SLOWLY VARYING CURRENTS

13.1.Transient and Steady-State Behavior

            It is convenient to discuss the behavior of circuits in two stages, according to whether the periodic or nonperiodic behavior is important. The periodic behavior is referred to as the steady-state behavior, while the nonperiodic behavior is known as the transient behavior. Both aspects are governed by the same basic integro-differential equations; however, the elementary techniques used in solving them are radically different in the two cases.

13.2.Kirchoff’s Laws

Kirchoff’s law I. The algebraic sum of the instantaneous currents flowing toward a junction is zero.

Kirchoff’s law II. The algebraic sum of the instantaneous applied voltages in a closed loop equals the algebraic sum of the instantaneous counter voltages in the loop.

13.3.Elementary Transient Behavior

            I(t) = De^-Rt/2L sin_nt

Where D is single real constant still to be evaluated. This evaluation is accomplished by nothing that at t = 0, Q and I are both zero and, hence, that

            │_t=o

Using this initial condition gives

            D =

The solution is now complete. The current oscillates at the natural frequency

           

13.4.Steady-State Behavior of A Simple Series Circuit

            Z = R + i

The impedance Z of the circuit consists of two parts: the real part or resistance (R), and the imaginary part or reactance (X). The reactance is further divided into the inductive reactance X_L = L and the capacitive reactance X_C = -1/C. The fact that the impedance is complex means that the current is not in phase with the applied voltage.

It is sometimes convenient to write the impedance in polar form:

            Z = │Z│

with

│Z│= [R² + ( - 1/)²]^1/2

And

           

using this form for the impedance, we may write the complex current as

           

And the physical current as

            I_p(t) =  cos

13.5.Series and Parallel Conection of Impedances

If two impedances are connected in series, then the same current flows through each of them. The voltages across the two impedances are V₁ = Z₁I and V₂ = Z₂I. The voltage across the combination is V₁ + V₂ = (Z₁ + Z₂)I. It is clear, then, that the connection of impedances in series adds impedances, that is,

            Z = Z₁ + Z₂ + Z₃ +                           (series connection)

                                              (parallel connection)

13.6.Power and Power Factors

If V(t) and I(t) are the complex voltage and current as shown, then the instantaneous power is

            P(t) = Re I(t)Re V(t)

The average power is a more important quantity, with the average being taken over either one full period or a very long time (many periods). If the phases are chosen so that V₀ is real and, as usual, Z = │Z│,

            P = Re I(t)Re V(t) =

13.7.Resonance

            A quantitative measure of the sharpness of the curve can be derived as follows. Let us define the “widht” of the resonance curve as the frequency interval between the “half-power frequencies,” which are the two frequencies where the power dissipated is one-half of the power dissipated at the peak frequecy ₀. Thus, we seek the values of  that satisfy

            () =  (₀)

           

Thus,

             = 2

or

           

13.8.Mutual Inductances in ac Circuits

            The approximations made seem rather extreme, but practical transformers exist that approximate ideal transformers over relatively wide frequency ranges. For such devices,

            ,                     V_L = aV₀

and

           

The last of these relationships shows that the transformer acts also as an impedance tranformer, with transformation ratio a^-2. It is left as an axercise to show that for very close coupling of the two windings a = N₂/N₁-that is, the turns ratio.

13.5.Mesh and Nodal Equations

            More complex ac circuits may be approached in two ways: one based on Kirchoff’s voltage law and known as mesh analysis and one based on Kirchoff’s current law and known as nodal analysis. Each method has its advantiages and disadvantiages. Since choosing the expedient method can greatly simplify some problems, both methods will be illustrated in this section.

13.10.Driving Point Impedance and Transfer Functions

The four transfer functions that relate the input variables to the output variables are defined as follows.

Transfer impedance

            Z_T = 

Tranfer admittance,

            Y_T =

Current gain,

            G_T =

Voltage gain,

            A_T =

It should be noted that the driving point impedance and the transfer functions depend on the source and load impedances as well as on the internal structure of the network. In many cases, however, adequate approximations can be made by using a zero impedance voltage source or an infinite impedance current source and an infinite impedance or zero impedance load.

13.11.Solving Network Equations with The Computer

            The third and final approach to solving network equations is to use commercial software. Most commonly used software is some version of SPICE (Simulation Program with Integrated Circuit Emphasis). As the name of the program indicates, SPICE is not limited to passive networks, but it deals with them very effectively.

Posted in: Foundations of Electromagnetic Theory