module04_general_formulation

Module 4: General Formulation of Electric Circuit Theory 4-2 4. General Formulation of Electric Circuit Theory All electromagnetic phenomena are described at a fundamental level by Maxwell's equations and the associated auxiliary relationships. For certain classes of problems, such as representing the behavior of electric circuits driven at low frequency, application of these relationships may be cumbersome. As a result, approximate techniques for the analysis of low-frequency circuits have been developed. These specializations are used to describe the "ideal" behavior of common circuit elements such as wires, resistors, capacitors, and inductors. However, when devices are operated in a regime, or an environment, which lies outside the range of validity of such approximations, a more fundamental description of electrical systems is required. When viewed in this more general context, what may initially appear to be unexpected behavior of a circuit element often reveals itself to be normal operation under a more complex set of rules. An understanding of this lies at the core of electromagnetically compatible designs. In this section, electric circuit theory will be presented in a general form, and the relationship between circuit theory and electromagnetic principles will be examined. The approximations associated with circuit theory and a discussion of the range of validity of these approximations will be included. It will be seen that effects due to radiation and induction are always present in systems immersed in time-varying fields, although under certain conditions these effects may be ignored. 4.1 Limitations of Kirchoff's laws The behavior of electric circuits is typically described through Kirchoff's voltage and current laws. Kirchoff's voltage law states that the sum of the voltages around any closed circuit path is zero M n V n 0 and Kirchoff's current law states that the sum of the currents flowing out of a circuit node is zero M N n � 1 I n 0 It is through application of these relationships that most descriptions of electric circuits proceed. However, both of these relationships are only valid under certain conditions: - The structures under consideration must be electrically small . At 60 Hz, the wavelength of a wave propagating through free space is 5 million meters, while at 300 MHz a wavelength in free space is 1m long. Radiation and induction effects arise when the current amplitude and phase vary at points along the conductor. - No variation exists along uninterrupted conductors. 4-3 - No delay time exists between sources and the rest of the circuit. Also all conductors are equipotential surfaces. - The loss of energy from the circuit, other than dissipation, is neglected. In reality, losses due to radiation may become significant at high frequency. In chapter 2, a time-dependent generalization of KVL was presented v(t) R source � R i(t) � L di(t) dt . Although this expression is valid for time-changing fields, it is assumed that the circuit elements are lumped, i.e, the resistance and inductance are concentrated in relatively small regions. This assumption begins to break down at frequencies where the circuit elements are a significant fraction of a wavelength long. In this regime, circuits must be described in terms of distributed parameters. Every part of the circuit has a certain impedance per unit length associated with it. This impedance may be both real (resistive) and imaginary (reactive). Also, interactions which occur in one part of the circuit may affect interactions which occur everywhere else in the circuit. In addition, the presence of other external circuits will affect interactions within a different circuit. Thus at high frequency, a circuit must be viewed as a single entity, not a collection of individual components, and multiple circuits must be viewed as composing a single, coupled system. 4.2 General formulation for a single RLC circuit The general formulation of electric circuit theory will begin with an analysis of a single circuit constructed with a conducting wire of radius a that may include a coil (inductor), a capacitor, and a resistor. It will be assumed that the radius of the wire is much smaller than a wavelength at the frequency of operation a /� o << 1 or � o a << 1 where � o 2Œ /� o . A current flows around the circuit. The tangential component of electric current 4-4 J s sˆ # 3J sˆ # (1 3E ) 1E s is driven by the tangential component of electric field along the wire, which is produced byE s charge and current in the circuit. Currents in the circuit are supported by a generator. The generator is a source region where a non-conservative (meaning that the potential rises in the direction of current flow) impressed electric field maintains a charge separation. This impressed field is due to an electrochemical3E e or other type of force, and is assumed to be independent of current and charge in the circuit. The charge separation supports an electric field (Coulomb field) within and external to the source3E region which gives rise to the current flowing in the circuit. Figure 1. Generalized electric circuit. In the regions external to the source, the impressed field does not exist, however within the3E e source both fields exist, therefore by Ohm's law the current density present at any point in the circuit is given by 3J 1 ( 3E � 3E e ) where varies from point to point. From this it is apparent that in order to drive the current1 density against the electric field which opposes the charge separation in the source region,3J 3E the impressed electric field must be such that 3E e > 3E . 4-5 Positions along the circuit are measured using a displacement variable s, having it's origin at the center of the generator. The unit vector parallel to the wire axis at any point along the circuit is . The tangential component of electric field along the conductor is thereforesˆ E s sˆ # 3E and the associated axial component of current density is J s sˆ # 3J sˆ # (1 3E) 1E s . The total current flowing through the conductor cross-section is then I s P c.s. J s ds . boundary conditions at the surface of the wire circuit& According to the boundary condition ˆt # ( 3E2 3E1) 0 the tangential component of electric field is continuous across an interface between materials. Application of this boundary condition at the surface of the wire conductor leads to E s (r a � ) E s (r a � ) , where E insides (s) Es(r a � ) is the field just inside the conductor at position s along the circuit, and E outsides (s) Es(r a � ) represents the field maintained at position s just outside the surface of the conductor by the current and charge in the circuit. Therefore the fundamental boundary condition employed in an 4-6 electromagnetic description of circuit theory is E insides (r a � ,s) E outsides (r a � ,s) . determination of & E insides (s) In order to apply the boundary condition above, the tangential electric field that exits at points just inside the surface of the circuit must be determined. This is not an easy task, because the impedance may differ in the various regions of the circuit. At any point along the conducting wire, including coils and resistors, the electric field is in general E insides (s) z i Is (s) where z i � E insides (s) I s (s) is the internal impedance per unit length of the region. - source region In the source region, both the impressed and induced electric fields exist, therefore the current density is J s 1 e (E s � E es ) where is the conductivity of the material in the source region, is the tangential1e E s component of electric field in the source region maintained by charge and current in the circuit, and is the tangential component of impressed electric field. From this, it isE es apparent that 4-7 E s J s 1 e E es J s S e 1 e S e E eS I s 1 e S e E es where is the cross-sectional area of the source region. In an ideal generator, the materialS e in the source region is perfectly conducting ( ), and has zero internal impedance. Thus 1e � � E s � E es in a good source generator. In general, in the source region E s (s) z ie Is (s) E es (s) where z i e 1 1 e S e is the internal impedance per unit length of the source region. - capacitor The tangential component of electric field at the edge of the capacitor and the currentE s flowing to the capacitor lie in the same direction. Therefore, within the capacitor E s (s) z ic Is (s) where is the internal impedance per unit thickness of the dielectric material contained inz ic the capacitor. The total potential difference across the capacitor is the line integral of the normal component of electric field which exists between the capacitor plates VAB VB Va P B A E s ds P A B E s ds P A B z i c Is(s)ds or VAB Is P A B z i c ds if it is assumed that a constant current Is flows to the capacitor. 4-8 Figure 2. Capacitor. The time-harmonic continuity equation states /# 3J j&! . Volume integration of both sides of this expression, and application of the divergence theorem yields Q S (nˆ # 3J )ds j&P V !dv resulting in I s j&Q where Q is the total charge contained on the positive capacitor plate. The total potential difference between the capacitor plates is then VAB j&Q P B A z i c ds 4-9 Figure 3. Single capacitor plate. but by the definition of capacitance C Q VAB therefore Q C j&Q P B A z i c ds . From this comes the expected expression for the impedance of a capacitor P A B z i c ds 1 j&C jXc . - arbitrary point along the surface of the circuit By combining the results for the three cases above, a general expression for the tangential component of electric field residing just inside the surface of the conductor at any point along the circuit is determined 4-10 E insides (s) z i(s) Is(s) E es (s) where is the internal impedance per unit length which is different for the variousz i(s) components of the circuit, and is the impressed electric field which is zero everywhereE es (s) outside of the source region. determination of & E outsides (s) In Chapter 2 it was shown that an electric field may be represented in terms of scalar and vector potentials. Thus at any point in space outside the electric circuit the electric field is 3E /- j& 3A where is the scalar potential maintained at the surface of the circuit by the charge present- in the circuit, and is the vector potential maintained at the surface of the circuit by the3A current flowing in the circuit. The well known Lorentz condition states that /# 3A � j k 2 & - 0 where k 2 &2µ0 1 j 1 &0 . In the free space outside the circuit , , and 0 thusµ µ o 0 0 o 1 k 2 �2o &2µo0o . Applying this to the Lorentz condition yields - j& � 2 o /# 3A which, upon substitution into the expression for electric field outside the circuit gives 4-11 3E j& � 2 o / (/# 3A ) � �2o 3A . The component of electric field tangent to the surface of the circuit is then given by E outsides (s) sˆ # 3E sˆ #/- j& (sˆ# 3A) j& � 2 o sˆ # / (/# 3A ) � �2o 3A or E outsides (s) 0- 0s j&A s j& � 2 o 0 0s (/# 3A ) � �2o As . satisfaction of the fundamental boundary condition& The boundary condition at the surface of the circuit states that the tangential component of electric field must be continuous, or E insides (s) E outsides (s) therefore the basic equation for circuit theory is E es (s) � z i(s) Is(s) 0 0s -(s ) j&A s (s ) j& � 2 o 0 0s /# 3A(s) � �2o As(s) . open and closed circuit expressions& From the development above, the expression for the impressed electric field is E es (s) 0 0s -(s) � j&A s (s) � z i (s)I s (s) . 4-12 Figure 4. General circuit. Integrating along a path C on the inner surface of the circuit from a point s1 to a point s2, which represent the ends of an open circuit, gives P s2 s1 E eS (s)ds P s2 s1 0 0s - (s)ds � j&P s2 s1 A s (s)ds � P s2 s1 z i(s)I s (s)ds but, because the impressed electric field exists only in the source region P s2 s1 E es (s)ds P B A E es (s)ds V eo where is the driving voltage. Now it can be seen thatV eo P s2 s1 0 0s -(s)ds P s2 s1 d- -(s2 ) -(s1 ) and thus the equation for an open circuit can be expressed 4-13 V e0 -(S2 ) -(s1 ) � j&P s2 s1 A s (s)ds � P s2 s1 z i(s) I s (s)ds . If the circuit is closed, then s1=s2 and -(s2)--(s1)=0. In this case, the circuit equation becomes V e0 Q C z i(s) I s (s)ds � j&Q C A s (s)ds . 4.3 General equations for coupled circuits Now the concepts developed above are extended to the case of two coupled circuits, each containing a generator, a resistor, a coil (inductor), and a capacitor. Circuit 1 will be referred to as the primary circuit, and circuit 2 will be referred to as the secondary circuit. This case is represented by a pair of coupled general circuit equations V e10 Q C1 z i 1 (s1) I1s(s1)ds1 � j&Q C1 3A11(s1) � 3A12(s1) # 3ds1 V e20 Q C2 z i 2 (s2 ) I2s(s2 )ds2 � j&Q C2 3A21(s2 ) � 3A22(s2 ) # 3ds2 . Here and are the magnetic vector potentials at the surface of the primary circuit3A11 3A12 maintained by the currents and in the primary and secondary circuits, given by I1s I2s 3A11(s1) µ o 4Œ C � 1 I1s(s � 1 ) e � j � oR11 R11(s1, s � 1 ) 3ds � 1 and 3A12(s1) µ o 4Œ C � 2 I2s(s � 2 ) e � j � oR12 R12(s1, s � 2 ) 3ds � 2 and and are the vector potentials at the surface of the secondary circuit maintained by3A21 3A22 the currents and in the primary and secondary circuits, given byI1s I2s 4-14 Figure 5. Generalized coupled circuits. 3A21(s2 ) µ o 4Œ C 1 I1s(s 1 ) e � j � oR21 R21(s2, s 1 ) 3ds 1 and 3A22(s2 ) µ o 4Œ C 2 I2s(s 2 ) e � j � oR22 R22(s2, s 2 ) 3ds 2 . Substituting these expressions for the various magnetic vector potentials into the general circuit equations above leads to V e10 C1 z i 1 (s1) I1s(s1)ds1 � j&µ o 4Œ C1 3ds1# C 1 I1s(s 1 ) e � j � oR11 R11(s1,s 1 ) 3ds 1 � j&µ o 4Œ C1 3ds1 # C 2 I2s (s 2 ) e � j � oR12 R12(s1,s 2 ) 3ds 2 4-15 V e20 C2 z i 2 (s2) I2s(s2)ds2 � j&µ o 4Œ C2 3ds2# C 1 I1s(s � 1 ) e � j � oR21 R21(s2,s � 1 ) 3ds � 1 � j&µ o 4Œ C2 3ds2 # C 2 I2s (s � 2 ) e � j � oR22 R22(s2,s � 2 ) 3ds � 2 . Note that and , lie along the inner periphery of the circuit while and lie along theC1 C2 C � 1 C � 2 centerline. When the circuit dimensions and , are specified, the equations above becomeV e10 V e 20 a pair of coupled simultaneous integral equations for the unknown currents and I1s(s1) I2s(s2 ) in the primary and secondary circuits. These equations are in general too complicated to be solved exactly. self and mutual impedances of electric circuits& Reference currents and are chosen at the locations of the generators in the primaryI10 I20 and secondary circuits, i.e., I10 I1s (s1 0) at the center of the primary circuit generator, and I20 I2s (s2 0) at the center of the secondary circuit generator. Now let the currents be represented by I1s (s1 ) I10 f1(s1 ) I2s (s2 ) I20 f2(s2 ) where , and , are complex distribution functions. The general circuitf1(0) f2(0) 1.0 f1 f2 equations formulated above may be expressed in terms of the reference currents as V e10 I10 Z11 � I20 Z12 V e20 I10 Z21 � I20 Z22 4-16 where = self-impedance of the primary circuit referenced to Z11 I10 = self-impedance of the secondary circuit referenced to Z22 I20 = mutual-impedance of the primary circuit referenced to Z12 I20 = mutual-impedance of the secondary circuit referenced to Z21 I10 and Z11 Z i 1 � Z e 1 Z22 Z i 2 � Z e 2 . Here is referred to as the internal self-impedance of the primary and secondary circuits. ThisZ i term depends primarily upon the internal impedance zi per unit length of the conductors present in the circuits, and includes effects due to capacitance and resistance. is referred to as theZ e external self-impedance of the primary and secondary circuits. This term depends entirely upon the interaction between currents in various parts of the circuit, and includes effects due to inductance. The various impedance terms are expressed as Z i1 Q C1 z i 1 (s1) f1(s1) ds1 Z i2 Q C2 z i 2 (s2) f2(s2) ds2 Z e1 j&µ o 4Œ C1 3ds1# C �1 f1(s � 1 ) e � j � oR11 R11(s1,s � 1 ) 3ds � 1 Z e2 j&µ o 4Œ C2 3ds2 # C �2 f2 (s � 2 ) e � j � oR22 R22(s2,s � 2 ) 3ds � 2 4-17 Z12 j&µ o 4Œ C1 3ds1 # C �2 f2 (s � 2 ) e � j � oR12 R12(s1,s � 2 ) 3ds � 2 Z21 j&µ o 4Œ C2 3ds2# C �1 f1(s � 1 ) e � j � oR21 R21(s2,s � 1 ) 3ds � 1 . It is noted that all of the circuit impedances depend in general on the current distribution functions and .f1 (s1) f2 (s2) driving point impedance, coupling coefficient, and induced voltage& Consider the case where only the primary circuit is driven by a generator excitation, i.e., V e20 0 In this case the general circuit equations become V e10 I10 Z11 � I20 Z12 0 I10 Z21 � I20 Z22 . This leads to a pair of equations I20 I10 Z21 Z22 V e10 I10 Z11 Z12 Z21 Z22 which can be solved to yield 4-18 I10 V e10 Z11 Z12 Z21 Z22 and I20 V e10 Z11 Z22 Z21 Z12 which are the reference currents in the generator regions. From these, it can be seen that the driving point impedance of the primary circuit is Z1 in V e10 I10 Z11 Z12 Z21 Z22 Z11 1 Z12Z21 Z11Z22 . For the case of a loosely coupled electric circuit having , and Z12 Z21 Z11 Z22 « 1 Z12 Z21 are both negligibly small, and therefore . For this case the expressions(Z1 ) in � Z11 V e10 I10 Z11 � I20 Z12 0 I10 Z21 � I20 Z22 . become V e10 � V i 12 I10 Z11 V i21 I20 Z22 where is the voltage induced in the primary circuit by the current in the secondaryV i12 I20 Z12 circuit, and is the voltage induced in the secondary circuit by the current in theV i21 I10 Z21 primary circuit. Because the circuits are considered to be loosely coupled V e10 » V i 12 4-19 and therefore the general circuit equations become V e10 x I10 Z11 V i21 I10Z21 I20 Z22 . near zone electric circuit& For cases in which a circuit is operated such that the circuit dimensions are much less than a wavelength, then the phase of the current does not change appreciably as it travels around the circuit. The associated EM fields are such that the circuit is confined to the near- or induction- zones. Most electric circuits used at power and low radio frequencies may be modeled this way. In this type of conventional circuit � o R11 « 1, �oR12 « 1, �oR21 « 1, �oR22 « 1 and therefore the following approximation may be made e � j � oR i j 1 � 2 o R 2 i j 2! � � 4 oR 4 i j 4! � ... j� o R i j 1 � 2 oR 2 i j 3! � � 4 o R 4 i j 5! � ... x 1 for i=1,2 and j=1,2. Because the circuit dimensions are small compared to a wavelength f1(s1) x f2(s2) x 1 and thus fi (si ) e � j � oR i j(si,s ffj ) Ri j(si,s fi j ) x 1 R i j(si,s fi j ) for i=1,2 and j=1,2. Substituting the approximations stated above into the expressions for the various circuit impedances leads to Z i1 Q C1 z i 1 (s1) ds1 Z i2 Q C2 z i 2 (s2) ds2 4-20 Z e1 jX e1 j&µ o 4Œ C1 3ds1# C fl1 3ds ffi 1 R11(s1,s ffi 1 ) Z e2 jX e2 j&µ o 4Œ C2 3ds2 # C fl2 3ds ffi 2 R22(s2,s ffi 2 ) Z12 jX12 j&µ o 4Œ C1 3ds1 # C fl2 3ds ffi 2 R12(s1,s ffi 2 ) Z21 jX21 j&µ o 4Œ C2 3ds2# C fl1 3ds ffi 1 R21(s2,s ffi 1 ) . Since the integrals above are frequency independent and functions only