Limitations

KCL and KVL both depend on the lumped element modelbeing applicable to the circuit in question. When the model is not applicable, the laws do not apply. KCL, in its usual form, is dependent on the assumption that current flows only in conductors, and that whenever current flows into one end of a conductor it immediately flows out the other end. This is not a safe assumption for high-frequency AC circuits, where the lumped element model is no longer applicable.[2]_{It is often possible to im-} prove the applicability of KCL by considering “parasitic capacitances” distributed along the conductors.[2]_Signif- icant violations of KCL can occur[3][4] _{even at 60Hz,} which is not a very high frequency.

In other words, KCL is valid only if the total electric charge,Q, remains constant in the region being consid-

ered. In practical cases this is always so when KCL is applied at a geometric point. When investigating a fi- nite region, however, it is possible that thecharge density within the region may change. Since charge is conserved, this can only come about by a flow of charge across the region boundary. This flow represents a net current, and KCL is violated.

KVL is based on the assumption that there is no fluctu- atingmagnetic fieldlinking the closed loop. This is not a safe assumption for high-frequency (short-wavelength) AC circuits.[2] _{In the presence of a changing magnetic} field the electric field is not aconservative vector field. Therefore the electric field cannot be the gradient of any potential. That is to say, theline integralof the electric field around the loop is not zero, directly contradicting KVL.

It is often possible to improve the applicability of KVL by considering “parasitic inductances” (including mutual in- ductances) distributed along the conductors.[2]_{These are} treated as imaginary circuit elements that produce a volt- age drop equal to the rate-of-change of the flux.

3.1.4 Example

Assume an electric network consisting of two voltage sources and three resistors.

According to the first law we have

R

R

R

i

i

i

ε

ε

s

s

The second law applied to the closed circuit s1gives

−R2i2+ ϵ1− R1i1= 0

The second law applied to the closed circuit s2gives

−R3i3− ϵ2− ϵ1+ R2i2= 0

Thus we get a linear system of equations in i1, i2, i3:      i1− i2− i3 = 0 −R2i2+ ϵ1− R1i1 = 0 −R3i3− ϵ2− ϵ1+ R2i2 = 0 Assuming R1= 100, R2= 200, R3= 300(ohms) ; ϵ1= 3, ϵ2= 4(volts) the solution is      i1= ₁₁₀₀1 i2= ₂₇₅4 i3=−₂₂₀3

i3has a negative sign, which means that the direction of i3 is opposite to the assumed direction (the direction defined in the picture).

3.1.5

See also

• Faraday’s law of induction • Kirchhoff’s laws (disambiguation) • Lumped matter discipline

3.1.6 References

[1] Oldham, Kalil T. Swain (2008). The doctrine of descrip-

tion: Gustav Kirchhoff, classical physics, and the “purpose of all science” in 19th-century Germany (Ph. D.). Univer-

sity of California, Berkeley. p. 52. Docket 3331743. [2] Ralph Morrison, Grounding and Shielding Techniques

in Instrumentation Wiley-Interscience (1986) ISBN 0471838055

[3] “High Voltage Cable Inspection”(video). |first1= missing |last1= in Authors list (help)

[4] Non-contact voltage detector

• Paul, Clayton R. (2001). Fundamentals of Electric Circuit Analysis. John Wiley & Sons. ISBN 0-471- 37195-5.

• Serway, Raymond A.; Jewett, John W. (2004). Physics for Scientists and Engineers (6th ed.). Brooks/Cole.ISBN 0-534-40842-7.

• Tipler, Paul (2004). Physics for Scientists and Engi- neers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.). W. H. Freeman. ISBN 0-7167-0810-8.

• Graham, Howard Johnson, Martin (2002). High- speed signal propagation : advanced black magic (10. printing. ed.). Upper Saddle River, NJ: Pren- tice Hall PTR.ISBN 0-13-084408-X.

3.1.7 External links

• MIT video lectureon the KVL and KCL methods • Faraday’s Law - Most Physics College Books have it

WRONG!by Walter H. G. Lewin, Ph.D., MIT

3.2 Norton’s theorem

This article is about the theorem in electrical circuits. For Norton’s theorem for queueing networks, seeflow- equivalent server method.

Known in Europe as the Mayer–Norton theorem, Nor- ton’s theorem holds, to illustrate in DC circuit theory terms, that (see image):

• Any linear electrical network with voltage and current sources and only resistances can be replaced at terminals A-B by an equivalent current source INO inparallelconnection with an equivalent resistance RNO.

• This equivalent current INO is the current obtained at terminals A-B of the network with terminals A-Bshort circuited.

3.2. NORTON’S THEOREM 87

Edward Lawry Norton

• This equivalent resistance RNO is the re- sistance obtained at terminals A-B of the network with all its voltage sourcesshort circuitedand all its current sourcesopen circuited.

For AC systems the theorem can be applied toreactive impedancesas well as resistances.

The Norton equivalent circuit is used to represent any network of linear sources and impedances at a given frequency.

Anyblack boxcontaining resistances only and voltage and cur- rent sources can be replaced by anequivalent circuitconsisting of an equivalent current source in parallel connection with an equivalent resistance.

Norton’s theorem and its dual,Thévenin’s theorem, are widely used for circuit analysis simplification and to study circuit’s initial-condition and steady-state response.

Norton’s theorem was independently derived in 1926 bySiemens & HalskeresearcherHans Ferdinand Mayer (1895–1980) andBell LabsengineerEdward Lawry Nor- ton(1898–1983).[1][2][3][4][5]

To find the equivalent,

1. Find the Norton current INₒ. Calculate the output current, IAB, with ashort circuitas theload(mean- ing 0 resistance between A and B). This is INₒ. 2. Find the Norton resistance RNₒ. When there are no

dependent sources(all current and voltage sources are independent), there are two methods of deter- mining the Norton impedance RNₒ.

• Calculate the output voltage, VAB, when in open circuit condition (i.e., no load resistor – meaning infinite load resis- tance). RNₒ equals this VAB divided by INₒ.

or

• Replace independent voltage sources with short circuits and independent current sources with open circuits. The to- tal resistance across the output port is the Norton impedance RNₒ.

This is equivalent to calculating the Thevenin resistance. However, when there are dependent sources, the more general method must be used. This method is not shown below in the diagrams.

• Connect a constant current source at the output terminals of the circuit with a value of 1 Ampere and calculate the volt- age at its terminals. This volt- age divided by the 1 A current is the Norton impedance RNₒ. This method must be used if the circuit contains dependent sources, but it can be used in all cases even when there are no dependent sources.

3.2.1 Example of a Norton equivalent cir-