# (amperes) = (coulombs) (3.1) (seconds) Time varying current. (volts) =

Size: px
Start display at page:

## Transcription

1 3 Electrical Circuits 3. Basic Concepts Electric charge coulomb of negative change contains electrons. Current ampere is a steady flow of coulomb of change pass a given point in a conductor in second. (amperes) (coulombs) (3.) (seconds) Time varying current ( + ) () () lim 0 (3.2) Voltage Voltage dierence between two points is the work in joules required to move coulomb of charge from one point to the other. (volts) (joules) (3.3) (coulombs) Power The rate at which something either absorbs or produces energy. The power absorbed by an electric element is the product of the voltage and the current. (Watts) (joules) (volts) (amperes) (3.4) (seconds) 3.2 Circuit Elements Ideal voltage source 7

2 Sec 3.2. Circuit Elements Resistor Capacitor Inductor i R vr i C v i C v Current Source Voltage Sourse DC Voltage Sourse GND i v 5V Figure 3.: Basic circuit elements. v v i (pure AC) v I V I (pure DC) v I V I + v i t Figure 3.2: DC signals versus AC signals. Voltage source provides a specified voltage across the two terminals and does not depend on the current flowing through the source. The output impedance of ideal voltage source is zero. The current flowing through an ideal voltage source is completely determined by the circuit connected to the source. Ideal current source Current source provides a specified current and does not depend on the voltage across the source. The output impedance of ideal current source is infinite. The voltage across an ideal current source is completely determined by the circuit connected to the source. DC signals v.s. AC signals Figure 3.2 shows the definitions of DC signals and AC signals. Resistor Resistance is the property of materials that resists the movement of electrons. 8

3 ecture 3. Electrical Circuits Ohm s aw (volts) (ohms) (amperes) Conductance is the inverse of resistance. For parallel resistors, (3.5) (3.6) When two resistors are connected in parallel, the equivalent resistance is smaller than any of the two resistors (3.7) For series resistors, (3.8) Capacitor Capacitance is the ability of a capacitor to store charges on its two conductors. (farad) (coulombs) (volts) (3.9) For time-varying voltage () across the capacitor The capacitor is an open circuit, i.e., () 0,when () isaconstant. The voltage across the capacitor can not jump. However, the current flowing through the capacitor does not have such constraint. () lim 0 () (3.0) For parallel capacitors, () Z () (3.) (3.2) For series capacitors, Inductor (3.3) 9

4 Sec 3.3. Circuit aws For time-varying current () flowing through the inductor Theinductorisashortcircuit, i.e., () 0,when () isaconstant. The current flowing through the inductor can not jump. However, the voltage across the inductor does not have such constraint. () (flux linkages) lim 0 lim 0 lim ( + ) () 0 () (3.4) For parallel inductors, () Z () (3.5) For series capacitors, (3.6) (3.7) Dual circuit Capacitor and Inductor formulas are the same except that the symbols dier. Capacitor () () + () (3.8) Inductor () () + () (3.9) 3.3 Circuit aws Kirchho s voltage law ( KV ) The algebraic sum of the voltages around any closed loop of a circuit is zero. Kirchho s current law ( KC ) The algebraic sum of the currents entering every node must be zero. 20

5 ecture 3. Electrical Circuits i S vo v i O S C R R Current Source Parallel v C R Voltage Source Serial i R Figure 3.3: Similarity between capacitor and inductor. 3.4 Network Theorems Definition 3. inear circuit is formed by interconnecting the terminals of independent sources, controlled sources, and linear passive elements to form one or more closed paths. inear passive elements include Resistor, Capacitor, andinductor. The characteristics of these elements satisfy the conditions of linearity. Resistor () () 2 () 2 () (3.20) ()+ 2 () ( ()+ 2 ()) Capacitor () ( ()) 2 () ( 2()) (3.2) ()+ 2 () ( ()+ 2 ()) Inductor () 2 () ()+ 2 () Z 0 Z 0 Z 0 () 2 () (3.22) ( ()+ 2 ()) Theorem 3.2 In a linear network containing multiple sources, the voltage across or current 2

6 Sec 3.4. Network Theorems R R2 V A I R3 A V R R2 (a) V A R 3 V + R R2 (b) I R A 3 V 2 (c) Figure 3.4: Example of linear circuit with multiple independent sources. through any passive element may be found as the algebraic sum of the individual voltages or currents due to each of the independent sources action along, with all other independent sources deactivated. Voltage source is deactivated by replacing it with a short circuit. Current source is deactivated by replacing it with an open circuit. Controlled sources remain active when the superposition theorem is applied. Example 3.3 Given the circuit in Figure 3.4, find the voltage across the resistor 3 using the superposition theorem of linear network.. The voltage across the resistor 3 is the superposition of the voltage when each independent source actions alone, as shown in Figure 3.4 (b) and (c). + 2 (3.23) 2. The contribution of the voltage source The current source is replaced with an open circuit. 3. The contribution of the current source (3.24) 22

7 ecture 3. Electrical Circuits The voltage source is replaced with an short circuit (3.25) 3.4. Equivalent Circuits of One-Port Networks inear Network A 2 (a) inear Network B Z TH V TH 2 (b) inear Network B I N YN 2 inear Network B (c) Figure 3.5: Equivalent circuits. (a) The original circuit. (b) Thevenin s equivalent. (c) Norton s equivalent. Equivalent circuit Areductionof a complex linear circuit into a simpler form. Amodelof a complex linear circuit contained in a black box. Theorem 3.4 Thevenin s theorem states that an arbitrary linear, one port network such as network A in Figure 3.5 (a) can be replaced at terminals 2 with an equivalent seriesconnected voltage source and impedance as in Figure 3.5 (b). is the open-circuit voltage of network A at terminals 2. is the ratio of the open-circuit voltage over short circuit current determined at terminals 2. The equivalent impedance looking into network A through terminals 2with all independent sources deactivated. Voltage sources are replaced by short circuits. Current sources are replaced by open circuits. 23

8 Sec 3.4. Network Theorems Theorem 3.5 Norton s theorem states that an arbitrary linear, one port network such as network A in Figure 3.5 (a) can be replaced at terminals 2 with an equivalent parallelconnected current source and admittance as in Figure 3.5 (c). is the short-circuit current flowing through terminals 2duetonetworkA. is the ratio of short-circuit current over open-circuit voltage at terminals 2. Conversion of equivalent circuits. Any method for determining is equally valid for finding (3.26) (3.27) (3.28) Example 3.6 In Figure 3.6, 4, 2, 2, 2 3, find the Thevenin s equivalent circuit and Norton s equivalent circuit for the network to the left of terminals 2 R R2 V A I A 2 Figure 3.6: Examples of Thevenin s and Norton s equivalent circuits.. Thevenin s equivalent is the open-circuit voltage at terminals (3.29) is the ratio of the open-circuit voltage over short circuit current determined at terminals 2 with network B disconnected. By the superposition of the short-circuit current caused by and the 24

9 ecture 3. Electrical Circuits short circuit current can be found. 2 ( )+( ) (3.30) 8 5 By definition, can be derived as 2 5. Alternatively, can be found as the equivalent impedance for the circuit to the left of terminals 2 is replaced with short circuit. is replaced with open circuit (3.3) 2. Norton s equivalent is the short-circuit current at terminals 2 whichcanbederivedasineq. (3.30). is the ratio of the short-circuit current over the open-circuit voltage with network B disconnected. From Eq. (3.29), the open circuit voltage is 8. Thus, Alternatively, Equivalent Circuits of Two-Port Networks A two-port network is an electrical circuit or device with two pairs of terminals. Figure 3.7 depicts a two-port linear network. Only two of the four variables, 2,, 2 can be independent. I I 2 + V - inear + V2 Network - I I 2 Figure 3.7: Two-port linear network. Characterization of Two-Port Networks parameters (, 2 depend on, 2 ) 25

10 Sec 3.4. Network Theorems The four parameters represent impedance. 2 and 2 are transfer impedances. Each of the parameters can be evaluated by open-circuiting an appropriate port of the network (3.32) parameters in matrix form (3.33) " # " #" # 2 (3.34) parameters (, 2 depend on, 2 ) The four parameters represent admittance. 2 and 2 are transfer admittances. Each of the parameters can be evaluated by short-circuiting an appropriate port of the network (3.35) (3.36)

11 ecture 3. Electrical Circuits parameters in matrix form. " # " 2 #" # (3.37) Relation between parameters and parameters. " # " # " parameters (, 2 depend on, 2 ) The represents impedance. The 22 represents admittance. Parameters and 2 are obtained by short-circuiting port 2. Parameters 2 and 22 are obtained by open-circuiting port. # (3.38) (3.39) (3.40) parameters in matrix form. " 2 # " #" 2 # (3.4) Example 3.7 In Figure 3.8, 0, 2 6, find the parameters and the parameters for the network 27

12 Sec 3.4. Network Theorems I R I V - 0.3I a I a R 2 V 2 - I I 2 Figure 3.8: Example of two-port linear network. From Eq. (3.33), the parameters can be obtained as follows. 2 0 (0 + 6) From Eq. (3.40), the parameters can be calculated as follows Equivalent Circuits of Special Two-Port Networks T-Model network. (Figure 3.9 (a)) 2and 3 can be derived from the parameters of a two-port networks. " # " # (3.42) Model network. (Figure 3.9 (b)) 28

13 ecture 3. Electrical Circuits and can be derived from the parameters of a two-port networks. " # " # 2 + (3.43) 2 22 Transformation from -Model to T-Model (3.44) Transformation from T-Model to -Model (3.45) I I 2 Z Z2 V Z3 V 2 (a) T-Model Network I Zc I 2 V Za Zb V2 (b) Model Network Figure 3.9: Equivalent circuits of two port networks. (a) T-Model Network. (b) Pi- Model Network. 29

14 Sec 3.5. aplace Transforms 3.5 aplace Transforms aplace Transform A mathematical tool for representing system transfer functions of causal TI systems. A generalization of Fourier transform. The result of Fourier transform can be obtained by evaluating the aplace transform along the imaginary axis. Definition 3.8 Given () with () 0for 0, its aplace Transform, which is defined as follows, is a complex function over a complex-number domain. () [()], Z 0 () Z 0 () (3.46) Definition 3.9 Correspondingly, the Inverse aplace Transform is as follows: () [ ()] I 2 () (3.47) Example 3.0 Given () &() (() () 0for 0) and their aplace Transforms () &(),the following shows the properties of aplace Transform. inearity () ()+() () ()+() (3.48) Time Derivatives () () () () (0 ) (3.49) () () Time Integrals () () (0 ) 2 (0 ) (0 ) () Z 0 () + (0 ) () () + (0 ) (3.50) Time Scaling () () ( ) where 0 (3.5) Time Delay () ( 0 ) () 0 () where 0 0 (3.52) 30

15 ecture 3. Electrical Circuits Multiplication () () () () () (3.53) Shift Convolution Product () () () () () () ()() () ( ) (3.54) () ()() (3.55) () I 2 ()( ) (3.56) Example 3. aplace Transform Pairs () 2 cos ( 0 + ) () () (! ) + cos sin ( + 0 ) + ( 0 ) 2 cos ( 0 tan ) ( ) Equivalent Circuits in aplace Domain Capacitor characteristic in aplace domain. Admittance of capacitor in aplace domain is Footnote () () () ( () (0 )) (3.57) Figure 3.0 (a) depicts the KC equivalent circuit in aplace domain. characteristic in aplace domain. 3

16 Sec 3.6. Equivalent Circuits in aplace Domain I C (s) / sc I C (s) V (s) C sc Cv( 0 ) C V C (s) v C (0 s ) (a) (b) I C (s) s I ( s) (s) V / s i (0 ) s (s) V i(0 ) (c) (d) Figure 3.0: (a) KC equivalent circuit of a capacitor in aplace domain. (b) KV equivalent circuit of a capacitor in aplace domain. (c) KC equivalent circuit of an inductor in aplace domain. (b) KV equivalent circuit of an inductor in aplace domain. Impedance of capacitor in aplace domain is () Z Z 0 () (3.58) () + (0 ) () ()+ (0 ) Figure 3.0 (b) depicts the KV equivalent circuit in aplace domain. Inductor characteristic in aplace domain. Admittance of inductor in aplace domain is () Z Z 0 () (3.59) () + (0 ) () ()+ (0 ) Figure 3.0 (c) depicts the KC equivalent circuit in aplace domain. characteristic in aplace domain. Impedance of inductor in aplace domain is () () () ( () (0 )) (3.60) Figure 3.0 (d) depicts the KV equivalent circuit of an inductor in aplace 32

17 ecture 3. Electrical Circuits domain. 3.7 System Transfer Function in Time and aplace Domains Two ways to deduce the system transfer function in time and aplace domains. Start in time domain and then take the aplace transform. Start in aplace domain and use Inverse aplace transform to return to time domain. Example 3.2 Consider the RC circuit in Figure 3., the input is the current source () and the output () is the voltage across the RC components. Find out the system transfer function and the outputs w.r.t. dierent inputs. is R C vo Figure 3.: The RC circuit to be solved in time domain and aplace domain. By writing the node equation with KC in time domain, we obtain the system equation. () () + () (3.6) By taking aplace transform to both sides, the system equation in aplace domain can be derived. () () + ( () (0 )) (3.62) The system transfer function in aplace domain. () () () (0 )0 +() + (3.63) 33

18 Sec 3.7. System Transfer Function in Time and aplace Domains Initial Value (Driving Free) Response Given () 0 () 0, the output can be obtained from Eq. (3.62). 0 () + ( () (0 )) () (0 )) +() (3.64) The time domain response can be calculated by taking Inverse aplace transform. () (0 ) (3.65) Impulse Response Given () (); (), the output can be obtained from Eq. (3.63). For simplicity, we do not consider initial condition, i.e., (0 )lim 0 () 0. () (3.66) + The time domain response can be calculated by taking Inverse aplace transform. () (3.67) Step Response Given () (); (), the output can be obtained from Eq. (3.63). For simplicity, we do not consider initial condition, i.e., (0 )lim 0 () 0. () ( +()) (3.68) +() The time domain response can be calculated by taking Inverse aplace transform. The capacitor is like an open circuit once it is fully charged. () ( ) (3.69) Sinusoidal Response Given () cos( 0 ) (); () ( ), the output can be obtained from Eq. (3.63). For simplicity, we do not consider initial condition, i.e., (0 )lim 0 () 0. () (3.70) The time domain response can be calculated by taking Inverse aplace transform. 34

19 ecture 3. Electrical Circuits The term ( + ) stands for transient response, which can be transformed into exponentially decaying time function. 2 +( 0 ) 2 The term (+ ) ( ) represent the steady state response, which shall yield the time-domain signal r ³ tan ( μ ) 0 ³ + () cos ( 0 )+ 0 sin ( 0 ) 0 2 ³ ³ μ ', ' 0 () cos ( 0 ) 0 cos ( 0 ) (3.7) At low frequency, the capacitor is similar to an open circuit. 0 À ' 0, ' () sin ( 0 0) cos At high frequency, the capacitor is similar to a short circuit. The circuit acts as a low-pass filter. Example 3.3 Consider the RC circuit in Figure 3.2, the input is the current source () and the output () is the voltage across the RC components. Find out the system transfer function and the outputs w.r.t. dierent inputs. is R C vo Figure 3.2: The RC circuit to be solved in time domain and aplace domain. Start in Time Domain 35

20 Sec 3.7. System Transfer Function in Time and aplace Domains i S (t) i R (t) i C (t) i (t) R C v o (t) Figure 3.3: Time domain analysis. By writing the node equation with KC in time domain, we obtain where () ()+ ()+ () (3.72) () () () () () Z () Z 0 () + (0 ) (3.73) The system equation in time domain. () () + () + Z Apply the aplace transform to both sides of Eq.(3.74). 0 () + (0 ) (3.74) () () + ( () (0 )) + ()+ (0 ) (3.75) where (), [ ()] and () [ ()]. The system transfer function in aplace domain. Assume no initial condition, i.e., (0 ) lim 0 () 0and (0 ) lim 0 () 0. () () ( + + ) (3.76) Start in aplace Domain 36

21 ecture 3. Electrical Circuits R s I S (s) sc (0 ) V o (s) Cv( 0 ) o i s Figure 3.4: aplace domain analysis. Replace all the circuit elements by their KC equivalent circuits in aplace domain. () (0 ) (3.77) ()+ (0 ) () ()+ ()+ () () + () (0 ) μ + ()+ (0 ) (3.78) The system transfer function. Impulse Response () ()+ ()+ () (0 )0 (0 )0 () + ()+ () () ( + + ) () (3.79) Impulse response is the system output w.r.t an input signal of impulse function. It is also the transfer function of the system which can be used to find system poles and zeros. () (); () Assume no initial condition. () ( + + ) () ( + + ) (3.80)

22 Sec 3.7. System Transfer Function in Time and aplace Domains System poles and zeros are the roots of denominator and numerator of the system transfer function. Zeros, the roots of (). Poles, the roots of (). () () (), () () (3.8) The simplest way to obtain () is by analyzing the system impulse response in aplace domain. () () () () () (3.82) The time-domain expression of (), [ ()] depends on the nature of the system poles, which are the roots of denominator. et the system characteristic equation be equal to 0. Rewrite it as The roots are (3.83) with,, (3.84) and 0 ± r 2 ± ( 2 )2 4 (3.85) 4 2 r 2 ± ( 2 )2 Thenatureoftheroots(, 0 ) shall depend on the value of the critical expression. 2, ( 2 )2 ( 2 )2 ( ) (3.86) ( 2 )2 38

23 ecture 3. Electrical Circuits Impulse Response of RC Circuit vo(t) Underdamped Criticall Damped Overdamped t Figure 3.5: Comparison of system impulse responses in dierent cases. Traditionally, we define two more parameters. Neper frequency, 2 2 Resonant frequency 0 (3.87) 0, r (3.88) We can now rewrite the roots of the characteristics equation. and 0 ± q (3.89) Three distinct cases are possible with respect to and 0 of and 0. Overdamped, p is real, i.e., and 0 are two real, distinctand negative values. () can be factorized as follows. ( 0 )and ( 0 ) depending on the values 39

24 Sec 3.7. System Transfer Function in Time and aplace Domains Both and are real numbers. () ( )( 0 ) (3.90) By taking the inverse aplace transform, the response of () (withoutinitial condition) can be formulized as follows: () + 0 (3.9) Figure 3.5 depicts the () in the overdamped case. Underdamped, p is imaginary, i.e., and 0 are two complex conjugate poles. The damped resonant frequency, p is real. and 0 ± (3.92) () can be factorized as follows. and are complex conjugates. () ( ( + )) ( ( )) ( + ) + ( ) (3.93) (3.94) Taking inverse aplace transform, 0 () canbeobtained. Note that ( + ) and( ) are real number since and are complex conjugates. () (+) + ( ) + [ (cos + sin )+ (cos sin )](3.95) [( + )cos + ( )sin ] Figure 3.5 depicts the () in the underdamped case. Critically damped, p , i.e.,

25 ecture 3. Electrical Circuits j Critically damped - s P s P ' 0 Overdamped Underdamped +j -j Figure 3.6: ocations of poles and zeros. and 0. () ( ()) Taking inverse aplace Transform, 0 () canbeobtained. ( + ) 2 (3.96) 0 () + (3.97) Figure 3.5 depicts the 0 () in the case of critically damped. Summary The positions of poles determine the nature of System Response. and 0 ± 2 2 with 2 and 0! Response is overdamped if and, 0 are two distinct real numbers. Response is underdamped if and, 0 are complex conjugate numbers. Response is critically damped if and, 0 are real and the same. Thepositionsofsystempolesmovealongthepath(knownasRootocus) shown in the following diagram. Sinusoidal Response Sinusoidal response is the system output w.r.t an input signal of sinusoidal function. () cos() () or () [ ()] ( ) 4

26 Sec3.8.BodePlots For simplicity, we do not consider initial condition. () 2 ( )( ) 2 ( )( 0 )(2 + 2 ) ( ) + ( 0 (3.98) ) + + ( ) Instead of solving for the coecients analytically by hand (which is a practically impossible task), we shall deduce the time and frequency domain responses by reasoning. Time-Domain Response The terms ( )and( 0 ) stand for transient response. Both can be transformed into damped response in time with exponentially decaying envelop. where (3.99) 2 The term ( + ) ( ) represent the steady state response, which shall yield the time-domain signal 0 cos( ) tan ( ) + ( ) ( ) + ( ) cos()+ sin() 0 cos( ) (3.00) 3.8 Bode Plots A graph used to show the frequency response of an TI system. To plot the magnitude in decibels (db) and use a log scale for The log scale helps to compress a wide range of data. Both the magnitude and the phase of the transfer function versus the angular frequency System transfer function () can be written as a product of factors of the following items.. Constant factor 2. Poles or zeros at the origin, ± 3. Real poles or zeros, (+) ± 4. Complex-conjugate poles or zeros, ( ) ±,where is the damping 42

27 ecture 3. Electrical Circuits ratio and 0 () () (3.0) () The magnitude of () in db allows us to plot the factors individually and sum the results to obtain the complete plot. Given () asfollows, () 20log 0 () (3.02) () ( + )( + 2 ) ( + )( + 2 ) (3.03) () can be obtained by plotting each factor individually () 20log 0 () 20log 0 ( + )( + 2 ) ( + )( + 2 ) (3.04) 20log 0 +20log log log log Bode plot for each of the factors.. Constant factor,. Magnitude The magnitude 20 log 0 is a constant. Phase The phase is a constant and equal to 0 ( 0 )or±80 ( ± ) depending on whether is positive or negative, respectively. 2. Poles or zeros at the origin, ± Magnitude The magnitude is a straight line that intersects the axis (0dB) at and has a slope of ±20dB/decade. Phase The phase is a constant and equal to ±90. () () ± ( 2 ) ± (3.05) () 20log 0 ± ±20 log 0 (3.06) 43

28 Sec3.8.BodePlots 3. Real poles or zeros, (+) ± Poles or zeros locate at ]() ±(2) ±90 (3.07) () ( +) ± ± (3.08) () 20log 0 μq +() 2 ± ±0 log 0 +() 2 (3.09) Magnitude ow frequency response ( ; ): A horizontal line of 0dB () ±0 log 0 () 0 db (3.0) High frequency response ( À ; À ): A straight line of slope ±20dB/decade that intersects 0dB when () ±0 log 0 () 2 μ ±20 log 0 μ ±20 log 0 log 0 (3.) Corner frequency response () ±0 log 0 2±3dB (3.2) Phase () 0 ± (negative) or 425 ( positive ) (3.3) ]() ± ± tan (3.4) ow frequency: ]() 0 Corner frequency: ]() ±45 High frequency: ]() ±90 4. Complex conjugate poles or zeros, ( ) ± For simplicity, we consider only the case of a single pair of complex conjugate poles. 44

29 ecture 3. Electrical Circuits If the poles are repeated by times, all coordinates on the curves will be multiplied by. If we have zeros instead of poles, curves are mirror images through the axis. () (3.5) () 20 log 0 ³ log 0 ³ (3.6) Magnitude ow frequency response ( ; ): A horizontal line of 0dB High frequency response ( À ; À ): A straight line of slope ±40dB/decade that intersects 0dB when ³ 2 () 0 log ³ 2 ' 0 log ' 40 log 0 () μ 40 log 0 log 0 (3.7) Phase 0 ]() tan ]() 80 + tan (3.8) (3.9) Example 3.4 Consider the transfer function () ( + ) where k and F, express its frequency responses including both the magnitude and the phase responses with Bode Plot. () can be firstly written as follows. () + (3.20) The factor Magnitude is a constant and equal to 20 log dB. Phase is a constant and equal to 0 The factor ( + ) 45

30 Sec3.8.BodePlots Magnitude is a constant and equal to 0dB. Phase is around 0 Magnitude is a straight line of slope 20dB/decade. Phase is around 45 when Phase is around 90 when À 60 Bode Diagram Magnitude (db) Phase (deg) Frequency (rad/sec) Figure 3.7: Bode plot for the transfer function () ( + ) Example 3.5 Consider the transfer function () ( + ) where k and F, express its frequency response including both the magnitude and the phase responses with Bode Plot. () can be firstly written as follows. () + (3.2) The factor Magnitude is a constant and equal to 20 log dB Phase is a constant and equal to 0 The factor Magnitude is a straight line of slope 20dB/decade and intersects the -axis at Phase is

31 ecture 3. Electrical Circuits The factor ( + ) Magnitude is a constant and equal to 0dB. Phase is around 0 Magnitude is a straight line of slope 20dB/decade. Phase is around 45 when Phase is around 90 when À 0 Bode Diagram Magnitude (db) Phase (deg) Frequency (rad/sec) Figure 3.8: Bode plot for the system transfer function () ( + ) Example 3.6 Consider the transfer function () ( ) where k, Fand, express its frequency response including both the magnitude and the phase responses with Bode Plot. () can be firstly written as follows. () ( ) (3.22) The factor Magnitude is a constant and equal to 20 log 0 0dB Phase is a constant and equal to 0 The factor Magnitude is a straight line that intersects the axis (0dB) at andhas a slope of 20dB/decade. 47

32 Sec3.8.BodePlots Phase is a constant and equal to 90. The factor ( 2 + +) Magnitude is a horizontal line of 0dB Phase is tan À Magnitude is a straight line of slope 40dB/decade that intersects 0dB when Phase is 80 + tan Bode Diagram 50 Magnitude (db) Phase (deg) Frequency (rad/sec) Figure 3.9: Bode plot for the system transfer function () ( ) 48

### Single-Time-Constant (STC) Circuits This lecture is given as a background that will be needed to determine the frequency response of the amplifiers.

Single-Time-Constant (STC) Circuits This lecture is given as a background that will be needed to determine the frequency response of the amplifiers. Objectives To analyze and understand STC circuits with

### 2. The following diagram illustrates that voltage represents what physical dimension?

BioE 1310 - Exam 1 2/20/2018 Answer Sheet - Correct answer is A for all questions 1. A particular voltage divider with 10 V across it consists of two resistors in series. One resistor is 7 KΩ and the other

### Networks and Systems Prof. V. G. K. Murti Department of Electrical Engineering Indian Institute of Technology, Madras

Networks and Systems Prof. V. G. K. Murti Department of Electrical Engineering Indian Institute of Technology, Madras Lecture - 34 Network Theorems (1) Superposition Theorem Substitution Theorem The next

### Introduction to AC Circuits (Capacitors and Inductors)

Introduction to AC Circuits (Capacitors and Inductors) Amin Electronics and Electrical Communications Engineering Department (EECE) Cairo University elc.n102.eng@gmail.com http://scholar.cu.edu.eg/refky/

### Basic. Theory. ircuit. Charles A. Desoer. Ernest S. Kuh. and. McGraw-Hill Book Company

Basic C m ш ircuit Theory Charles A. Desoer and Ernest S. Kuh Department of Electrical Engineering and Computer Sciences University of California, Berkeley McGraw-Hill Book Company New York St. Louis San

### Some of the different forms of a signal, obtained by transformations, are shown in the figure. jwt e z. jwt z e

Transform methods Some of the different forms of a signal, obtained by transformations, are shown in the figure. X(s) X(t) L - L F - F jw s s jw X(jw) X*(t) F - F X*(jw) jwt e z jwt z e X(nT) Z - Z X(z)

### Dynamic circuits: Frequency domain analysis

Electronic Circuits 1 Dynamic circuits: Contents Free oscillation and natural frequency Transfer functions Frequency response Bode plots 1 System behaviour: overview 2 System behaviour : review solution

### ECE2262 Electric Circuits. Chapter 1: Basic Concepts. Overview of the material discussed in ENG 1450

ECE2262 Electric Circuits Chapter 1: Basic Concepts Overview of the material discussed in ENG 1450 1 Circuit Analysis 2 Lab -ECE 2262 3 LN - ECE 2262 Basic Quantities: Current, Voltage, Energy, Power The

### Electrical Engineering Fundamentals for Non-Electrical Engineers

Electrical Engineering Fundamentals for Non-Electrical Engineers by Brad Meyer, PE Contents Introduction... 3 Definitions... 3 Power Sources... 4 Series vs. Parallel... 9 Current Behavior at a Node...

### QUESTION BANK SUBJECT: NETWORK ANALYSIS (10ES34)

QUESTION BANK SUBJECT: NETWORK ANALYSIS (10ES34) NOTE: FOR NUMERICAL PROBLEMS FOR ALL UNITS EXCEPT UNIT 5 REFER THE E-BOOK ENGINEERING CIRCUIT ANALYSIS, 7 th EDITION HAYT AND KIMMERLY. PAGE NUMBERS OF

### Chapter 2. Engr228 Circuit Analysis. Dr Curtis Nelson

Chapter 2 Engr228 Circuit Analysis Dr Curtis Nelson Chapter 2 Objectives Understand symbols and behavior of the following circuit elements: Independent voltage and current sources; Dependent voltage and

### Physics 116A Notes Fall 2004

Physics 116A Notes Fall 2004 David E. Pellett Draft v.0.9 Notes Copyright 2004 David E. Pellett unless stated otherwise. References: Text for course: Fundamentals of Electrical Engineering, second edition,

### Refinements to Incremental Transistor Model

Refinements to Incremental Transistor Model This section presents modifications to the incremental models that account for non-ideal transistor behavior Incremental output port resistance Incremental changes

### Basic Electronics. Introductory Lecture Course for. Technology and Instrumentation in Particle Physics Chicago, Illinois June 9-14, 2011

Basic Electronics Introductory Lecture Course for Technology and Instrumentation in Particle Physics 2011 Chicago, Illinois June 9-14, 2011 Presented By Gary Drake Argonne National Laboratory drake@anl.gov

### Sinusoidal Steady State Analysis (AC Analysis) Part II

Sinusoidal Steady State Analysis (AC Analysis) Part II Amin Electronics and Electrical Communications Engineering Department (EECE) Cairo University elc.n102.eng@gmail.com http://scholar.cu.edu.eg/refky/

### Lecture #3. Review: Power

Lecture #3 OUTLINE Power calculations Circuit elements Voltage and current sources Electrical resistance (Ohm s law) Kirchhoff s laws Reading Chapter 2 Lecture 3, Slide 1 Review: Power If an element is

### AC Circuits. The Capacitor

The Capacitor Two conductors in close proximity (and electrically isolated from one another) form a capacitor. An electric field is produced by charge differences between the conductors. The capacitance

### EIT Review 1. FE/EIT Review. Circuits. John A. Camara, Electrical Engineering Reference Manual, 6 th edition, Professional Publications, Inc, 2002.

FE/EIT eview Circuits Instructor: uss Tatro eferences John A. Camara, Electrical Engineering eference Manual, 6 th edition, Professional Publications, Inc, 00. John A. Camara, Practice Problems for the

### Charge The most basic quantity in an electric circuit is the electric charge. Charge is an electrical property of the atomic particles of which matter

Basic Concepts of DC Circuits Introduction An electric circuit is an interconnection of electrical elements. Systems of Units 1 Charge The most basic quantity in an electric circuit is the electric charge.

### Notes on Electricity (Circuits)

A circuit is defined to be a collection of energy-givers (batteries) and energy-takers (resistors, light bulbs, radios, etc.) that form a closed path (or complete path) through which electrical current

### ELECTRONICS E # 1 FUNDAMENTALS 2/2/2011

FE Review 1 ELECTRONICS E # 1 FUNDAMENTALS Electric Charge 2 In an electric circuit it there is a conservation of charge. The net electric charge is constant. There are positive and negative charges. Like

### Notes for course EE1.1 Circuit Analysis TOPIC 10 2-PORT CIRCUITS

Objectives: Introduction Notes for course EE1.1 Circuit Analysis 4-5 Re-examination of 1-port sub-circuits Admittance parameters for -port circuits TOPIC 1 -PORT CIRCUITS Gain and port impedance from -port

### ECE2262 Electric Circuits. Chapter 6: Capacitance and Inductance

ECE2262 Electric Circuits Chapter 6: Capacitance and Inductance Capacitors Inductors Capacitor and Inductor Combinations Op-Amp Integrator and Op-Amp Differentiator 1 CAPACITANCE AND INDUCTANCE Introduces

### EE221 Circuits II. Chapter 14 Frequency Response

EE22 Circuits II Chapter 4 Frequency Response Frequency Response Chapter 4 4. Introduction 4.2 Transfer Function 4.3 Bode Plots 4.4 Series Resonance 4.5 Parallel Resonance 4.6 Passive Filters 4.7 Active

### ECE 255, Frequency Response

ECE 255, Frequency Response 19 April 2018 1 Introduction In this lecture, we address the frequency response of amplifiers. This was touched upon briefly in our previous lecture in Section 7.5 of the textbook.

### R. W. Erickson. Department of Electrical, Computer, and Energy Engineering University of Colorado, Boulder

R. W. Erickson Department of Electrical, Computer, and Energy Engineering University of Colorado, Boulder 8.1. Review of Bode plots Decibels Table 8.1. Expressing magnitudes in decibels G db = 0 log 10

### Chapter 10: Sinusoidal Steady-State Analysis

Chapter 10: Sinusoidal Steady-State Analysis 10.1 10.2 10.3 10.4 10.5 10.6 10.9 Basic Approach Nodal Analysis Mesh Analysis Superposition Theorem Source Transformation Thevenin & Norton Equivalent Circuits

Chapter 4 Sinusoidal Steady-State Analysis In this unit, we consider circuits in which the sources are sinusoidal in nature. The review section of this unit covers most of section 9.1 9.9 of the text.

### Index. Index. More information. in this web service Cambridge University Press

A-type elements, 4 7, 18, 31, 168, 198, 202, 219, 220, 222, 225 A-type variables. See Across variable ac current, 172, 251 ac induction motor, 251 Acceleration rotational, 30 translational, 16 Accumulator,

### IMPERIAL COLLEGE OF SCIENCE, TECHNOLOGY AND MEDICINE UNIVERSITY OF LONDON DEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING EXAMINATIONS 2010

Paper Number(s): E1.1 IMPERIAL COLLEGE OF SCIENCE, TECHNOLOGY AND MEDICINE UNIVERSITY OF LONDON DEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING EXAMINATIONS 2010 EEE/ISE PART I: MEng, BEng and ACGI

### Review of Linear Time-Invariant Network Analysis

D1 APPENDIX D Review of Linear Time-Invariant Network Analysis Consider a network with input x(t) and output y(t) as shown in Figure D-1. If an input x 1 (t) produces an output y 1 (t), and an input x

### Sinusoidal Steady State Analysis (AC Analysis) Part I

Sinusoidal Steady State Analysis (AC Analysis) Part I Amin Electronics and Electrical Communications Engineering Department (EECE) Cairo University elc.n102.eng@gmail.com http://scholar.cu.edu.eg/refky/

### NETWORK ANALYSIS WITH APPLICATIONS

NETWORK ANALYSIS WITH APPLICATIONS Third Edition William D. Stanley Old Dominion University Prentice Hall Upper Saddle River, New Jersey I Columbus, Ohio CONTENTS 1 BASIC CIRCUIT LAWS 1 1-1 General Plan

### Sophomore Physics Laboratory (PH005/105)

CALIFORNIA INSTITUTE OF TECHNOLOGY PHYSICS MATHEMATICS AND ASTRONOMY DIVISION Sophomore Physics Laboratory (PH5/15) Analog Electronics Active Filters Copyright c Virgínio de Oliveira Sannibale, 23 (Revision

### Circuit Analysis. by John M. Santiago, Jr., PhD FOR. Professor of Electrical and Systems Engineering, Colonel (Ret) USAF. A Wiley Brand FOR-

Circuit Analysis FOR A Wiley Brand by John M. Santiago, Jr., PhD Professor of Electrical and Systems Engineering, Colonel (Ret) USAF FOR- A Wiley Brand Table of Contents. ' : '" '! " ' ' '... ',. 1 Introduction

### Prof. Anyes Taffard. Physics 120/220. Voltage Divider Capacitor RC circuits

Prof. Anyes Taffard Physics 120/220 Voltage Divider Capacitor RC circuits Voltage Divider The figure is called a voltage divider. It s one of the most useful and important circuit elements we will encounter.

### BME/ISE 3511 Bioelectronics - Test Six Course Notes Fall 2016

BME/ISE 35 Bioelectronics - Test Six ourse Notes Fall 06 Alternating urrent apacitive & Inductive Reactance and omplex Impedance R & R ircuit Analyses (D Transients, Time onstants, Steady State) Electrical

### Learnabout Electronics - AC Theory

Learnabout Electronics - AC Theory Facts & Formulae for AC Theory www.learnabout-electronics.org Contents AC Wave Values... 2 Capacitance... 2 Charge on a Capacitor... 2 Total Capacitance... 2 Inductance...

### EE221 Circuits II. Chapter 14 Frequency Response

EE22 Circuits II Chapter 4 Frequency Response Frequency Response Chapter 4 4. Introduction 4.2 Transfer Function 4.3 Bode Plots 4.4 Series Resonance 4.5 Parallel Resonance 4.6 Passive Filters 4.7 Active

### First and Second Order Circuits. Claudio Talarico, Gonzaga University Spring 2015

First and Second Order Circuits Claudio Talarico, Gonzaga University Spring 2015 Capacitors and Inductors intuition: bucket of charge q = Cv i = C dv dt Resist change of voltage DC open circuit Store voltage

### To find the step response of an RC circuit

To find the step response of an RC circuit v( t) v( ) [ v( t) v( )] e tt The time constant = RC The final capacitor voltage v() The initial capacitor voltage v(t ) To find the step response of an RL circuit

### 2. Basic Components and Electrical Circuits

1 2. Basic Components and Electrical Circuits 2.1 Units and Scales The International System of Units (SI) defines 6 principal units from which the units of all other physical quantities can be derived

### Basic Electrical Circuits Analysis ECE 221

Basic Electrical Circuits Analysis ECE 221 PhD. Khodr Saaifan http://trsys.faculty.jacobs-university.de k.saaifan@jacobs-university.de 1 2 Reference: Electric Circuits, 8th Edition James W. Nilsson, and

### EE 3120 Electric Energy Systems Study Guide for Prerequisite Test Wednesday, Jan 18, pm, Room TBA

EE 3120 Electric Energy Systems Study Guide for Prerequisite Test Wednesday, Jan 18, 2006 6-7 pm, Room TBA First retrieve your EE2110 final and other course papers and notes! The test will be closed book

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

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

### Circuits Advanced Topics by Dr. Colton (Fall 2016)

ircuits Advanced Topics by Dr. olton (Fall 06). Time dependence of general and L problems General and L problems can always be cast into first order ODEs. You can solve these via the particular solution

### ENGG 225. David Ng. Winter January 9, Circuits, Currents, and Voltages... 5

ENGG 225 David Ng Winter 2017 Contents 1 January 9, 2017 5 1.1 Circuits, Currents, and Voltages.................... 5 2 January 11, 2017 6 2.1 Ideal Basic Circuit Elements....................... 6 3 January

### .. Use of non-programmable scientific calculator is permitted.

This question paper contains 8+3 printed pages] Roll No. S. No. of Question Paper 7981 Cnique Paper Code 1\;ame of the Paper ~ame of the Course Semester Duration : 3 Hours 2511102 Circuit Analysis [DC-1.1]

### EE348L Lecture 1. EE348L Lecture 1. Complex Numbers, KCL, KVL, Impedance,Steady State Sinusoidal Analysis. Motivation

EE348L Lecture 1 Complex Numbers, KCL, KVL, Impedance,Steady State Sinusoidal Analysis 1 EE348L Lecture 1 Motivation Example CMOS 10Gb/s amplifier Differential in,differential out, 5 stage dccoupled,broadband

### ELEC 250: LINEAR CIRCUITS I COURSE OVERHEADS. These overheads are adapted from the Elec 250 Course Pack developed by Dr. Fayez Guibaly.

Elec 250: Linear Circuits I 5/4/08 ELEC 250: LINEAR CIRCUITS I COURSE OVERHEADS These overheads are adapted from the Elec 250 Course Pack developed by Dr. Fayez Guibaly. S.W. Neville Elec 250: Linear Circuits

### Fundamental of Electrical circuits

Fundamental of Electrical circuits 1 Course Description: Electrical units and definitions: Voltage, current, power, energy, circuit elements: resistors, capacitors, inductors, independent and dependent

### 1 Phasors and Alternating Currents

Physics 4 Chapter : Alternating Current 0/5 Phasors and Alternating Currents alternating current: current that varies sinusoidally with time ac source: any device that supplies a sinusoidally varying potential

### Basic Electronics. Introductory Lecture Course for. Technology and Instrumentation in Particle Physics Chicago, Illinois June 9-14, 2011

Basic Electronics Introductory Lecture Course for Technology and Instrumentation in Particle Physics 2011 Chicago, Illinois June 9-14, 2011 Presented By Gary Drake Argonne National Laboratory Session 2

### Electric Current. Note: Current has polarity. EECS 42, Spring 2005 Week 2a 1

Electric Current Definition: rate of positive charge flow Symbol: i Units: Coulombs per second Amperes (A) i = dq/dt where q = charge (in Coulombs), t = time (in seconds) Note: Current has polarity. EECS

### Notes on Electricity (Circuits)

A circuit is defined to be a collection of energy-givers (active elements) and energy-takers (passive elements) that form a closed path (or complete path) through which electrical current can flow. The

### BFF1303: ELECTRICAL / ELECTRONICS ENGINEERING

BFF1303: ELECTRICAL / ELECTRONICS ENGINEERING Introduction Ismail Mohd Khairuddin, Zulkifil Md Yusof Faculty of Manufacturing Engineering Universiti Malaysia Pahang Introduction BFF1303 ELECTRICAL/ELECTRONICS

### Electromagnetic Oscillations and Alternating Current. 1. Electromagnetic oscillations and LC circuit 2. Alternating Current 3.

Electromagnetic Oscillations and Alternating Current 1. Electromagnetic oscillations and LC circuit 2. Alternating Current 3. RLC circuit in AC 1 RL and RC circuits RL RC Charging Discharging I = emf R

### GATE 20 Years. Contents. Chapters Topics Page No.

GATE 0 Years Contents Chapters Topics Page No. Chapter- Chapter- Chapter- Chapter-4 Chapter-5 GATE Syllabus for this Chapter Topic elated to Syllabus Previous 0-Years GATE Questions Previous 0-Years GATE

### a + b Time Domain i(τ)dτ.

R, C, and L Elements and their v and i relationships We deal with three essential elements in circuit analysis: Resistance R Capacitance C Inductance L Their v and i relationships are summarized below.

### DEPARTMENT OF COMPUTER ENGINEERING UNIVERSITY OF LAHORE

DEPARTMENT OF COMPUTER ENGINEERING UNIVERSITY OF LAHORE NAME. Section 1 2 3 UNIVERSITY OF LAHORE Department of Computer engineering Linear Circuit Analysis Laboratory Manual 2 Compiled by Engr. Ahmad Bilal

### ENERGY AND TIME CONSTANTS IN RC CIRCUITS By: Iwana Loveu Student No Lab Section: 0003 Date: February 8, 2004

ENERGY AND TIME CONSTANTS IN RC CIRCUITS By: Iwana Loveu Student No. 416 614 5543 Lab Section: 0003 Date: February 8, 2004 Abstract: Two charged conductors consisting of equal and opposite charges forms

### Handout 11: AC circuit. AC generator

Handout : AC circuit AC generator Figure compares the voltage across the directcurrent (DC) generator and that across the alternatingcurrent (AC) generator For DC generator, the voltage is constant For

### Chapter 8: Converter Transfer Functions

Chapter 8. Converter Transfer Functions 8.1. Review of Bode plots 8.1.1. Single pole response 8.1.2. Single zero response 8.1.3. Right half-plane zero 8.1.4. Frequency inversion 8.1.5. Combinations 8.1.6.

### 8.1.6 Quadratic pole response: resonance

8.1.6 Quadratic pole response: resonance Example G(s)= v (s) v 1 (s) = 1 1+s L R + s LC L + Second-order denominator, of the form 1+a 1 s + a s v 1 (s) + C R Two-pole low-pass filter example v (s) with

### Some Important Electrical Units

Some Important Electrical Units Quantity Unit Symbol Current Charge Voltage Resistance Power Ampere Coulomb Volt Ohm Watt A C V W W These derived units are based on fundamental units from the meterkilogram-second

### ECE 202 Fall 2013 Final Exam

ECE 202 Fall 2013 Final Exam December 12, 2013 Circle your division: Division 0101: Furgason (8:30 am) Division 0201: Bermel (9:30 am) Name (Last, First) Purdue ID # There are 18 multiple choice problems

### CS 436 HCI Technology Basic Electricity/Electronics Review

CS 436 HCI Technology Basic Electricity/Electronics Review *Copyright 1997-2008, Perry R. Cook, Princeton University August 27, 2008 1 Basic Quantities and Units 1.1 Charge Number of electrons or units

### FE Review 2/2/2011. Electric Charge. Electric Energy ELECTRONICS # 1 FUNDAMENTALS

FE eview ELECONICS # FUNDAMENALS Electric Charge 2 In an electric circuit there is a conservation of charge. he net electric charge is constant. here are positive and negative charges. Like charges repel

### Frequency Dependent Aspects of Op-amps

Frequency Dependent Aspects of Op-amps Frequency dependent feedback circuits The arguments that lead to expressions describing the circuit gain of inverting and non-inverting amplifier circuits with resistive

### Electric Circuit Theory

Electric Circuit Theory Nam Ki Min nkmin@korea.ac.kr 010-9419-2320 Chapter 11 Sinusoidal Steady-State Analysis Nam Ki Min nkmin@korea.ac.kr 010-9419-2320 Contents and Objectives 3 Chapter Contents 11.1

### Conventional Paper-I Part A. 1. (a) Define intrinsic wave impedance for a medium and derive the equation for intrinsic vy

EE-Conventional Paper-I IES-01 www.gateforum.com Conventional Paper-I-01 Part A 1. (a) Define intrinsic wave impedance for a medium and derive the equation for intrinsic vy impedance for a lossy dielectric

### ECE2262 Electric Circuits. Chapter 6: Capacitance and Inductance

ECE2262 Electric Circuits Chapter 6: Capacitance and Inductance Capacitors Inductors Capacitor and Inductor Combinations 1 CAPACITANCE AND INDUCTANCE Introduces two passive, energy storing devices: Capacitors

### EXP. NO. 3 Power on (resistive inductive & capacitive) load Series connection

OBJECT: To examine the power distribution on (R, L, C) series circuit. APPARATUS 1-signal function generator 2- Oscilloscope, A.V.O meter 3- Resisters & inductor &capacitor THEORY the following form for

### 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

### PHYSICS FORM 5 ELECTRICAL QUANTITES

QUANTITY SYMBOL UNIT SYMBOL Current I Amperes A Voltage (P.D.) V Volts V Resistance R Ohm Ω Charge (electric) Q Coulomb C Power P Watt W Energy E Joule J Time T seconds s Quantity of a Charge, Q Q = It

### CHAPTER ONE. 1.1 International System of Units and scientific notation : Basic Units: Quantity Basic unit Symbol as shown in table 1

CHAPTER ONE 1.1 International System of Units and scientific notation : 1.1.1 Basic Units: Quantity Basic unit Symbol as shown in table 1 Table 1 1.1.2 Some scientific notations : as shown in table 2 Table

### Sinusoids and Phasors

CHAPTER 9 Sinusoids and Phasors We now begins the analysis of circuits in which the voltage or current sources are time-varying. In this chapter, we are particularly interested in sinusoidally time-varying

### RC, RL, and LCR Circuits

RC, RL, and LCR Circuits EK307 Lab Note: This is a two week lab. Most students complete part A in week one and part B in week two. Introduction: Inductors and capacitors are energy storage devices. They

### NETWORK ANALYSIS ( ) 2012 pattern

PRACTICAL WORK BOOK For Academic Session 0 NETWORK ANALYSIS ( 0347 ) 0 pattern For S.E. (Electrical Engineering) Department of Electrical Engineering (University of Pune) SHREE RAMCHANDRA COLLEGE OF ENGG.

### 18 - ELECTROMAGNETIC INDUCTION AND ALTERNATING CURRENTS ( Answers at the end of all questions ) Page 1

( Answers at the end of all questions ) Page ) The self inductance of the motor of an electric fan is 0 H. In order to impart maximum power at 50 Hz, it should be connected to a capacitance of 8 µ F (

Sinusoidal Steady-State Analysis Almost all electrical systems, whether signal or power, operate with alternating currents and voltages. We have seen that when any circuit is disturbed (switched on or

### Electric Circuit Theory

Electric Circuit Theory Nam Ki Min nkmin@korea.ac.kr 010-9419-2320 Chapter 8 Natural and Step Responses of RLC Circuits Nam Ki Min nkmin@korea.ac.kr 010-9419-2320 8.1 Introduction to the Natural Response

### ES250: Electrical Science. HW1: Electric Circuit Variables, Elements and Kirchhoff s Laws

ES250: Electrical Science HW1: Electric Circuit Variables, Elements and Kirchhoff s Laws Introduction Engineers use electric circuits to solve problems that are important to modern society, such as: 1.

### Source-Free RC Circuit

First Order Circuits Source-Free RC Circuit Initial charge on capacitor q = Cv(0) so that voltage at time 0 is v(0). What is v(t)? Prof Carruthers (ECE @ BU) EK307 Notes Summer 2018 150 / 264 First Order

### Electric Circuit Theory

Electric Circuit Theory Nam Ki Min nkmin@korea.ac.kr 010-9419-2320 Chapter 18 Two-Port Circuits Nam Ki Min nkmin@korea.ac.kr 010-9419-2320 Contents and Objectives 3 Chapter Contents 18.1 The Terminal Equations

### 11. AC Circuit Power Analysis

. AC Circuit Power Analysis Often an integral part of circuit analysis is the determination of either power delivered or power absorbed (or both). In this chapter First, we begin by considering instantaneous

### Solving a RLC Circuit using Convolution with DERIVE for Windows

Solving a RLC Circuit using Convolution with DERIVE for Windows Michel Beaudin École de technologie supérieure, rue Notre-Dame Ouest Montréal (Québec) Canada, H3C K3 mbeaudin@seg.etsmtl.ca - Introduction

### EE313 Fall 2013 Exam #1 (100 pts) Thursday, September 26, 2013 Name. 1) [6 pts] Convert the following time-domain circuit to the RMS Phasor Domain.

Name If you have any questions ask them. Remember to include all units on your answers (V, A, etc). Clearly indicate your answers. All angles must be in the range 0 to +180 or 0 to 180 degrees. 1) [6 pts]

### Gen. Phys. II Exam 2 - Chs. 21,22,23 - Circuits, Magnetism, EM Induction Mar. 5, 2018

Gen. Phys. II Exam 2 - Chs. 21,22,23 - Circuits, Magnetism, EM Induction Mar. 5, 2018 Rec. Time Name For full credit, make your work clear. Show formulas used, essential steps, and results with correct

### REUNotes08-CircuitBasics May 28, 2008

Chapter One Circuits (... introduction here... ) 1.1 CIRCUIT BASICS Objects may possess a property known as electric charge. By convention, an electron has one negative charge ( 1) and a proton has one

### ECE 2100 Circuit Analysis

ECE 2100 Circuit Analysis Lesson 3 Chapter 2 Ohm s Law Network Topology: nodes, branches, and loops Daniel M. Litynski, Ph.D. http://homepages.wmich.edu/~dlitynsk/ esistance ESISTANCE = Physical property

### ENGR 2405 Class No Electric Circuits I

ENGR 2405 Class No. 48056 Electric Circuits I Dr. R. Williams Ph.D. rube.williams@hccs.edu Electric Circuit An electric circuit is an interconnec9on of electrical elements Charge Charge is an electrical

### Direct Current (DC) Circuits

Direct Current (DC) Circuits NOTE: There are short answer analysis questions in the Participation section the informal lab report. emember to include these answers in your lab notebook as they will be

### Electrical Eng. fundamental Lecture 1

Electrical Eng. fundamental Lecture 1 Contact details: h-elhelw@staffs.ac.uk Introduction Electrical systems pervade our lives; they are found in home, school, workplaces, factories,

### Fundamentals of Engineering Exam Review Electromagnetic Physics

Dr. Gregory J. Mazzaro Spring 2018 Fundamentals of Engineering Exam Review Electromagnetic Physics (currently 5-7% of FE exam) THE CITADEL, THE MILITARY COLLEGE OF SOUTH CAROLINA 171 Moultrie Street, Charleston,

### EIT Review. Electrical Circuits DC Circuits. Lecturer: Russ Tatro. Presented by Tau Beta Pi The Engineering Honor Society 10/3/2006 1

EIT Review Electrical Circuits DC Circuits Lecturer: Russ Tatro Presented by Tau Beta Pi The Engineering Honor Society 10/3/2006 1 Session Outline Basic Concepts Basic Laws Methods of Analysis Circuit

### Energy Storage Elements: Capacitors and Inductors

CHAPTER 6 Energy Storage Elements: Capacitors and Inductors To this point in our study of electronic circuits, time has not been important. The analysis and designs we have performed so far have been static,

### Lecture 21. Resonance and power in AC circuits. Physics 212 Lecture 21, Slide 1

Physics 1 ecture 1 esonance and power in A circuits Physics 1 ecture 1, Slide 1 I max X X = w I max X w e max I max X X = 1/w I max I max I max X e max = I max Z I max I max (X -X ) f X -X Physics 1 ecture