formulation in terms of total charge and current si units name...

This discussion revisits ideas several times to make the points clear. (Updated from inductor.docx) Maxwell’s Eq. are the fundamental underpinnings of electronics. We will not incorporate these equations in detail towards solving problems in analog electronics but we begin by recognizing that electric and magnetic fields play an important role in understanding. Indeed by choosing models for components that incorporate these underpinnings we can simplify the approach. Deriving the properties of components will require an understanding of some of Maxwell’s relationships. View the graphic illustrations from Beyond the Mechanical Universe to begin to develop the type of understanding that is helpful in understanding circuits.

Formulation in terms of total charge and current SI UNITS Name Differential form Integral form

1 Gauss's law

2 Gauss's law for B

3 Maxwell–Faraday equation

4 Ampère's circuital law

Energy and energy flow are important. To illustrate the formulation flow, consider charge flow with ρ as charge per unit volume and 𝐽 the flow of charge or current. 𝐽 is the current density è amperes per square meter. ∇ ∙ 𝐽 is the divergence of the flow. It measures how the flow of charge changes. If the divergence is 0, ∇ ∙ 𝐽 = 0 then there is no build up of charge. When it is non zero it means that the charge density will increase or decrease because more is flowing in/out as out/in. Conservation of charge says that a time varying charge at some point must be due to the movement of charge into/out of a region. Equal flow in an out does not build up charge so it must be related to the divergence of the flow. !"!"+ ∇ ∙ 𝐽 = 0 Conservation of charge

Poynting vector

𝑆

rate of energy transport by E&M fields per unit area is described by the vecto

Poynting theorem

Time rate of change of the energy density of an E&M field at a point plus the divergence (related to the difference in the transport of energy in vs energy out) of the pointing vector (flow of energy). Must be equal to the transfer of energy to/from electric charges by the work done by the Electric field.

More simply for circuits under conditions where radiation can be ignored

• Electric fields are produced by charge build up. Electrostatic fields. Fields start and end on charge.

• No magnetic charges (N/S poles) so the magnetic field lines have no starting or ending points.

• To build a non conservative E field you pull the energy from the current by a changing magnetic field (dB/dt has an associated E)(Lenz’s law)

• Magnetic fields are generated by currents

Ralph Morrison [Fields of Electronics (pg 77-‐78)]è Paraphrased The true field patterns for various elements extend out into space. In the circuit view of capacitors and inductors the field energy does not leave the confines of the components. As components get smaller a larger fraction of their field energy is in the space around the component. This energy must leave and return. For an AC circuit this is twice per cycle so the time scale is 1/f. Field energy travels at roughly c (1 ns/ft). Thus the field extension and the frequency set the scale for when the energy cannot return to the component fast enough. If the field energy cannot return in phase with the current or voltage the energy is radiated. The statement above tries to establish when radiation can be ignored from a geometric and speed argument. The point is important. Antennae are designed to radiate energy as their fundamental electrical function. Protoboard circuits are not supposed to radiate and they don’t at the frequencies used in the lab. There are however fields set up in space and the energy transport is throught these fields. On wiki I discovered this nice images. They depict the external fields and show energy propagation.

energy density u 𝑢! =!"!#$%!"#$%&

= !!𝜀!𝐸!, 𝑢! =

!"!#$%!"#$%&

= !!!!

!!

http://physics.stackexchange.com/questions/17085/is-‐there-‐something-‐like-‐the-‐poynting-‐vector-‐for-‐hydraulic-‐circuits

www.furryelephant.com/content/electricity/visualizing-‐electric-‐current/surface-‐charges-‐poynting-‐vector/ red electric, green magnetic, arrows energy flow

Now you can add specific constraints and assumptions

• Add in conducting materials and special geometry to get circuits. • Conductors immediately cancel field (V=0 inside) • driving frequencies slow • wait until transients are damped out • Ohm’s law for resistors. • capacitors store all energy in E and neglect B • wires and resitors neglect effects of B • Inductors store energy in B, ideal èmagnetic fields only energy storage. • Power supplies can drive circuits at fixed frequency and fixed amplitude.

So the field eq. with the geometric and material properties have a steady state solution and the voltages across the components can be characterized as below. Voltage (general and electrostatics) In electrostatics the fields and charge distribution’s are independent of time. In this

case

reduces to ∇×𝐸 = 0 and under these conditions the electric field can be defined in terms of the gradient of a scalar potential V(x,y,z). Once time dependent fields are allowed the full content of Maxwells equations must include both 𝐸, 𝐵 and their relationship. The result is that an expanded potential function 𝐴,Φ must be used. Here 𝐴 is the vector potential and Φ is the scalar potential which is a generalized version of V. The most general choice for 𝐴,Φ defines them up to a gauge transformation. That is any 𝐴,Φ related by 𝐴⟹ 𝐴 + ∇Λ è you can add the gradient of a scalar function Λ to 𝐴 and Maxwell’s equations remain unchanged. Φ⟹ Φ− !

!!!!! as long as Λ satisfies ∇!Λ− !

!!!!!!!!

= 0 Of interest is the choice of Coulomb gauge where the scalar potential is chosen based on the condition that ∇ ∙ 𝐴 = 0 which leads to ∇!Φ = 𝜌 𝜖! . This defines the potential as that due instantaneously to the charge distribution at time t. This formalism is counter intuitive. One expects the relativistic constraint that fields should set up in a manner consistent with a propagation time less than or equal to c should also apply to these potential functions. However it is possible to define the scalar potential in this manner and preserve the nature of the E and B derived because of the corresponding 𝐴 . Take home message For circuits in this course, the usual definition of the potential derived from electrostatics will be fine. However in reviewing general E&M one finds that V only applies to electrostatics. Circuit theory must deal with both the steady state and the transient aspects of circuitry. A charging capacitor is an example of a time dependent circuit response. Treating circuit responses that are on the 10’s nanosecond scale (100 MHz) do not require complex extensions from electrostatics. However one must realize that in general there may be effects due to the time dependent nature of the circuit response that requires the more complete formulation. When circuits and time periods get small then quantum effects and the full set of Maxwell’s equation may need to be considered. When analyzing the sources of noise small voltages may be generated by pickup of time dependent fields in the environment.

First let us carefully consider voltage, V, and EMFs, εxxx There are two types of electric fields based on the way they are generated.

1. 𝐸! generated by charges in space Q. Based on Max. Eq 1. conservative electrostatic field 𝐸!

2. 𝐸! generated by change 𝐵 →!"!" i.e. Faraday’s law. Based on Max. Eq. 3.

These fields have non zero integral on a closed path (non-‐conservative).

Φ! = 𝐵 ∙ 𝑑𝐴

is the magnetic flux through a surface.

𝐸! ∙𝑑𝑙 = −𝑑Φ𝑑𝑡

The electric field 𝐸! is generated such that integral around the line that encloses the surface is due to a changing flux

Convention è VBA means the "V" from A to B, at B relative to A, or with respect to A] Some distribution of static charge creates an electric field 𝐸! and you move a small test charge through this field between points AèB and you can define the voltage at B wrt A as the work done against the field to move the test charge.

𝑉!" = − 𝐸! ∙ 𝑑𝑙!

!

The voltage in a circuit is the work per unit charge to move through the field of the first type along some path.

By defining EMFs wrt closed paths only the non-‐conservative forces are counted

This careful separation is required because in electric circuits the charges are moved by various forces through some path along a circuit. In a battery, for example, there may be chemical potentials that drive the charges in one direction and a build up of charge that creates an electric field 𝐸!.

The battery above builds a charge distribution so that the electric field inside the battery balances the chemical forces pushing + charges up. The work/per unit charge done by the chemical process is extracted and stored in the idealized field shown. Voltage is defined as the work done moving a charge through the electric field and εbat is the work done by the chemical forces per unit charge. This is the obvious case since there are two distinct forms of energy: chemical and electrostatic. The electrostatic field shown above only represents the field inside the battery. This field also extends out through the circuit, is modified by additional charge build up and drives charges through, for example, resistors. If charges are free to flow they can be transported from the positive to the negative terminal extracting energy from this electric field. Charges extract energy qV in this transport. The chemical field only resides inside the battery. And in any closed loop there is a gain qεbat on each cycle through the circuit. For the battery εbat=Vbat

The field that permeates the circuit due to the charges on the PS terminals (shown as the two charged metal plates above) can be described by a potential. All of the potentials used to analyze a circuit are expressed in terms of the work done moving through the circuit by these fields. The chemical energy and the induced E can be considered to generate these fields by moving the charges to the ends of the

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components (PS, inductor) . It is this build up of charge that generates the voltage, field through the circuit. Interior to the PS and the inductor this field is equal and opposite to the induced field for inductors or the way the chemical process does work for a battery. Again the charges are pushed against a field to build up charge at the ends until the battery can no longer move charge to the terminals or the inductors induced field can no longer move charge to set up the electrostatic field 𝐸!. WIKIPEDIA (with some additional edits) EMF is classified as the external work expended per unit of charge to produce an electric potential difference across two open-‐circuited terminals. For an open ended inductor charge builds at the ends which develop a field 𝐸! to balance the file 𝐸! . 𝐸! = −𝐸!

Voltage is equal to the work which would have to be done, per unit charge, against a static electric field 𝐸! to move the charge between two points.

Inside a source of emf that is open-circuited, the conservative electrostatic field created by separation of charge exactly cancels the forces producing the emf. Thus, the emf has the same value but opposite sign as the integral of the electric field aligned with an internal path between two terminals A and B of a source of emf in open-circuit condition (the path is taken from the negative terminal to the positive terminal to yield a positive emf, indicating work done on the electrons moving in the circuit).[22] Mathematically:

where Ecs is the conservative electrostatic field 𝐸!created by the charge separation associated with the emf, dℓ is an element of the path from terminal A to terminal B. This equation applies only to locations A and B that are terminals, and does not apply to paths between points A and B with portions outside the source of emf. This equation involves the electrostatic electric field due to charge separation Ecs and does not involve (for example) any non-conservative component of electric field due to Faraday's law of induction.

In the case of a closed path in the presence of a varying magnetic field, the integral of the electric field around a closed loop may be nonzero; one common application of the concept of emf, known as "induced emf" is the voltage induced in a such a loop.[24] The "induced emf" around a stationary closed path C is:

where now E is the entire electric field, conservative and non-conservative, and the integral is around an arbitrary but stationary closed curve C through which there is a varying magnetic field. Note that the electrostatic field does not contribute to the net emf around a circuit because the electrostatic portion of the electric field is conservative (that is, the work done against the field around a closed path is zero).

This definition can be extended to arbitrary sources of emf and moving paths C:[25]

which is a conceptual equation mainly, because the determination of the "effective forces" is difficult. POWER SUPPLY The power supply effectively puts charge on its plates (act simply like a capacitor) with a given magnitude and frequency. Driving the circuit with this charged CAP (PS terminals) produces fields across all the components.

A complete picture takes this into account when there is significant energy radiated by the PS. For an AC PS at a fixed frequency 𝜔 = 2𝜋𝑓 [1 Hz = 2𝜋 radians/s]. All processes will reach an equilibrium oscillatory behavior with frequency 𝜔. The differences will be in amplitude and phase. We fix the phase of the current from the PS as follows: 𝑖!" 𝑡 = 𝐼!"!𝑠𝑖𝑛 𝜔𝑡 = 𝐼!"!𝑐𝑜𝑠 𝜔𝑡 − !

!

𝑣!" 𝑡 = 𝑉!"!𝑠𝑖𝑛 𝜔𝑡 + 𝜑!" phase depends on circuit Resistors In R there is a field and the charges move through this field and the field does work on the charges which is released as heat. Field is always in phase with the current. Work per unit charge is the voltage. Voltage on the resistor is minus the work described. So the potential or V is the work you would do to lift a test charge from bottom to the top (i.e. moving opposite the current direction). When you go down

hill you gain energy gravity does positive work but the potential of gravity drops. Potentials increase by pushing stuff against the appropriate field. Resistor is in phase with the current 𝑉! = 𝑖𝑅 = 𝐼!𝑅 sin 𝜔𝑡 = 𝐼!𝑅 cos 𝜔𝑡 −

𝜋2

Since the resistor voltage is in phase with the current energy is always being supplied to the resistor. Thus a resistor is a energy dissapator. All through the circuit you calculate ∆𝑉 across components as:

∆𝑉 = 𝑉! − 𝑉! = − 𝐸 ∙ 𝑑𝑙

However there is a special field that you consider and you do not include fields that drive the plate charging in the PS or the induced fields of the inductor. In both these devices there are two parts that push charge. In the battery it might be chemical or electrical but this mechanism charges up the PS plates. The overall force experienced by the charge is zero. [Same as when you lift a ball at constant speed. No overall force but person’s work is stored in the field and the field extracts this work by doing negative work.] The battery pushes the charge with a force that matches the E-‐field. In the inductor there is an induced E field that sucks the energy out and into the B or decreasing B cause an E to push charge and give the charge energy. So when you calculate the voltage/field across the inductor you don’t use the E induced which is always equal and opposite E circuit. Just as in the battery the steady state solution has the total force as zero in the battery and the inductor. There are two fields that balance. Again, the induced E field is equal and opposite electrostatic field of the circuit in the inductor. Current again assumes -‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐ 𝑖 = 𝐼! sin 𝜔𝑡 = 𝐼! cos 𝜔𝑡 − !

!

Capacitor -‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐ Produces an electric field from+ plate to the minus plate both in between the plates and through the circuit. This field will drive charges and can be characterized by a voltage V =Q/C. [top high, bottom low]

𝑉! =𝑄𝐶 =

1𝐶

𝑑𝑖𝑑𝑡 = −

𝐼!𝜔𝐶 cos (𝜔𝑡)

Of course when viewing the full cycle the capacitor field changes direction and sometimes the current moves in the direction of the field (charging) but there are also times that the current is moving opposite the direction of the field (discharging). These correspond to parts of the cycle where energy is stored and released respectively. Inductor -‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐ Inductor has an induced field Eind that does not permeate the circuits this wraps around the wires in a closed loop. This field will drive charge along the field lines (opposite of the way the capacitor works) until a voltage is built up to oppose this field. You must imagine a capacitor in parallel that receives the charges that are pushed through thereby creating a field of type 𝐸! in the inductor opposite the induced field. This field can be characterized using the potential rather than a field. Eind is an 𝐸! type field.

phasor

direction at t=0

Current is increasing in the direction shown. Faraday’s law states and EMF will be generated to oppose this increase. The inductor is shown in the middle frame (non standard representation) where the induced field blue is balanced by the build up of charge which generates the black field and a corresponding voltage equal to the EMF just as in the battery.

𝑉! = 𝐿𝑑𝑖𝑑𝑡 = 𝜔𝐿𝐼!cos (𝜔𝑡)


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Inductor & Power Supply restated-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐ The power supply and the inductor are similar in that there is an alternate source of the field generated around the loop. The power supply uses, for example, chemical energy to drive the charges onto a capacitor, its terminals. The inductor uses the magnetic field energy transferred to the current by the changing flux. Eind is an 𝐸! type field.

These additional Emfs push charge through the element. For an inductor since the induced field pushes the charge as they move through the inductor a voltage must be maintained to counter this push. The hidden capacitance of the circuit will therefore charge up as shown above in the same way that a battery charges up the ends (terminals) which generates an electric field in the circuit. At times of increasing positive current there must be a supplied voltage to push the charges in the direction of the current. At times of decreasing positive current the EMF is pushing the current and the voltage is opposite the current direction. During this part of the cycle the EMF does positive work on the charges and the electrostatic fields related to voltage are opposite the motion so they extract energy (negative work). So electrostatic fields are setup across the various components by the buildup of charge in the circuit. Since the components can be adequately described in terms of the voltage drops and not the fields one only needs to find the work done crossing each element. Thus EMFs and Voltages are used to characterize the circuit behavior. Each element has been idealized to encompass one aspect of the problem. A conducting wire will adjust charge to eliminate internal fields. A capacitor builds conservative electric fields, 𝐸!, with no significant magnetic field. An inductor builds magnetic fields and subsequent non-‐conservative 𝐸! that generates the conservative electric fields. To treat general situations these basic elements are combined. derivatives function same derivative integral Phasor [real] sin(θ) cos(θ-‐π/2) cos(θ) -‐cos(θ) down cos(θ) sin(θ+π/2) -‐sin(θ) sin(θ) right -‐sin(θ) cos(θ+π/2) -‐cos(θ) cos(θ) up


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-‐cos(θ) sin(θ-‐π/2) sin(θ) -‐sin(θ) left

green is sine blue is sine phase of +pi/2 red is phase –pi/2 Let phase angle be defined wrt x-‐axis in that we project rotating arrows (counter clockwise, angular speed 𝜔) i.e. look at their cosine projection. For t=0 cosine is 1. For 𝜔𝑡 = !

! cosine is 0.

The blue oscillation above is a cosine oscillation with phase of 0. Green is a sine function with 0 phase or a cosine with !

! , cos 𝜔𝑡 − !

!= sin (𝜔𝑡)

Red is cosine with !!

180o=-‐π 0o -‐90o=-‐π/2 You can also imagine a mass spring system with a driving force. The diff. eq. will have a steady state solution as an analog to the electric circuit. See Coax cable.doc -‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐other ideas -‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐

-‐1.5

-‐1

-‐0.5

0

0.5

1

1.5

0 20 40 60 80 100 120

L

C

R

Looking for solutions to configurations of components coupled to driving mechanisms or power supplies. There are mechanical analogs to passive circuits that involves springs, masses, damping and driving forces. To find solutions we develop the differential equation. One approach is:

1. demand that the voltage (due to 𝐸! fields) around a closed loop is zero.

2. Current into a junction is the same as the current out.

𝑉!" 𝑡 = 𝑉! 𝑡 + 𝑉! 𝑡 + 𝑉! 𝑡 𝑖!" 𝑡 = 𝐼! 𝑡 + 𝐼! 𝑡 Kirchoff’s Laws

Series RLC circuit

𝑉!" 𝑡 power supply 𝑉! 𝑡 = 𝑅𝑖 𝑡 capacitor 𝑉! 𝑡 = ! !

! capacitor

𝑉! 𝑡 = 𝐿 !" !

!" capacitor

𝑉!" 𝑡 = 𝑅𝑖 𝑡 + ! !

!+ 𝐿 !" !

!"

𝑉!" 𝑡 = 𝑅𝑑𝑄 𝑡𝑑𝑡 +

𝑄 𝑡𝐶 + 𝐿

𝑑!𝑄 𝑡𝑑𝑡!

0 = 𝑅𝑑𝑄 𝑡𝑑𝑡 +

𝑄 𝑡𝐶 + 𝐿

𝑑!𝑄 𝑡𝑑𝑡!

𝑎𝑦!! + 𝑏𝑦! + 𝑐𝑦 = 0

The solution is of the form:

𝑦 𝑡 = 𝐴!𝑒!" ; 𝑟 =−𝑏 ± 𝑏! − 4𝑎𝑐

2𝑎 There are three possibilities: critically damped, overdamped, underdamped. When you build a circuit and then apply power the circuit must transition from the initial state to the steady state. For well behaved circuits one can often just wait for the transients to damp out and look for the steady state. However, some circuits may have remarkable responses to certain transitions. One classic example is the inductor. Large voltages can be generated if one simply opens a switch in a circuit with large inductance. This is of course how spark plug voltages were generated in cars with a distributor with rotor and gap. The link on the web page to an RLC lab explores this in a nice way. For now we ignore the transient behavior and look for solutions only when the driving voltage: 𝑉!" 𝑡 = 𝑉! sin 𝜔𝑡 + 𝜑 Since any function 𝑉!" 𝑡 can be written as a sum of sinusoidal functions (Fourier transform) and since the equation above is linear, the response of the RLC circuit to a genera sinusoidal voltage can be used to construct the response to a general driving potential. To find the solution we represent the voltages across components by a complex impedance Z and then V=IZ. However Z is complex. The complex nature and the form of the equation preserve the frequency but allow for a phase and an amplitude that depend on the component. These factors are included in the discussion above. The complex impedance is an arrow in the complex plane. These arrows or phasors add like vectors. Considering the equivalent impedance of a LC parallel circuit.

Voltages are equal and the current is the sum. In order to include the phase differences we find. 𝑉 = 𝐼𝑍 Z is the impedance 𝑉 𝑡 = 𝑉! 𝑡 = 𝑉! 𝑡 parallel elements have the same voltage and different current. = 𝐼!"𝜔𝐿 sin 𝜔𝑡 +

𝜋2 = 𝐼!"𝜔𝐿𝑗 sin 𝜔𝑡

= 𝐼!"1𝜔𝐶 sin 𝜔𝑡 −

𝜋2 = 𝐼!"

1𝑗𝜔𝐶 sin 𝜔𝑡

The fact that they are 180o out of phase means their amplitudes subtract and this is handled by the imaginary factors j. So we add the two currents to get total

𝐼! =𝑉!𝑗𝜔𝐿 + 𝑉!𝑗𝜔𝐶 = 𝑗𝑉!(𝜔𝐶 −

1𝜔𝐿)

𝑍!"#$% =1

𝜔𝐶 − 1𝜔𝐿

=𝜔𝐿

𝜔!𝐿𝐶 − 1

Define 𝜔! =

!!"; 𝜔!! =

!!"

𝑍!"#$% =

!"!!!

!!!!!! at the “magic frequency the impedance is infinite [broken]

The voltage is up the two currents point left and right. When the two currents are equal then the charge feeds the inductor and the inductor charges the capacitor in such a manner that no additional current is required to maintain the voltage. therefore as soon as the steady state is reached the cap and the inductor will maintain the voltage of the power supply with i=0. This similar to a capacitor in a DC circuit. It charges up and then maintains the PS voltage without current. This will occur when 𝜔 = !

!" for higher frequencies we find that 𝜔𝐶 > !

!" in this case the

impedance 𝑍 = !!" and therefore requires more current to maintain the voltage on

the the capacitor. This arrangement makes the overall current have the appropriate phase as a capacitor. It is good to understand that in the limit of the frequency 0 or ∞ the components have interesting response. impedance 𝜔

0 𝜔

∞

inductor 𝑗𝜔𝐿 short circuit open circuit capacitor 1

𝑗𝜔𝐶 open circuit short circuit

To improve basic understanding here are some impedance values I found. Check to see if you agree.

Practical Electronics Handbook, Ian Sinclair

Notice that in dealing with a AC-‐PS, Max. eq 4 is not invoked so the fields generated by !!

!" are ignored.

formulation in terms of total charge and current si units name...

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