1 ECE 222 Electric Circuit Analysis II Chapter 2 Physics Rules for Circuits Herbert G. Mayer, PSU Status 4/21/2016 For use at CCUT Spring 2016

2 Syllabus SI Units SI Units Change of Current in L Change of Current in L Change of Voltage in C Change of Voltage in C Definitions Definitions Passive Sign Convention Passive Sign Convention Displacement Current in Capacitor Displacement Current in Capacitor Unit of Farad Unit of Farad Unit of Henry Unit of Henry Equations Equations Bibliography Bibliography

3 Only 7 SI Base Units

4

5 Units Derived from 7 SI

6 For example, take Hertz, Hz, [Hz] For example, take Hertz, Hz, [Hz] Second row, rightmost column Second row, rightmost column Unit is s -1 AKA Hz Unit is s -1 AKA Hz Is derived from SI unit second, for time: s Is derived from SI unit second, for time: s Careful, two different meanings of word “second” Careful, two different meanings of word “second”

7 SI Units m: a meter is the length of light traveled in 1/299,792,458 th of a second m: a meter is the length of light traveled in 1/299,792,458 th of a second kg: kilogram is equal to the reference prototype, i.e. a defined cube; will likely change 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 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 * newton per meter of conductor A: One ampere is the current which in 2 parallel conductors 1 meter apart in a vacuum produces a force of 2 * newton per meter of conductor See completely different definition for A on p. 12! See completely different definition for A on p. 12!

8 SI Units K: Kelvin – thermodynamic temperature unit of the 1/ fraction of water temperature at triple point K: Kelvin – thermodynamic temperature unit of the 1/ 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 kilogram of carbon 12; entities can be atoms, molecules, electrons mol: mole – is amount of substance of a system which contains as many elementary entities as there are atoms in 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 specified elementary entities Old definition: the mole is the amount of substance that contains 6.022,141,79 x specified elementary entities cd: candela – is luminous intensity of a source that emits monochromatic radiation of frequency 540 * hertz, plus some further constraints cd: candela – is luminous intensity of a source that emits monochromatic radiation of frequency 540 * hertz, plus some further constraints

9 Change of Current in L Current i cannot change instantaneously within an inductor Current i cannot change instantaneously within an inductor If i would suddenly change, the voltage would grow toward infinity; physically not possible 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 Or infinite voltage would be required to accomplish that; physically not possible See formula for v(t), with L being inductivity in Henry 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 But yes, voltage can change instantaneously in an inductor; i.e. an inductor can raise sudden voltage across its terminals Never forget: inductivity unit Henry [H] in SI = [V s A -1 ] Never forget: inductivity unit Henry [H] in SI = [V s A -1 ]

10 Change of Current in L Current instead has to grow gradually, from i = 0 A to different value in a quiet system; quiet here mans: initially no current Current instead has to grow gradually, from i = 0 A to different value in a quiet system; quiet here mans: initially no current Or from i (t=0) = i 0 with i 0 ≠ 0 A in a system with past electric history Or from i (t=0) = i 0 with i 0 ≠ 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 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, when a DC source is connected to an inductor, is caused by the field being built up, or being changed; after some time no more resistance for DC Initial resistance, when a DC source is connected to an inductor, is caused by the field being built up, or being changed; after some time no more resistance for DC Thus one best not place an inductor directly parallel to a constant voltage source; instead have a resistor in series Thus one best not place an inductor directly parallel to a constant voltage source; instead have a resistor in series

11 Change of Voltage in C Voltage v cannot change instantaneously in a capacitor Voltage v cannot change instantaneously in a capacitor To change v in 0 time, an infinitely strong current would be required; physically not possible 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 See formula for i(t), with C being Capacity in Farad i(t) = C * dv / dt But current i can change instantaneously in a capacitor, i.e. a capacitor can have a sudden displacement current But current i can change instantaneously in a capacitor, i.e. a capacitor can have a sudden displacement current Voltage grows gradually, from v = 0 in an initial system Voltage grows gradually, from v = 0 in an initial system Or voltage grows from v t0 = v 0, with v 0 > 0 V in a system with past electric history Or voltage grows from v t0 = v 0, with v 0 > 0 V in a system with past electric history Never forget: capacity unit Farad [F] in SI = [A s V -1 ] Never forget: capacity unit Farad [F] in SI = [A s V -1 ]

12 Def: Ampere Ampere is the unit of current; one of the 7 base units of the SI Ampere is the unit of current; one of the 7 base units of the SI Named after André Marie Ampère, French physicist 1775 – 1836 Named after André Marie Ampère, French physicist 1775 – 1836 When * electrons stream though a conductor in a second, the amount of charge moved is 1 C and the current 1 A; AKA “amp”. (Old definition) When * electrons stream though a conductor in a second, the amount of charge moved is 1 C and the current 1 A; AKA “amp”. (Old definition) i = dq / dt 1 A = 1 C / s C here: Coulomb! Not capacitance

13 Def: Capacitance Electrical capacitance represented by letter C, measured in Farad F. Capacitor does not directly conduct current; insulator separates its 2 plates 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, even pF 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, even pF i ~ dv / dt i = C * dv / dt ithe resulting current in A, caused by the changing voltage Cthe capacitor’s capacitance, measured in Farad dvthe change in voltage across the 2 plates

14 Def: Capacitor A capacitor’s power p and energy w? p = v * i p = C * v * dv / dt w = C * v 2 / 2 wenergy in Joule ppower v measured in Watt ithe displacement current, in A Cis the capacitor’s capacitance, measured in Farad dvthe change in voltage across the 2 conducting plates

15 Def: Capacitor SI Unit Farad is not one of the 7 elementary SI units 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: To remember, one of the many ways that [F] is expressed by simpler SI units, remember the secret and fictitious equation: Q = Q Q = Q AKA: Q = CU Where Q is charge in Coulomb. SI units are: Ampere times seconds [A s] Where Q is charge in Coulomb. SI units are: Ampere times seconds [A s] CU is really capacity times Volt, i.e. SI units [F V] CU is really capacity times Volt, i.e. SI units [F V] With [A s] = [F V] we derive Farad: With [A s] = [F V] we derive Farad: [F] = [A s V -1 ] Volt in turn can be substituted by other SI units Volt in turn can be substituted by other SI units

16 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 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… ×10 18 electrons A coulomb is equal to a charge of approximately 6.241… ×10 18 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 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 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 Charge and gravity are magic Charge and gravity are magic

17 Def: Electron An electron? Subatomic particle with electric charge; we call that charge negative; part of lepton family 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 Called an elementary particle, since it seems to have no sub-particles Has mass of approx. 1/1836 of a proton Has mass of approx. 1/1836 of a proton Yet electrons have properties of particles AND waves Yet electrons have properties of particles AND waves

18 Def: Inductance Electrical inductance and related power and energy? p = i * v p = i * L * di / dt w = ( L / 2 ) * i 2 wthe energy in Joule pthe power measured in Watt Lthe inductance in Henry H ithe current in A dithe change of current over time, in A

19 Def: Inductance Electrical inductance? A charge in motion (e.g. some current) creates a magnetic field around its conductor 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 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 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 vmeasured in Volt V Linductance in Henry H dithe change in current A

20 Def: Magnetic Coupling Assumes circuit with 2 inductors, one in circuit c1 and a second inductor in circuit c2 Assumes circuit with 2 inductors, one in circuit c1 and a second inductor in circuit c2 Inductor c1 and c2 are not connected, but close to one another; close means, the magnetic field of one permeates the other unit; and vice versa Inductor c1 and c2 are not connected, but close to one another; close means, the magnetic field of one permeates the other unit; and vice versa When voltage in c2 is induced by a current in c1, we say, c1 and c2 are magnetically coupled When voltage in c2 is induced by a current in c1, we say, c1 and c2 are magnetically coupled AKA mutual inductance AKA mutual inductance

21 Def: Resistance Electrical resistance? A material’s opposition to the free flow of electrons Electrical resistance? A material’s opposition to the free flow of electrons Read about negative resistance in [5] Read about negative resistance in [5] In an insulator, such as vacuum or porcelain, resistivity is very large, typically >> 1 MΩ (Mega Ohm) In an insulator, such as vacuum or porcelain, resistivity is very large, typically >> 1 MΩ (Mega Ohm) R ~ k i * length / Area A I

22 Def: Resistance Resistance Continued: In a conductor, such as silver, carbon (graphene) or copper or gold, resistivity is very small 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 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; k i being a material constant! Resistance grows proportional to the length l of conducting material, and decreases inversely proportional to the diameter A of the conductor; k i being a material constant! R ~ l / A R = k i * l / A k i being a constant depending on material lbeing the length Abeing the diameter of the conducting material --not ampere!

23 Def: Volt A Volt is the SI unit of electrical force to push one ampere of current against a one Ω resistance 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 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 A Volt is AKA the potential difference between 2 planes that are 1 m apart with an electric field of 1 newton / coulomb NOT one of the 7 SI base units on page 3 or 4! NOT one of the 7 SI base units on page 3 or 4! In the mks system the dimension is a derived unit: In the mks system the dimension is a derived unit: [V] = [kg m 2 A -1 s -3 ]

24 Def: Volt A Volt is named named in honor of the Italian physicist Alessandro Volta ( ), inventor of the first voltaic pile (chemical battery) A Volt is named named in honor of the Italian physicist Alessandro Volta ( ), inventor of the first voltaic pile (chemical battery) A Volt is Amperes times Ohm, Watts per Ampere, or Joules per Coulomb: 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

25 Passive Element Capacitors, Inductors, Resistors are passive elements Capacitors, Inductors, Resistors are passive elements They cannot generate energy They cannot generate energy Inductors and Capacitors can store energy Inductors and Capacitors can store energy Capacitor stores electric energy Capacitor stores electric energy Inductor stores magnetic energy Inductor stores magnetic energy

26 Passive Sign Convention Assigning a reference direction for current or voltage in a circuit is arbitrary Assigning a reference direction for current or voltage in a circuit is arbitrary Used consistently, any method works out fine Used consistently, any method works out fine The most widely used method is the Passive Sign Convention: 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 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 Else use the – sign That we call the Passive Sign Convention That we call the Passive Sign Convention

27 Displacement Current in Capacitor Graphic symbol for capacitor is dual vertical bar, or one side rounded; alludes to two conducting plates Graphic symbol for capacitor is dual vertical bar, or one side rounded; alludes to two conducting plates Capacitor cannot directly transport current, due to the separating dielectric, AKA insulator 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 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 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 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 At its terminals, a capacitor’s displacement current is indistinguishable from a conduction current

28 Unit of Farad Starting with fundamental equation involving capacity C: Starting with fundamental equation involving 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 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 ] With the units [F] = [A s] [V -1 ] Equivalently (plus many more): Equivalently (plus many more): [F] = [A s V -1 ] = [s Ω -1 ] = [Q V -1 ] – Q: Coulomb

29 Unit of Henry Starting with fundamental equation, involving inductivity: Starting with fundamental equation, involving inductivity: 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 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 ] With the units [H] = [V s] [A -1 ] For [H] we derive the units below –and many others: For [H] we derive the units below –and many others: [H] = [V s A -1 ] = [s Ω] = [J A -2 ] – J: Joule

30 Equations SI unit of [H] = [V s A -1 ] SI unit of [H] = [V s A -1 ] SI unit of [F] = [A s V -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: 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: 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 I 0 : KCL for Natural Response in parallel R L C circuit with single R, L, C and voltage v(t) = v, and initial current I 0 : v / R + 1/L v dt + C dv/dt + I 0 = 0

31 Equations Verify the SI units (dimension) for KCL equation Verify the SI units (dimension) for KCL equation v / R + 1/L v dt + C dv/dt + I 0 = 0 All summands must be of units [A] All summands must be of units [A] SI units of v / R= [V A V -1 ]= [A] SI units of v / R= [V A V -1 ]= [A] SI units of 1/L v dt= [V -1 s -1 A V s]= [A] 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] SI units of C dv / dt= [A s V -1 V s -1 ]= [A] Confirmed! Confirmed!

32 Bibliography 1. 1.Electric Circuits, 10 nd edition, Nilsson and Riedel, Pearsons Publishers, © 2015 ISBN-13: SI Units from NIST: NIST Special Publication 330, © 2008 Edition, by Taylor and Thompson, lists the SI units 4.Redefining the SI Base Units 4.Peter Mohr, NIST Publication “Redefining the SI Base Units”, November 2.,