4/27 Friday. I have all the old homework if you need to collect them.

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1 4/27 Friday Last HW: do not need to turn it. Solution will be posted on the web. I have all the old homework if you need to collect them. Final exam: 7-9pm, Monday, 4/30 at Lambert Fieldhouse F101 Calculator OK TA will have regular office hour on Monday (9am-4:30pm) I will have office hour 10am-12pm on Monday Grading: Total 550 points HW: 50 points (worst 4 grades dropped) Test 1, 2, 3: each for 100 points Final: 200 points Historical data: average grade of this course is

2 Voltage, Current, Power, Energy Current i(t) is the rate of charge flowing through where q(t) is the amount of charge having flown through from time t 0 to t Instantaneous power absorbed by an element is (passive sign convention) where W(t) is total amount of energy absorbed from time t 0 to t

3 Circuit Elements Resistance (in ) Inductance (in H) Capacitance (in F) (ohm s law) Memoryless With Memory With Memory doesn t jump doesn t jump Absorb/dissipate but doesn t store energy Store energy Store energy series parallel series parallel series parallel

4 Circuit Elements + - Independent voltage source Independent current source Dependent voltage source Dependent current source V: fixed I: arbitrary I: fixed V: arbitrary V: controlled by another signal I: arbitrary I: controlled by another signal V: arbitrary Op-amp is a VCVS with a very high gain A + -

Basic Circuit Laws KCL: the total current flowing into (out of) a node is zero Generalization: the total current flowing into (out of) a Gaussian surface is

5 Basic Circuit Laws KCL: the total current flowing into (out of) a node is zero Generalization: the total current flowing into (out of) a Gaussian surface is zero. KVL: the total voltage drop around a loop is zero These are the basic circuit laws that apply to any circuit Voltage division formula Current division formula

6 Nodal and Loop (Mesh) Analysis Both are analysis tools for an arbitrary circuit Together with the Ohm s law, one can determine all the currents and voltages in a circuit with independent (dependent) sources and resistors. Sometimes it is easier to apply one of them than applying the other Nodal Analysis Loop (Mesh) Analysis Variables Node voltages Loop currents Circuit law used KCL KVL Simplified cases Grounded voltage source Current source belonging to a single loop Complicated cases Floating voltage source (super node) Current source belonging to more than one loop

7 Analyzing Op-Amp Circuits Assume v C (0)=0 Always use the ideal op-amp property: V + =V - and i + =i - =0 First find the voltage at one terminal, say, V + Use V + =V - to find the voltage at the other terminal, say, V - Apply KCL to + terminal, using the fact that i + =0

8 Techniques for Simplifying Linear Circuit Analysis Linearity Principle In a linear circuit, the response/output is a linear function of all the independent sources Superposition principle To get the response, one can set all except one independent source to zero, compute the result, and then add them up Setting an independent voltage source to zero -> short circuit Setting an independent current source to zero -> open circuit Source transformation +

9 Thevenin and Norton Equivalent Circuits Linear two-terminal network Thevenin equivalent circuit Linear two-terminal network Norton equivalent circuit

10 To Compute Equivalent Circuits Find V oc, the open circuit voltage. V oc =V s Find I sc, the short circuit current. I sc =V s /R th Then set all the independent sources to zero, and the equivalent resistance is R th A more general way, especially when there are dependent sources (including op-amps) within the two-terminal network, is to compute its v-i relation directly. Apply an arbitrary current I Compute the voltage drop as a function of I (pay attention to the polarity) Compare with V=V s +IR th

11 Maximum Power Transfer Theorem Theorem: for the simple two-terminal network show above in the box connected to a variable load R L, the maximum instantaneous power is transferred to the load when R L =R th and the maximum instantaneous power is given by

12 First Order Circuits Linear network (resistors, independent and dependent sources) Linear network (resistors, independent and dependent sources) Time constant =L/R th Time constant =R th C

13 First Order Circuits The currents and voltages in a 1st order circuit are of the form Elapsed time Final value of x Initial value at time t 0 Time constant Initial value is often given by the condition the switch has been closed (open) for a long time before it is opened (closed) at time t 0. One needs to use the condition that v C and i L doesn t jump to get the initial value after the switching. For final value, replace the capacitor by open circuit, inductor by close circuit, and compute the final value.

Second Order Circuits The voltage/current follows an inhomogeneous differential equation: with and given Its homogeneous version has general solution (case 1)

14 Second Order Circuits The voltage/current follows an inhomogeneous differential equation: with and given Its homogeneous version has general solution (case 1) (case 2) (case 3) depending on the roots of the characteristic equation Inhomogeneous equation has one particular solution General solution to the inhomogeneous equation

Second Order Circuit The key is to write the differential equation and/or characteristic equation for the voltage/current of interest with and given For series/parallel RLC circuit, you can use the

15 Second Order Circuit The key is to write the differential equation and/or characteristic equation for the voltage/current of interest with and given For series/parallel RLC circuit, you can use the formula sheet directly. Sometimes by setting independent sources to zero, you can convert a 2 nd -order circuit to series/parallel RLC circuit. Initial value is often given by the condition the switch has been closed (open) for a long time before it is opened (closed) at time t 0. One needs to use the condition that v C and i L doesn t jump to get the initial value after the switching. For final value, replace the capacitor by open circuit, inductor by close circuit, and compute the final value.

16 Sinusoidal Steady State Analysis When all the excitations of a circuit are sinusoidal with the same frequency, all the steady state responses are sinusoidal with frequency Phasor is a convenient tool for the SSS analysis of linear circuits consisting of R, L, and C Phasor of x(t)=cos( t+ ): Phasor Ohm s Law: impedance admittance

17 Phasor Analysis Basic circuit laws for phasors: KCL and KVL Ohm s law in terms of impedance Nodal and mesh analysis Thevenin and Norton equivalent circuits Frequency response: low/high pass filter band pass/reject filter

Effective Value of Periodic Signals The effective value (or root-mean square) of a periodic signal f(t) is: For sinusoidal voltages/currents, the effective

18 Effective Value of Periodic Signals The effective value (or root-mean square) of a periodic signal f(t) is: For sinusoidal voltages/currents, the effective value is the peak value divided by the square root of 2 V eff V m 2 Average power consumed by the element is: P V I cos( ) av eff eff v i V m 2 I m cos( ) v i

19 Effective Phasors Average power consumed by the element is: where S is the complex power defined by

20 Complex power P pf v i P 1 pf 2 ( ) 1 Conservation of (complex) power - No conservation law for apparent power Power factor: (leading or lagging)

21 Maximum power transfer: P L V ( R R ) ( X X ) S 2 S 2 2 L s L R L The maximal average power that Z L =R L +jx L can absorb is achieved when And the maximal average power is and (DC version becomes a special case)

Chapter 10 AC Analysis Using Phasors

Chapter 10 AC Analysis Using Phasors 10.1 Introduction We would like to use our linear circuit theorems (Nodal analysis, Mesh analysis, Thevenin and Norton equivalent circuits, Superposition, etc.) to