, yet
for DC it completely insulates
 At its terminals, a capacitor’s displacement current is
indistinguishable from a conduction current
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Unit of Farad
 Starting with fundamental equation with capacity C:
i = C * dv / dt
 Where C is measured in Farad [F], we can bring C to
one side and get its dimension from the other parts
C = i * dt / dv
 With the units [F] = [A s] [V-1]
 Equivalently (plus many more):
[F] = [A s V-1] = [s Ω-1] = [Q V-1] – Q: Coulomb
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Unit of Henry
 Starting with fundamental equation:
v = L * di / dt
 Where L is measured in Henry [H], we can bring L to
one side and get its dimension from the other parts
L = v * dt / di
 With the units [H] = [V s] [A-1]
 For [H] we derive the units below –and many others:
[H] = [V s A-1] = [s Ω] = [J A-2] – J: Joule
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Equations
 SI unit of [H] = [V s A-1]
 SI unit of [F] = [A s V-1]
 KVL for Natural Response in R L circuit with single R,
L, and current i(t) = i:
L di / dt + R i = 0
 KCL for Natural Response in R C circuit with single R,
C, and voltage v(t) = v:
C dv / dt + v / R = 0
 KCL for Natural Response in parallel R L C circuit with
single R, L, C and voltage v(t) = v, and initial current I0:
v / R + 1/L v dt + C dv/dt + I0 = 0
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Equations
 Verify the SI units (dimension) for KCL equation
v / R + 1/L v dt + C dv/dt + I0 = 0
 All summands must be of units [A]
 SI units of v / R
= [V A V-1]
 SI units of 1/L v dt
= [V-1 s-1 A V s] = [A]
 SI units of C dv / dt
= [A s V-1 V s-1] = [A]
 Confirmed!
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= [A]
Bibliography
1. Electric Circuits, 10nd edition, Nilsson and Riedel,
Pearsons Publishers, © 2015 ISBN-13: 978-0-13376003-3
2. SI Units from NIST:
http://physics.nist.gov/cuu/Units/units.html
3. NIST Special Publication 330, © 2008 Edition, by
Taylor and Thompson, lists the SI units
4. Peter Mohr, NIST Publication “Redefining the SI Base
Units”, November 2., 2011
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