• Course Info

    EE 202 - Circuit Theory II

    Course Content

    This is the 2nd and final course in the chain of Circuit Theory courses. The course is accompanied with a separate laboratory course (EE214).

    The course starts with the discussion of coupled inductors. The dot-convention, power/energy relations are introduced and the passivity condition is given.

    In the next part of the course, the time-domain description of N’th order circuits is studied. The main goal is the introduction of the state equation formalism for this purpose. (Other analysis methods such as node/mesh and modified node analysis methods are also given.) The solution of N’th order constant coefficient differential equations is studied and zero-input and zero-state solutions are developed. Particular solutions for the exponential inputs are studied. The Laplace transformation based solution are studied.

    The next topic is the AC analysis. After the introduction of phasors, the phasor domain analysis of LTI dynamic circuits is given. The average power, reactive power and complex power definitions are given. Single-phase power factor compensation problems are studied. AC analysis topic is continued with the analysis of balanced 3-phase systems. 3-phase power calculations are emphasized.

    In the last part of the course, Laplace domain (s-domain) circuit analysis is studied. The network functions are introduced. The zero-state responses are studied. The frequency response for 1st and 2nd order circuits are studied. The 2nd order resonance circuit is examined. The approximate magnitude and  phase response plots (Bode plots) are studied.

    This course builds a foundation to many courses in system theory, circuit design and power systems.

    YouTube Playlist for all EE 202 Lectures:

    Teacher

  • Lecture 1

    • 00:00 Coupled/Mutual Inductors
      04:40 Terminal Equations
      10:45 Pairwise coupling and L matrix symmetry
      14:00 Dot Convention
      18:20 Example 1 (page1) - dots aligned
      23:18 Example 2 (page2) - dots at opposite sides
      24:38 Example 3 (page2) - inductors separated
      27:15 Example 4 (page2) - equivalent inductance calc.
      36:25 Example 5 (page2) - equivalent T network
      41:00 Example 6 (page3) - more than 2 inductors
       
       

  • Lecture 2

    • 00:00 Example 7 (page3) - Mesh equations with mutual inductors
      08:55 Power & Energy relations of mutual inductors
      12:22 Total Power
      15:00 Energy Stored at time t
      22:20 How to check Positive Semi-Definiteness (PSD)
      25:00 Terminal Equations in Integral Form
      35:00 Initial Condition Models
      42:10 Exercise (page5) - verify IC models (take-home!)
      44:25 Example (page5) - reducing the problem to an RL circuit
      47:10 Example (page5) - Step 1 - equivalent L calc.
      52:48 Example (page5) - Step 2 - solving RL circuit
       
       

  • Lecture 3

    • 00:00 State Equations
      05:35 Example (page7) - Series RLC
      14:33 What is "State" - Definition
      21:55 Graph Theory for obtaining the state eqns. systematically
      22:43 Tree - definition
      24:25 Example 1 (page8) - tree construction
      29:36 Cotree - definition
      30:28 Example 2 (Page8) - tree and its cotree
      33:10 Cutset - definition
      37:25 Fundamental Cutset - definition
      42:05 Fundamental Loop - definition
      45:00 Example (page9) - tree and its fun. cutsets, fun. loops
      52:10 Procedure to obtain State Equations
       
       

  • Lecture 4

    • 00:00 Example (page10) - Obtaining state eqn. representation
      03:20 Example (page10) - Step 1 - Proper tree
      12:38 Example (page10) - Step 2 - state variables
      16:30 Example (page10) - Step 3 & 4 - Fun.Cutset/Loop in terms of state var.'s
      26:55 Example (page12) - State eqn. rep. (transformer)
      45:20 Example (page12) - Remark - Order of a circuit/system
      46:50 Example (page12) - Remark Explanation - Proper tree and minimal representation
       
       

  • Lecture 5

    • 00:00 Modeling Circuits with LTV/Nonlinear Elements
      01:00 Example (page14)
      06:05 -Terminal equations in flux and charge
      08:42 -Circuit graph
      10:40 -State variables
      12:55 -State eqn.: ɸ1dot
      16:30 -State eqn.: ɸ2dot
      19:45 -State eqn.: q_cdot
      21:12 -State eqn. system in matrix form
      26:35 Example (Nonlinear elements)(page15)
      29:44 -State variables
      31:45 -State eqn.: q1dot
      34:22 -State eqn.: v2dot
      35:45 -State eqn.: iLdot
      36:50 -State eqn. system in matrix form
       
       

  • Lecture 6

    • 00:00 Solution to Homogeneous State Equation
      10:14 Fund. Matrix: M(t)=e^(At)
      16:15 Properties of e^(At)
      21:12 Computing e^At by Laplace Transform
      27:15 Example (Lossless LC Circuit)
      34:05 -Alternative solution by using State Transition Matrix
      42:15 General solution to x'=Ax+Bw
      49:15 Example (p.19) - Impulse resp. calc.
       
       

  • Lecture 7

    • 00:00 Alternative Formulations of LTI Circuits
      00:45 Node Analysis
      03:56 Example (p.20)
      08:00 -Expressing i1, i2, ic in terms of node voltages
      17:00 -Node Analysis eqn. set in matrix form
      20:17 The General Form Yn(D)e=ins
      22:34 Mesh Analysis
      23:14 Example (p.21)
      28:55 -Expressing v1, v2 and vc in terms of mesh currents
      30:44 -Mesh Analysis eqn. set in matrix form
      32:50 The General Form Zm(D)im=Vms
      35:05 Modified Node Analysis
      36:40 Example (p.22)
      44:00 -Node Analysis
      46:50 -Modified Node Analysis
       
       

  • Lecture 8

    • 00:00 LTI Circuit Analysis Example by State Eqn.
      00:25 Example (p.23)
      03:05 -part a) State Eqn. set
      15:45 -part b) Natural Freq.
      20:08 -part c) Particular soln. to 2exp(-3t) input
       
       

  • Lecture 9

    • 00:00 Natural Response
      01:30 Example (page25)
      04:24 -Soln. part a (node equation formulation)
      10:10 -Soln. part b (nat. freq)
      20:50 -Soln. part b (mode with s1=-1/3)
      24:45 -Soln. part b (mode with s2=-1)
      26:03 -Soln. part b (arbitrary I.C.)
      27:54 -Soln. part b (arbitrary I.C., c1, c2=?)
      32:17 -Soln. part b (final soln. for node voltage vector)
      32:52 -Soln. part b (deriving branch voltages from node voltage vector)
      35:25 Notes
      35:42 - Note #1: All modes may not be visible at all branches (observability)
      37:00 - Note #2: Exciting only a single mode of the circuit (mode excitation)
       
       

  • Lecture 10

    • 00:00 Determining the natural frequencies
      00:35 -Node formulation
      04:28 -Mesh formulation
      06:00 -State formulation
      10:20 Remark: Nat. freq. are independent of formulation
      11:55 Example (p.28) (Two caps. and R - all in series)
      12:55 - Soln. : Mesh Eqn. form. (Missing mode!)
      18:45 - Soln. : State Eqn. form. (All modes recovered!)
      24:40 Remark: Node/Mesh Eqn. may miss modes with s=0!
       
       

  • Lecture 11

    • 00:00 Mode Excitation
      04:15 Forms of natural response given nat. freq.
      07:35 Some Mode Excitation Facts (Natural response, no ext. input)
      07:40 Fact 1: All branch voltages/currents are in form K exp(\lambda t)
      10:55 Fact 2: Eigenvectors of A excite the modes of the system
      15:10 Example (p.29)
      19:00 -Soln. a (State Eqn.)
      22:34 -Soln. a (char. poly, nat. freq.)
      25:45 -Soln. a (general soln.)
      30:55 -Soln. b (Exciting only non-oscillatory modes)
      32:10 -Soln. b (Finding the eig. vector for s = -1)
      39:17 -Soln. b (Expressing IC's to excite mode /w s=-1)
       
       

  • Lecture 12

    • 00:00 Particular solutions for complex exponential inputs
      01:50 Circuit to illustrate the topic
      05:00 Classic approach via diff. eqn.
      07:15 New approach (intro. to phasors)
      12:45 Cap. terminal eqn. for input = alpha exp(s_0t)
      14:41 Connection between cap. volt/current for input = alpha exp(s_0t)
      19:01 Replacing dynamic comp. with generalized R-like comp.
      19:27 Going back to circuit for topic illustration
      22:25 KVL/KCL implications
      29:10 Particular soln's for different inputs
      34:12 Remark: This method is applicable if s_0 is not an nat. freq.
      38:56 Example 2 (p.32) - Case of complex valued s_0
       
       

  • Lecture 13

    • 00:00 Laplace Transform
      05:05 Integral form of Laplace Transform
      10:35 Example (p. 33) (Unit step function)
      14:38 Example (p. 34) (e^-at)
      17:20 Example (p. 34) (Unit impulse function)
      20:50 Linearity Property
      22:05 Example (p. 34)( L {sin \omega t} )
      25:56 Integration Property
      27:05 Proof of Integration Property
      32:40 Example (p. 35) (Unit Ramp)
      36:05 Complex Differentiation
      37:53 Differentiation in time domain
      42:50 Example (p. 36) ( L{cos \omega t} )
      47:09 s-Domain Translation Property
      47:58 Example (p. 37) ( L{ e^(-at) cos \omega t } )
      49:02 Time Domain Translation Property
      50:00 Example (p. 37) (translated unit ramp)
      51:45 Scale change
      53:17 Initial and Final Value Theorems
       
       

  • Lecture 14

    • 00:00 Pole-Zero Diagram
      07:45 Example (p. 38) (poles-zeros and their multiplicity)
      13:38 - Drawing pole-zero diagram
      16:22 Inverse Laplace Transform by Partial Frac. Expansion
      18:36 Proper rational function (definition)
      22:57 Residue (definition)
      19:20 Special Case: Simple Poles
      26:15 Example (p. 39) L^{-1} { (2s+6)/(s^3+3s^2+s) }
      34:55 Example (p. 40) -case of complex-valued poles -
       
       

  • Lecture 15

    • 00:00 Summary
      01:56 Inverse Laplace Transform by Partial Fraction Expansion
      05:42 Residue calc. for simple poles
      06:35 Example (p. 41) (revisited example from Lec. 14)
      15:30 -A 2nd way for the case of complex valued poles
      25:00 General Case (Simple/repeated poles)
      31:40 How to find k_i,m? (coefficients of part. frac. exp.)
      35:40 Example (p. 43) L^-1 ( { 1/ ( (s+1)^3 s^2 ) )
       
       

  • Lecture 16

    • 00:00 Circuit Response Using Laplace Transform
      00:20 Example (p. 44)
      03:00 -Solution by Node Analysis
      16:50 -Obtaining Zero State & Zero Input Responses
      19:20 -Obtaining Zero State Response
      23:55 -Particular and Homogeneous Solutions
      26:30 -Solution by Mesh Analysis
      39:05 -Solution by State Equations
       
       

  • Lecture 17

    • 00:00 Sinusoidal Steady State (SSS) analysis
      04:25 Example on existence of SSS
      10:05 Some useful identities
      15:45 Circuit branch variables at SSS
      20:10 Definition of phasor
      24:50 Phasor diagram
      29:55 Example (p.48) (phasor representation)
      31:45 -Branch variables in time-domain
      34:35 -Branch variables in phasor-domain
      38:44 -Phasor diagram for branch currents
      40:37 -Phasor diagram for branch voltages
      43:45 Claim & Proof: Phasors of current obey KCL/KVL
       
       

  • Lecture 18

    • 00:00 Terminal Equations in Phasor Domain
      00:40 Resistor
      02:43 -Voltage phasor
      05:12 -Phasor dom. rel. between V_r and I_r
      07:25 -Phasor diagram (resistor volt./current)
      08:40 -In phase signals (definition)
      10:05 Inductor
      11:41 -Phasor dom. rel. between V_L and I_L
      15:15 -Phasor diagram (inductor volt./current)
      17:20 -Current phasor lags volt. phasor (inductor)
      19:15 Capacitor
      20:11 -Phasor dom. rel. between V_C and I_C
      20:55 -Phasor diagram (capacitor volt./current)
      21:40 -Current phasor leads volt. phasor (capacitor)
      23:50 Impedance & Admittance
      27:02 Impedance (definition)
      29:10 Remark on impedance
      31:37 Z = R + j X (resistance and reactance defn.)
      32:00 Y = G + j B (conductance and susceptance defn.)
      33:03 Warning: R \neq 1/G, X \neq 1/B (in general)
      33:55 Impedance - Admittance div (RLC)
      36:30 Series & Parallel connections
      37:17 -Equivalent impedance
      39:05 Example (p.52) equivalent impedance calc.
      47:12 Ind. sources (AC sources) in phasor domain
      50:10 Dependent sources in phasor domain
      52:50 Coupled inductors in phasor domain
      57:02 Ideal transformer in phasor domain
       
       

  • Lecture 19

    • 00:00 Phasor Domain Analysis
      03:28 -Step 1 : Transform circuit to phasor domain
      05:25 -Step 2 : Solution in phasor circuit
      07:10 -Step 3 : Transform soln. back to time-domain
      08:15 Ex. (p.54): Series RC at SSS
      10:45 -Solution (example)
      22:35 Ex. (p.55): Circuit with two meshes
      25:30 -Solution (example)
       
       

  • Lecture 20

    • 00:00 Ex. (p.56): Node analysis in phasor domain
      02:40 -Solution
      13:40 Ex. (p.57): Node analysis in phasor domain (ideal op-amp)
      17:00 -Solution
      26:40 -Question on example: What if Vs(t)=5cos(3t+23') V?
      28:52 -Answer: Use linearity
      34:44 Ex. (p.58): Mesh analysis in phasor domain
      37:20 -Solution
       
       

  • Lecture 21

    • 00:00 Superposition of ind. sources at AC steady-state
      00:25 Example (p.59)
      04:25 - Superposition in time-domain
      08:32 - Superposition circuit #1 (omega = 2 rad/sec.) phasor-domain
      16:47 - Superposition circuit #2 (omega = 3 rad/sec.) phasor-domain
      19:01- Final solution in time-domain
      20:50 Example (p.60)
      23:40 - Superposition circuit #1 (omega = 2 rad/sec.) phasor-domain
      30:53 - Superposition circuit #2 (omega = 0 rad/sec.) phasor-domain
      34:10 - Final solution in time-domain
      35:02 - Conclusion on superposition of ind. sources at AC steady-state
       
       

  • Lecture 22

    • 00:00 Thevenin/Norton equivalent circuit
      03:35 Example (p.61)
      07:00 -Solution: find Voc (mesh analysis)
      14:15 -Solution: find Isc (node analysis)
      19:58 -Solution: Zth = Voc/Isc calc.
      21:15 -Solution: Another way for Zth calc. (input impedance calc.)
      23:10 -Solution: Equivalent circuit configurations
      24:50 Example 1 (p.63) - Impedance Reflection Rule Ideal Transformer
      29:40 Example 2 (p.63) - Input impedance calculation
      43:00 Example (p.64) - Finding AC steady-state soln. given state equation representation
      47:40 - Solution: State eqn.
      48:25 - Solution: Stability
      48:47 - Solution: Finding particular soln. (AC steady-state) via phasors
      01:00:55 - Solution: Solution starts from steady-state, no transient!
       
       

  • Lecture 23

    • 00:00 Effective (RMS) Value
      07:34 Definition of RMS
      09:40 RMS scaling factor is 1/sqrt(2) for sinusoids
      11:40 Instantaneous & average power in SSS
      12:07 - P(t), Pav for resistor
      20:45 - P(t), Pav for capacitor
      28:46 -- E(t), Eav for capacitor
      31:50 - P(t), Pav for inductor
      33:48 -- E(t), Eav for inductor
      35:16 Power into one port circuit: P(t) and Pav
      40:45 Power factor term
      43:05 Pav computation of 1-port with input impedance Zin
       
       

  • Lecture 24

    • 00:00 Classification of 1-ports: Resistive, Inductive, Capacitive
      04:35 Recall (Resistor, Inductor, Capacitor)
      09:18 Definition (Pav, E^L_av, E^C_av)
      13:42 Passivity implication
      14:20 Tellegen's theorem in phasor domain
      18:04 Exercise on Tellegen's (also see Lec. 26 - cons. of complex power)
      18:31 Implications of Tellegen's theorem at SSS
      28:05 1-port power relation implied by Tellegen's theorem (cons. of power)
      30:20 Remark on the imaginary part of Z*I_eff^2
      31:50 Classification of 1-ports (Resistive, Inductive, Capacitive)
      33:17 -Classification div
      37:05 Example (p.71) Z(j\omega) acting as a resistive, ind., cap. 1-port
       
       

  • Lecture 25

    • 00:00 Complex power
      02:45 S = 1/2 x V x conj(I) : Complex power
      04:17 S : in terms of current and volt. phasors
      06:48 Pav = P = Re{S} : Avg. Power
      10:01 Q = Im{S} : Reactive Power
      11:05 mag(S) : Apparent power
      14:15 Complex power calculation of 1-port
      26:05 - Impedance and power triangles
      29:00 - Case 1: Resistive 1-port (unity p.f.)
      31:28 - Case 2: Inductive 1-port (lagging p.f.)
      34:25 - Case 3: Capacitive 1-port (leading p.f.)
      36:28 - Case 4: Purely inductive 1-port (p.f. = 0 lagging)
      37:37 - Case 5: Purely capacitive -port (p.f. = 0 leading)
      38:45 Example (p.74) 10 kVA load at 0.8 p.f. leading
       
       

  • Lecture 26

    • 00:00 Example (p.74): 1-port at SSS. Given v(t), i(t), compute S.
      01:45 - Solution \theta_i = -20 deg.
      05:24 - Solution \theta_i = 140 deg.
      08:58 Example (p.75) : Load at 100 Vrms, 5kW and p.f. 0.8 leading
      19:02 Conservation of Complex Power
      24:59 Total complex power of the circuit
      31:55 Ex.(p.76) Find generator side voltage given load config.
      38:00 Solution by conservation of power (recommended way!)
      44:29 Big Mistake Warning: |S_1| + |S_2| \neq |S_1 + S_2|
       
       

  • Lecture 27

    • 00:00 Superposition in power calculations
      06:10 Instantaneous power and superposition
      07:40 Average power and superposition
      19:00 Conclusion: Avg. Power calc. with sources having distinct frequencies
      22:00 Ex. (p.78): Avg. Power calc. (DC and AC sources together)
      24:40 - Solution - DC circuit (AC: OFF)
      27:49 - Solution - AC circuit (DC: OFF)
      34:30 - Solution - Avg. power on R when both sources are ON
      35:55 - Exercise(p.78) - Find avg. stored energy in L
      37:40 Average stored energy of coupled inductors
      40:30 - Instantaneous stored energy in mut. ind.
      41:55 - Average stored energy in mut. ind.
       
       

  • Lecture 28

    • 00:00 Power factor correction
      02:25 - Practical loads operate at a fixed/nominal voltage
      06:01 - I_rms of each load is fixed (nominal operation cond.)
      06:30 - Power loss over transmission line/feeder
      09:10 - Higher pf reduces P_loss (pf=1 is the best!)
      10:30 - Industrial applications: pf correction by load capacitance
      14:30 Ex. (p.81): Increasing pf from 0.75 to 0.95 (lagging)
      19:08 Soln. - a) Finding complex power of 2nd load
      31:50 Soln. - b) Finding the compensating capacitance
      44:45 Ex. (p.82): Efficiency (P_loss reduction after compensation)
      46:36 - Percent efficiency (definition)
      52:41 - Using high voltage transmission lines to increase efficiency
       
       

  • Lecture 29

    • 00:00 Maximum power transfer
      02:52 - Optimization prob.: Given/fixed Thevenin equiv. Find Z_Load to maximize P_Load.
      05:00 - Solution by differentiation
      12:40 Remark on general maximization/optimization problem
      14:49 - Exercise: Maximum power transfer by adjusting turns-ratio
      19:26 Ex. (p.84) - Using maximum power transfer condition
      23:00 - Solution (Optimality cond. Z_L = conj(Z_th) requires 2 deg. of freedom on adjusting real/imag parts which we have in this problem)
      34:05 Ex. (p.85) : Finding source volt. of a ladder network terminated with a load
      36:07 - Solution by conservation of power method *recommended method*
       
       

  • Lecture 30

    • 00:00 Balanced three phase circuits
      04:30 Phasor diagram (3 synchronous generators with +/- 120 deg. phase lag)
      08:16 Configuration #1: Y connection (a,b,c and n terminals)
      10:46 Y-connected 3-Φ generator
      12:01 Line-to-line voltage (definition)
      12:45 Phase voltage definition for Y-connection
      14:45 Phase voltages: Y-connection
      16:12 Vab - Line-to-line voltage calculation
      19:40 Vbc - Line-to-line voltage calculation
      22:25 Vca - Line-to-line voltage calculation
      24:50 Phasor diagram of phase/line voltages
      26:00 Vab - line-to-line volt. calc. on phasor diagram
      29:20 Y to Y circuit (balanced)
      32:55 -Source side (Y-connected)
      33:05 -Load side (Y-connected)
      34:11 -Phase currents
      35:14 -Line-to-line or simply line voltage
      34:25 -Line and Line currents
      36:40 -Summary of definitions (Y-to-Y connection)
      41:04 Balanced system (definition)
      42:45 Remark: Current on neutral-to-neural line (I_nN)
      48:05 Per-phase circuit (single phase equivalent)
      49:57-Solution of phase-a
      50:40-Transferring phase-a solution to the other phases
       
       

  • Lecture 31

    • 00:00 Per-phase circuit (Y-to-Y connection, continued)
      04:49 -Complex power delivered to a single phase of Y-load
      10:00 Per-phase and total power for 3-Φ systems
      15:00 Example: Y-to-Y connected system
      18:40 -Drawing circuit conf. from given info.
      24:18 -Per-phase analysis
      24:25 -Drawing per-phase equiv. circuit conf. (Y-to-Y)
      26:13 -Line-current a calculation (I_a)
      28:35 -Line-currents b/c derived from I_a (balanced system)
      29:50 -Power calculations
      33:33 Instantaneous power (delivered to the load)
      37:45 -P_A(t): Instantaneous power delivered to phase-a
      40:35 -P_B(t), P_C(t): Instant. power delivered to phases-b/c (from P_A(t))
      43:53 -Total instantaneous power delivered to load
       
       

  • Lecture 32

    • 00:00 Delta (Δ) connection
      02:52 -Phase currents (Δ connection)
      03:05 -Phase voltages (Δ connection)
      05:06 Phase- and line- currents (Δ connection)
      06:40 -Line-current calculation via phasor diagram (Δ connection)
      13:10 Line/phase currents & Line/phase voltage for Y connection
      14:55 Line/phase currents & Line/phase voltage for Δ connection
      16:08 Line/phase rms quantities and sqrt(3) factor: Y-connection
      17:15 Line/phase rms quantities and sqrt(3) factor: Δ-connection
      18:27 Total (3-Φ) power in terms of line quantities
      23:00 Remark: Δ-Y transformation to use per-phase circuit
      27:00 Equiv. per-phase circuit after Δ-Y transformation on load side
      28:35 Example (Y-to-Δ connected system)
      33:20 -Drawing 3-Φ circuit config
      37:24 -Drawing per-phase circuit
      39:25 -Solution of per-phase equivalent circuit
      39:58 -Deriving phase currents of Δ-load from line current (phasor diag.)
      44:54 -Phase voltage calc. of Δ-load
      47:00 -Total real-power of load
       
       

  • Lecture 33

    • 00:00 Example (p.95) (Y-to-Δ system)
      07:25 -Drawing per-phase equivalent circuit
      10:24 -Finding total (3-Φ) complex power supplied by generator
      15:35 -Finding efficiency of the system
      17:44 -Finding Z_line
      20:45 -Finding I_line (phasor diagram)
      25:40 -Finding Vbc(t)
      32:40 Example (p.100) (Delta and Y loads in parallel)
      38:38 -Solution by conservation of power
      45:50 -Finding rms-line current from apparent power
      46:52 -Finding supplied power (conservation of power)
      48:00 -Finding Van (rms) from apparent power
      49:35 -Finding 3-Φ power supplied by the source
       
       

  • Lecture 34

    • 00:00 Two wattmeter method for balanced 3Φ power measurement
      01:53 Wattmeter representation in the circuit diagram
      05:14 Operation of wattmeter (wattmeter reading)
      11:00 Phasor diagram for voltages
      11:10 - Vxn: Reference voltage phasor
      13:40 - Vyn, Vzn with respect to Vxn
      18:01 - Vxn, Vyn, Vzn: Phasor diagram (line-to-neutral)
      19:00 - Vxy, Vyz, Vzx : Phasor diagram (line-to-line)
      23:43 Line currents: Ix, Iy, Iz
      25:50 Calculating Wattmeter #1 and #2 readings (P1, P2)
      28:40 Combining P1 and P2 to get total real power of 3Φ load
      33:40 Combining P1 and P2 to get total reactive power of 3Φ load
      36:15 Remark: The sign of total reactive power is found only if you know +/- sequencing of the 3Φ circuit.
       
       

  • Lecture 35

    • 00:00 s-domain circuit analysis
      01:55 Steps of s-domain circuit analysis (outline)
      06:15 An example for s-domain analysis (outline)
      13:55 Circuit laws in s-domain (KCL and KVL)
      15:40 Components in s-domain
      15:40 - Resistor
      19:14 - Capacitor
      29:50 - Inductor
      34:42 - Dependent sources
      36:45 - Ideal Transformer
      38:55 Impedance in s-domain (Definition)
      43:05 s-domain impedance relations for R, L, C components.
       
       

  • Lecture 36

    • 00:00 Series/Parallel connection of components (s-domain)
      04:30 Source transformation in s-domain
      06:30 Example: Initial condition models for capacitor via source-transformation
      09:18 Coupled inductor in s-domain
      13:15 Coupled inductor initial condition models (s-domain)
      14:35 -Coupled inductor IC model with current sources (s-domain)
      19:53 -Coupled inductor IC model with voltage sources (s-domain)
      22:13 Example (p.106): Analysis of an RLC circuit in s-domain
      25:10 -Transform the circuit to s-domain
      28:25 -Select an analysis method and apply
      29:47 -Writing the node equation
      31:19 -Solving for the node voltage in s-domain
      31:35 -Finding branch current given the node voltage expression
      32:20 -Evaluating Vs(s) and finalizing the solution in s-domain
      33:57 -Applying inv. Laplace trans. and finalizing the time-domain solution.
      36:15 -Comment: Both natural frequencies are visible in the solution for branch current
       
       

  • Lecture 37

    • 00:00 Superposition principle (LTI resistive circuits)
      04:22 Superposition principle extends to s-domain for LTI *dynamic* circuits
      05:38 Sources of excitation
      06:24 -Ind. sources due to external inputs/driving forces (s-domain)
      06:59 -Ind. sources due to initial conditions (s-domain)
      09:43 Zero-input and zero-state solutions in s-domain
      11:40 Example (p.107): Parallel RC to RLC (2nd order circuit with IC under DC input )
      16:32 -Solution for zero-state and zero-input solution (part a)
      30:25 -Numerical solution for given component values (part b)
      35:11 Example (p.109): Mesh analysis in s-domain
      37:15 -Transform circuit to s-domain
      42:21 -Equivalent circuit after source-transformation (one less mesh)
      44:21 -Writing mesh equations
      47:05 -Expressing mesh equations in matrix form
      49:48 -Inverting the matrix equation to get mesh currents
      52:58 -Computing the mesh current I2 in s-domain
      53:15 -Expressing the branch voltage given the mesh currents
      55:06 -Inverse Laplace transform via residue method and coef. comparison
      58:35 -Expressing the solution in time-domain
       
       

  • Lecture 38

    • 00:00 Example (p.110) (Node Analysis Example)
      03:40 -Solution: Before switching
      07:00 -Solution: After switching
      10:45 -Solution: KCL at node 1
      12:30 -Solution: KCL at node 2
      14:04 -Solution: Solving for E_2(s)
      17:15 -Solution: Partial frac. expansion of E_2(s)
      19:44 -Solution: Time-domain expression for the answer
      21:15 Example (p.112) (Thevenin Equiv.)
      25:05 -Solution: Thevenin Equivalent Calculation
      31:18 -Solution (part a): Load = Resistor
      35:15 -Solution (part b): Load = Capacitor
       
       

  • Lecture 39

    • 00:00 Transfer Function
      05:14 Example 1: Series RC Circuit (basic example)
      07:25 Comment on poles of Y(s) (the system output)
      11:45 Comment on homogeneous & particular solutions
      12:38 Example 2 (p.113): Series RC with AC input
      15:45 TF of one-port & two-port circuits
      17:20 -Driving point impedance Z(s)
      19:25 -Voltage transfer function H_v(s)
      20:26 -Current transfer function H_i(s)
      21:11- Transfer admittance func. H_y(s)
      21:59 -Transfer impedance func. H_z(s)
      23:52 Example (p.115) (find impedance parameters of 2-port)
      27:13 -Solution via mesh analysis
      29:35 Example (p.115) A 2nd order circuit
      33:28 -Solution (part a): Find Z(s)
      35:26 -Solution (part b): Find H_v(s)
      37:15 -Solution (part c): Find poles and zeros of H_v(s)
      40:35 Example (p.116) Ideal op-amp in linear region
      43:35 -Solution (part a): Find H_v(s)
      46:30 -Solution (part b): Find poles and zeros of H_v(s)
       
       

  • Lecture 40

    • 00:00 Relation between transfer function & impulse response
      03:55 Example (p.117) A 2nd order circuit
      06:30 -Solution: Find impulse response for the output v_2(t)
      12:20 Step response
      17:09 Impulse response & Convolution
      21:39 Example (p.118) Given h(t), find ramp response
      22:53 -Solution in s-domain
      27:16 -Solution in time-domain (via convolution integral)
      33:00 Example (p.122) Series RLC
      35:15 -Solution (part a): Find the diff. eqn. satisfied by i_L(t)
      38:04 -Solution (part b): Obtain the diff. eqn. from H(s)
      41:17-Solution (parts a and b): Comparison of time-/Laplace- domain approaches
       
       

  • Lecture 41

    • 00:00 Relation between transfer func. & sinusoidal steady-state response
      04:11 Complete solution to AC input (s-domain)
      06:10 Steady-state solution for sdiv circuits
      09:30 Deriving y_ss(t) from transfer function (partial frac. exp.)
      16:53 y_ss(t) of an arbitrary sdiv LTI circuit (final result)
      19:22 Conclusion: Steady-state output of a sdiv LTI system with AC input
      22:20 Example (p.121) Given h(t), find ss solution for AC inputs
      25:01 -Solution: H(s) calculation
      26:50 -Solution (part a): input = 5cos(1000t)
      31:25 -Solution (part b): input = 5cos(3000t + pi/4)
       
       

  • Lecture 42

    • 00:00 Frequency response
      02:20 Fund. relation between LTI systems and sinusoidal inputs
      07:15 Gain & Phase function
      09:50 Example (p.123) -explaining freq. resp.-
      15:55 Stopband definition
      16:50 Passband definition
      17:50 Cutoff frequency definition
      19:38 Bandwidth definition (3dB bandwidth)
      20:40 Frequency response plot
      24:22 Prototype gain characterisitic
      24:24 -Low-pass filter
      26:55 -High-pass filter
      29:08 -Band-pass filter
      31:28 -Band-stop filter
       
       

  • Lecture 43

    • 00:00 1st order circuit frequency response
      00:23 1st order low-pass filter
      05:25 -General form of H(j&#969) (low-pass)
      06:30 -Gain function (low-pass)
      07:25 -Phase function (low-pass)
      09:10 -Gain & Phase plots (low-pass)
      17:28 1st order high-pass filter
      20:50 -General form of H(j&#969) (high-pass)
      21:55 -Gain function (high-pass)
      23:05 -Phase function (high-pass)
      24:58 -Gain & Phase plots (high-pass)
      31:36 Example (p.127) - 1st order Active Filter-
      34:50 -part a: H(s)
      37:30 -part b: Filter type
      37:52 -part c: Passband gain
      39:35 -part d: Design circuit for fc=500 Hz.
      41:12 -part e: Phase shitf at cut-off freq.
       
       

  • Lecture 44

    • 00:00 Second order band-pass filter
      00:36 -General form of H(jω)
      03:32 -Gain function
      11:45 -Gain plot
      15:08 -Cut-off frequencies
      20:22 -Bandwidth
      21:01 -Quality factor
      23:40 -Phase function
      29:10 -Phase plot
      31:52 Example (p.130)
      37:40 Physical meaning of Q
       
       

  • Lecture 45

    • 00:00 Ex.1 (p.131) - Series RLC: Bandstop Filter -
      08:08 Ex.2 (p.131) - 2nd Order Bandpass -
      10:20 -soln.part a: H(s)
      17:05 -soln.part b: Gain and phase plots
      18:40 -soln. part c: Bandwidth, Q, cut-off freq.
      23:40 Ex.3 (p.132) - Active 2nd order bandpass filter -
      27:14 -soln.part a: H(s)
      34:15 -soln.part b: Type of filter
      34:47 -soln.part c: Filter design for a given spec.
       
       

  • Lecture 46

    • 00:00 Second order low-pass filter
      00:54 -H(s) of 2nd order low pass filter
      04:24 -Max. amplitude of H(j&#969) (low/high Q cases)
      08:35 -Gain plots with respect to alpha (damping factor)
      11:00 Second order high-pass filter
      11:35 -H(s) of 2nd order high pass filter
      17:47 -Gain plots with respect to alpha (damping factor)
      20:08 Resonant frequency (definition)
      23:14 Ex. (impedance cancellation due to L and C at resonance)
      26:00 Example (p.136) - Parallel RLC resonance freq. calc. -
      33:50 Cascade connection of networks (p.134)
      45:25 -Q. How to eliminate loading effect in cascade comb.
       
       

  • Lecture 47

    • 00:00 Finite-Q inductors/capacitors
      03:56 -L (with internal resistance) and C parallel comb. and its approximation
      08:03 -Selecting R2 in the approximate circuit
      11:40 -Remark: Relation between Q of non-ideal parallel LC and R2
      13:36 -Impedance comparison: Z1(j&#969) and Z2(j&#969)
      18:18 -Conclusion: Finite-Q inductor (L in series r) app. around resonance
      20:55 Dual Case: Finite-Q capacitor (C // R)
       
       

  • Lecture 48

    • 00:00 Filter Design by Scaling
      00:23 -Magnitude scaling
      05:58 -Frequency scaling
      12:49 Ex. (p.139) - Filter design by scaling prototype filter -
      16:33 -Transfer function of scaled filter
      25:13 -Final solution
       
       

  • Lecture 49

    • 00:00 Example (p.141) (frequency resp.)
      01:41 -Soln. a - Pole-zero diagram
      05:30 -Soln. b - Magnitude response
      17:44 -Soln. b - Phase response
      24:16 Ex. (p.142) - Freq. resp. and pole-zero diagram
      29:11 Example (p.143) (2nd order bandpass filter design)
      33:10 -Soln. - Filter #1 - prototype filter design
      39:00 -Soln. - Filter #1 - scaling of prototype filter
      41:55 -Soln. - Filter #2 - prototype filter design and scaling
       
       

  • Lecture 50

    • 00:00 Bode Plots (Bode Approximation)
      02:00 -Bringing H(s) or H(jω) into the standard form
      03:16 -Magnitude resp. calc. for H(s) in standard form.
      03:16 -Phase resp. calc. for H(s) in standard form.
      07:30 Note: Phase resp. terms add-up; mag. resp. terms form a product
      09:50 Decibel (dB) definition, mag. resp. terms expressed additively in dB
      12:08 -Term #1 : Constant term
      19:58 -Term #2 : jω term
      23:15 Decade definition in dB plots
      31:25 -Term #3 : 1st order term: 1 + j&#969/&#945 or 1/(1 + j&#969/&#945)
      42:08 -Term #3 : Bode app. for phase response of 1st order term
       
       

  • Lecture 51

    • 00:00 Example (p.148)
      02:25 -Soln. a) Bring H(s) into standard form
      06:02 -Soln. a) Asymptotic gain plot (bode plot)
      07:13 -Soln. a) how to mark points on log scaled x-axis
      18:20 -Soln. b) Evaluating points on mag. plot in dB scale
      22:05 -Soln. c) Asymptotic phase plot
      35:30 Example (p.149)
      36:25 -Soln. a) Bring H(s) into standard form
      39:30 -Soln. a) Bode app. for mag. resp.
      48:50 -Soln. a) how to determine points on log scaled x/y - axes
       
       

  • Lecture 52

    • 00:00 Example (p.150) Active Filter Design
      04:35 -Solution - Obtaining H(s)
      08:17 -Solution - Bode approximation to mag. plot
      14:41 Bode approximation for 2nd order term
      16:43 Ex. (p.152)
      16:45 -Soln - How to bring a 2nd order term to the standard form
      18:35 Gain and phase resp. for the 2nd order term
      21:50 -Bode approximation for gain function
      29:44 -Bode approximation for phase function