Whites, EE 8 Lecture 8 Page of 9 Lecture 8: Maxwell s Equations an Electrical ircuits. Electrical circuit analysis is usually presente as a theory unto itself. However, the basis of electrical circuit analysis actually comes from electromagnetics, i.e., Maxwell s equations. It is important to recognize this since electrical circuit theory is really only an approximation an uner the right conitions, it can fail. We will illustrate how electrical circuit analysis is erive from Maxwell s equations by consiering the following physical circuit (i.e., an electrical circuit that has physical imensions): I(t) R V s (t) s' c L 0 Keith W. Whites
Whites, EE 8 Lecture 8 Page of 9 First, we will apply Faraay s law to the contour c E t l B t s cs () sc Between any two ajacent terminals a an b, we will efine voltage as b a Vba t E t l () To make the LHS of () fit (), we will move the minus sign in () so that E t l B t s cs () sc As we apply () to the physical circuit in the figure, we will choose to ignore the effects of the leas an the wires connecting the elements. (We can come back later an a these effects in, if we wish.) The integral on the LHS of () will be broken up into four subsections as El El El El B s () sc V V V V We will consier separately each of the five terms in ().
Whites, EE 8 Lecture 8 Page of 9 Source Voltage V This is the source voltage (or emf): E t l V s t (5) Resistor Voltage V This is the resistor voltage: E t l V t You consiere resistors supporting irect current previously in EE 8. For sinusoial steay state, we will assume here that the resistors are electrically small. That is, their imensions are much smaller than a wavelength ( c/ f ). Insie a conuctive material, by Ohm s law J t Et Therefore, J t V t E t l l. Assuming the frequency f is small enough so that J is nearly uniform over the cross section (which is not true at high f), then
Whites, EE 8 Lecture 8 Page of 9 where I t V t l I t R (6) A R l A This is the typical formula for compute the D resistance of a conuctor that has a uniform current ensity over its cross section. Why the minus sign in (6)? Because of the assume polarity: I(t) V apacitor Voltage V This is the capacitor voltage: E t l V t We consiere the capacitor in etail earlier in Lecture 5 in connection with isplacement current. Uner the quasistatic Qt V t then assumption
Whites, EE 8 Lecture 8 Page 5 of 9 Q t It V t Forming the antierivative of this expression gives t V t I t V In terms of V, we can write this result as t V t V t I t (7) We see here once again that the basic circuit operation of the capacitor arises because of isplacement current, as we iscusse earlier in Lecture 5. Inuctor Voltage V This is the inuctor voltage: onsier the contour c E t l V t
Whites, EE 8 Lecture 8 Page 6 of 9 I(t) c An apply Faraay s law to this contour El B t s cs sc We can separate the line integral into two parts El El B t s coil coil 0 The secon term on the LHS equals zero if there is no resistance in the wires of the inuctor. Otherwise, there woul be an R L term similar to V earlier. For the RHS of (8) B t s t Nm t (9) coil where (t) is the flux linkage through the surface forme by the coil of wire. Substituting (9) into (8) gives t E t l (0) (8)
Whites, EE 8 Lecture 8 Page 7 of 9 For magnetostatic fiels, you saw in EE 8 that LI () We will assume here that the frequency is small enough (i.e., is quasistatic ) that t It L LI L I t 0 () I t L Substituting () into (0) gives I t V L I t or V V L () We see that the basic operation of the inuctor arises irectly from Faraay s law! ollecting the Results Now, we ll pull all of this together. Substituting (), (7), (6), an (5) into () gives t I t Vs t I t R It L B s () s This result was erive irectly from Maxwell s equations!
Whites, EE 8 Lecture 8 Page 8 of 9 In (), the surface s is the open surface boune by c, minus the surface boune by the inuctor. The effect of the magnetic flux through the inuctor surface is accounte for in the inuctor term in (). Next, let s apply KVL to the lumpe element representation for this physical circuit: I(t) V R R V s (t) V L V L I t Vs t RI t It L (5) This result was erive irectly from circuit theory. t omparing () an (5) we see that if the RHS of () is negligible, then these two equations are equal! We have erive the terminal (VI) characteristics for the lumpe element circuit from Maxwell s equations. This is a BIG accomplishment.
Whites, EE 8 Lecture 8 Page 9 of 9 In effect, every electrical circuit is a little electromagnetic test be: Resistor: behavior governe by Ohm s law, apacitor: behavior governe by Maxwell s law, Inuctor: behavior governe by Faraay s law. Missing Term in KVL? What is the extra term B s in ()? It is the (negative) emf s contribution from the magnetic flux linking the surface s of the entire circuit (i.e., the entire surface s minus the inuctor): At low f, we can often safely ignore this term. At high f an overall physical circuit imensions approaching tens of centimeters or larger, this term may not be negligible.