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ECE 342 1. Networks & Systems

ECE 342 1. Networks & Systems. Jose E. Schutt-Aine Electrical & Computer Engineering University of Illinois jschutt@emlab.uiuc.edu. What is Capacitance?. 3. 1. 2. Voltage = 6V Charge=Q No Current. Voltage build up Charge build up Transient Current. Voltage=0 No Charge

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ECE 342 1. Networks & Systems

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  1. ECE 342 1. Networks & Systems Jose E. Schutt-Aine Electrical & Computer Engineering University of Illinois jschutt@emlab.uiuc.edu

  2. What is Capacitance? 3 1 2 Voltage = 6V Charge=Q No Current Voltage build up Charge build up Transient Current Voltage=0 No Charge No Current

  3. What is Capacitance? 4 6 5 Voltage=0 No Charge No Current Voltage decaying Charge decaying Transient Current Voltage=6V Charge=Q No Current

  4. Capacitance Relation: Q = Cv Q: charge stored by capacitor C: capacitance v: voltage across capacitor i: current into capacitor

  5. What is Inductance? • Current in wire produces magnetic field • Flux is magnetic field integrated over area

  6. Inductance

  7. Inductance Relation:Y = Li Y: flux stored by inductor L: inductance i: current through inductor v: voltage across inductor

  8. Low-Pass Circuit Need to solve for vo(t)

  9. Low-Pass Circuit Need to solve for vo(t)

  10. Reactive Circuit Need to solve for vx(t)

  11. Laplace Transforms The Laplace transform F(s) of a function f(t) is defined as: To a mathematician, this is very meaningful; however circuit engineers seldom use that integral The Laplace transform provides a conversion from the time domain into a new domain, the s domain

  12. Laplace Transforms The Laplace transform of the derivative of a function f(t) is given by Differentiation in the time domain becomes a multiplication in the s domain. That is why Laplace transforms are useful to circuit engineers s = jw

  13. Low-Pass Circuit Time domain s domain Time domain s domain

  14. Pole exists at Low-Pass Circuit - Solution Assume that Vi(s)=Vd/s Inversion

  15. Reactive Circuit

  16. Reactive Circuit - Solution Exercise: Use inversion to solve for vx(t)

  17. Table of Laplace Transforms

  18. Capacitance – s Domain Assume time-harmonic behavior of voltage and current variables jw = s The capacitor is a reactive element

  19. Inductance – s Domain Assume time-harmonic behavior of voltage and current variables jw = s The inductor is a reactive element

  20. Thevenin Equivalent • Principle • Any linear two-terminal network consisting of current or voltage sources and impedances can be replaced by an equivalent circuit containing a single voltage source in series with a single impedance. • Application • To find the Thevenin equivalent voltage at a pair of terminals, the load is first removed leaving an open circuit. The open circuit voltage across this terminal pair is the Thevenin equivalent voltage. • The equivalent resistance is found by replacing each independent voltage source with a short circuit (zeroing the voltage source), replacing each independent current source with an open circuit (zeroing the current source) and calculating the resistance between the terminals of interest. Dependent sources are not replaced and can have an effect on the value of the equivalent resistance.

  21. Norton Equivalent • Principle • Any linear two-terminal network consisting of current or voltage sources and impedances can be replaced by an equivalent circuit containing a single current source in parallel with a single impedance. • Application • To find the Norton equivalent current at a pair of terminals, the load is first removed and replaced with a short circuit. The short-circuit current through that branch is the Norton equivalent current. • The equivalent resistance is found by replacing each independent voltage source with a short circuit (zeroing the voltage source), replacing each independent current source with an open circuit (zeroing the current source) and calculating the resistance between the terminals of interest. Dependent sources are not replaced and can have an effect on the value of the equivalent resistance.

  22. Thevenin Equivalent - Example Circuit for calculating impedance Calculating Thevenin voltage KCL law at node A’ gives 1+Iy = Ix Also KVL gives 10=2Iy+3Ix Combining these equations gives Ix = 2.4 mA From which we calculate Vx = Vth=3(2.4)=7.2 V

  23. Thevenin Example Find voltage across R4 From which

  24. Thevenin Equivalent Circuit for calculating impedance (current source is replaced with open circuit)

  25. Circuit with Thevenin Equivalent ZTH VTH

  26. Thevenin Example Need to find the current i1(t) in resistor R3

  27. Thevenin Example Thevenin voltage calculation

  28. Thevenin Example Calculating the impedance

  29. Thevenin Example

  30. Thevenin Example Expanding using partial fraction With K1 = 3.33,K2= -5, K3= 1.67 The time-domain current i1(t) is:

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