Coupling Element and Coupled circuits Coupled inductor Ideal transformer Controlled sources

Coupling Element and Coupled circuits Coupled elements have more that one branch and branch voltages or branch currents depend on other branches. The characteristics and properties of coupling element will be considered. Coupled inductor Two coils in a close proximity is shown in Fig.1 Fig.1 Coupled coil and reference directions

Coupled inductor Magnetic flux is produced by each coil by the functions Where and are nonlinear function of and By Faraday’s law

Coupled inductor Linear time-invariant coupled inductor If the flux is a linear function of currents and In sinusoid steady-state Note that the signs of and are positive but the sign for M can be

Coupled inductor Dots are often used in the circuit to indicate the sign of M Fig. 2 Positive value of M

Coupled inductor Coefficient of coupling The coupling coefficient is If the coils are distance away k is very small and close to zero and equal to 1 for a very tight coupling such for a transformer.

Coupled inductor Multi-winding Inductors and inductance Matrix For more windings the flux in each coil are are self inductances and are mutual inductances In matrix form

Coupled inductor Fig 3 Three-winding inductor

Coupled inductor Induced voltage The induced voltage in term current vector and the inductance matrix is Example 1 Fig. 4 shows 3 coils wound on a common core. The reference direction of current and voltage are as shown in the figure. Since and has the same direction but are not therefore is positive while and are negative. Fig. 4

Coupled inductor It is useful to define a reciprocal inductance matrix which makes where Thus the currents are

Coupled inductor In sinusoid steady-state Series and parallel connections of coupled inductors Equivalent inductance of series and parallel connections of coupled inductors can be determined as shown in the example 2.

Coupled inductor Example 2 Fig. 5 shows two coupled inductors connected in series. Determine the Equivalent inductance between the input terminals. Fig. 5 H

Coupled inductor Example 3 Fig. 6 shows two coupled inductors connected in series. Determine the Equivalent inductance between the input terminals. Fig. 6 H Note for series inductors

Coupled inductor Example 4 Two coupled inductors are connected in parallel in Fig 6. Determine the Equivalent inductance. Fig 6

Coupled inductor The currents are KVL By integration of voltage Therefore H Notefor parallel inductors

Ideal transformer Ideal transformer is very useful for circuit calculation. Ideal transformer Is a coupled inductor with the properties  dissipate no energy  No leakage flux and the coupling coefficient is unity  Infinite self inductances Two-winding ideal transformer Fig. 7

Ideal transformer Figure 7 shows an ideal two-winding transformer. Coils are wound on ideal Magnetic core to produce flux. Voltages is Induced on each winding. If is the flux of a one-turn coil then Since and we have In terms of magnetomotive force (mmf) and magnetic reluctance

Ideal transformer If the permeability is infinite becomes zero then and From (1) and (2) The voltage does not depend on or but it depends only on

Ideal transformer For multiple windings (equal volt/ turn) Fig. 8

Ideal transformer Impedance transformation

In sinusoid stead state Fig. 9

Controlled sources Controlled sources are used in electronic device modeling. There four kinds of controlled source.  Current controlled current source  Voltage controlled current source  Voltage controlled voltage source  Current controlled voltage source Fig. 10

Controlled sources Current controlled current source : Current ratio Voltage controlled current source :Transconductance Voltage controlled voltage source :Voltage ratio Current controlled voltage source :Transresistance

Controlled sources Example1 Determine the output voltage from the circuit of Fig.11 Fig.11 Mesh 1 Mesh 2

Controlled sources Example 2 Determine the node voltage from the circuit of Fig.12 Fig.12 KCL

Controlled sources Diff. (3) from (1) then

Controlled sources The initial conditions From (3) From (5) and (6) andcan be solved

Controlled sources Other properties The instantaneous power entering the two port is Since either or is zero thus If is connected at port 2 Therefore Power entering a two port is always negative

Controlled sources Example 3 Consider the circuit of Fig. 13 in sinusoid steady-state. Find the input impedance of the circuit. Fig. 13

Controlled sources Note if the input impedance can be negative and this two port Network becomes a negative impedance converter.