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BME372New

Lecture#1ElectricalBasics

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Circuit Analysis •  Circuit Elements

–  Passive Devices –  Active Devices

•  Circuit Analysis Tools –  Ohms Law –  Kirchhoff’s Law –  Impedances –  Mesh and Nodal Analysis –  Superposition

•  Examples

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Characterize Circuit Elements

•  Passive Devices: dissipates or stores energy – Linear – Non-linear

•  Active Devices: Provider of energy or supports power gain – Linear – Non-linear

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Circuit Elements – Linear Passive Devices

•  Linear: supports a linear relationship between the voltage across the device and the current through it. – Resistor: supports a voltage and current which

are proportional, device dissipates heat, and is governed by Ohm’s Law, units: resistance or ohms Ω

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resistor with theassociated resistance theof value theis R where)( (t)RItV RR =

6

Circuit Elements – Linear Passive Devices

– Capacitor: supports a current which is proportional to its changing voltage, device stores an electric field between its plates, and is governed by Gauss’ Law, units: capacitance or farads, f

capacitor with theassociated ecapacitanc theof value theis C where)( dt(t)dVCtI C

C =

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Circuit Elements – Linear Passive Devices

– Inductor: supports a voltage which is proportional to its changing current, device stores a magnetic field through its coils and is governed by Faraday’s Law, units: inductance or henries, h

inductor with theassociated inductance theof value theis L where)( dt(t)dILtV L

L =

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Circuit Elements - Passive Devices Continued

•  Non-linear: supports a non-linear relationship among the currents and voltages associated with it – Diodes: supports current flowing through it in

only one direction

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Circuit Elements - Active Devices •  Linear

–  Sources •  Voltage Source: a device which supplies a voltage

as a function of time at its terminals which is independent of the current flowing through it, units: Volts –  DC, AC, Pulse Trains, Square Waves, Triangular

Waves

•  Current Source: a device which supplies a current as a function of time out of its terminals which is independent of the voltage across it, units: Amperes –  DC, AC, Pulse Trains, Square Waves, Triangular

Waves

+

-- Vab(t)

a

b

Iab(t)

a

b

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Circuit Elements - Active Devices Continued

– Ideal Sources vs Practical Sources •  An ideal source is one which only depends on the

type of source (i.e., current or voltage) •  A practical source is one where other circuit

elements are associated with it (e.g., resistance, inductance, etc. ) – A practical voltage source consists of an ideal voltage

source connected in series with passive circuit elements such as a resistor

– A practical current source consists of an ideal current source connected in parallel with passive circuit elements such as a resistor

+

--

Vab(t)

a

b

Rs

Vs(t)

Is(t)

a

b

Rs Iab(t)

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Circuit Elements - Active Devices Continued

– Independent vs Dependent Sources •  An independent source is one where the output

voltage or current is not dependent on other voltages or currents in the device •  A dependent source is one where the output voltage

or current is a function of another voltage or current in the device (e.g., a BJT transistor may be viewed as having an output current source which is dependent on the input current)

+

-- Vab(t)

a

b

vs vo vi

Ro

Ri

Rs

RL Avovi

io ii

+ +

- -

+

-

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Circuit Elements - Active Devices Continued

•  Non-Linear – Transistors: three or more terminal devices

where its output voltage and current characteristics are a function on its input voltage and/or current characteristics, several types BJT, FETs, etc.

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Circuits •  A circuit is a grouping of passive and active

elements •  Elements may be connecting is series,

parallel or combinations of both

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Circuits Continued •  Series Connection: Same current through the devices

–  The resultant resistance of two or more Resistors connected in series is the sum of the resistance

–  The resultant inductance of two or more Inductors connected in series is the sum of the inductances

–  The resultant capacitance of two or more Capacitors connected in series is the inverse of the sum of the inverse capacitances

–  The resultant voltage of two or more Ideal Voltage Sources connected in series is the sum of the voltages

–  Two of more Ideal Current sources can not be connected in series

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Series Circuits

•  Resistors

•  Inductors

•  Capacitors

R1 R2

RT = R1+ R2

L1 L2

LT = L1+ L2

C1 C2

21

21

21

111

CCCC

CC

CT +=+

=

T

bcabac

IRRRIIRIRVVV

=+=+=+=

)( 21

21

dtdILdt

dILL

dtdILdt

dILVVV

T

bcabac

=+=

+=+=

)( 21

21

∫∫

∫∫=+=

+=+=

IdtCIdtCC

IdtCIdtCVVV

T

bcabac

1)11(

11

21

21

a b c

a b c

a b c

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Series Circuits

•  Resistors 20Ω 50Ω

RT = 20+ 50 = 70Ω

70)5020(5020

IIIIVVV bcabac

=+=+=+=

a b c

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Series Circuits

•  Inductors 25h 100h

LT =25+ 100 = 125h

dtdI

dtdI

dtdI

dtdIVVV bcabac

125)10025(

10025

=+=

+=+=

a b c

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Series Circuits

•  Capacitors 5f 10f

fCT 33.3310

1550

105105

101

511 ===

+×=

+=

∫∫

∫∫=+=

+=+=

IdtIdt

IdtIdtVVV bcabac

103)

101

51(

101

51

a b c

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Series Circuits

•  Capacitors 10f 10f

fCT 5210

20100

10101010

101

1011 ===

+×=

+=

∫∫∫

∫∫==+=

+=+=

IdtIdtIdt

IdtIdtVVV bcabac

51

102)

101

101(

101

101

a b c

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Circuits Continued •  Parallel Connection: Same Voltage across the devices

–  The resultant resistance of two or more Resistors connected in parallel is the inverse of the sum of the inverse resistances

–  The resultant inductance of two or more Inductors connected in parallel is the inverse of the sum of the inverse inductances

–  The resultant capacitance of two or more Capacitors connected in parallel is the sum of the capacitances

–  The resultant current of two or more Ideal Current Sources connected in parallel is the sum of the currents

–  Two of more Ideal Voltage sources can not be connected in parallel

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Parallel Circuits

•  Resistors

•  Inductors

•  Capacitors

R1 R2

CT = C1+ C2

L1 L2

C1 C2

21

21

21

111

LLLL

LL

LT +=+

=

21

21

21

111

RRRR

RR

RT +=+

=

T

ab

RVVRR

RV

RVIII

=+=

+=+=

)11(21

2121

∫∫

∫∫=+=

+=+=

VdtLVdtLL

VdtLVdtLIII

T

ab

1)11(

11

21

2121

dtdVCdt

dVCC

dtdVCdt

dVCIII

T

ab

=+=

+=+=

)( 21

2121

a

b

Iab

I1 I2

Iab

I1 I2

a

b

a

b

Iab

I1 I2

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Combining Circuit Elements Kirchhoff’s Laws

•  Kirchhoff Voltage Law: The sum of the voltages around a loop must equal zero

•  Kirchhoff Current Law: The sum of the currents leaving (entering) a node must equal zero

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Combining Rs, Ls and Cs

•  We can use KVL or KCL to write and solve an equation associated with the circuit. – Example: a series Resistive Circuit

V(t) = I(t)R1 + I(t)R2

V(t) = I(t)(R1 + R2)

R1+

-- V(t)

I(t) R2

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Combining Rs, Ls, and Cs

•  We can use KVL or KCL to write and solve an equation associated with the circuit. – Example: a series Resistive Circuit

I1(t)+I2(t)+I3(t)=0 I2(t)=-V(t)/R1; I3(t)=-V(t)/R2; I1(t) - V(t)/R1 - V(t)/R2 =0 I1(t)=V(t)/R1 +V(t)/R2=V(t)[1/R1 +1/R2]

24

R1V(t) R2+

-

I1(t)

I2(t)

I3(t)

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Impedances •  Our special case, signals of the form: V(t) or I(t) =Aest

where s can be a real or complex number

•  This is only one portion of the solution and does not include the transient response.

t

t

tttt

tt

tt

t

etIA

eA

eeeAe

dtAedtdAeAee

dttIdttdItIe

5

5

5555

55

55

5

3610)(;

3610

)36(

)5

55510(10

551010

)(2.1)(510)(10

==

=

+×+=

++=

++=

∫

∫

t

t

AetI

fChLRetV

dttICdt

tdILRtItV

5

1115

111

)(: trysLet'

2.;5;10;10)(:assume sLet'

)(1)()()(

=

==Ω==

++= ∫

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Impedances •  Since the derivative [and integral] of Aest =

sAest [=(1/s)Aest], we can define the impedance of a circuit element as Z(s)=V/I where Z is only a function of s since the time dependency drops out.

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Impedances

RAe

RAeIVsZ

RAetRItV Ae tI

sCsCAeAe

IVsZ

CsAedt

tdVCtI Ae tV

sLAe

sLAeIVsZ

LsAedt

tdILtV Ae tI

st

st

stst

st

st

stst

st

st

stst

===

===

===

===

===

===

)(

;)()( then ;)( assume slet' resistor, aFor

1)(

;)()( then ;)( assume slet' capacitor, aFor

)(

;)()( then ;)( assume slet' inductor,an For

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Impedances •  What about signals of the type: cos(ωt+θ);

•  Recall Euler’s formula e jθ = cos θ +j sin θ where j is the imaginary number =

•  A special case of our special case is for sinusoidal inputs, where s=jω

1−

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Sinusoidal Steady State Continued

Real Numbers

Imaginary Numbers

θ

Inductor Imaginary Numbers

θ

Capacitor Imaginary Numbers

θ

Resistor

− For an inductor, ZL = jωL⇒ωL∠π2

− For a capacitor, ZC = 1jωC

⇒ 1ωC

∠− π2

.

− For a resistor, ZR = R⇒ R∠0

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Sinusoidal Steady State Continued For V(t)=Acosωt, using phasor notation for V(t) è and I(t) è I, our equation can be re-written:

IIIV1

11

111

10

tionrepresentaPhasor toConverting

)(1)()()(

CjLjRA

dttICdt

tdILRtItV

ωω ++=∠=

++= ∫

V = A∠0

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Sinusoidal Steady State Continued

]))1(

[tancos()1(

)(

tion,representa time theback to Converting

])1(

[tan)1()1(

01

0

1

11

1

2

11

21

1

11

1

2

11

21

111

111

RC

Lt

CLR

AtI

RC

L

CLR

A

CLjR

A

CjLjR

A

ωω

ω

ωω

ωω

ωωω

ωω

ω

−−

−+=

−−∠

−+=

−+

∠=++

∠=

−

−I

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Bode Plots

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I = A

R12 + (ωL1 −

1ωC1

)2

∠− tan−1[(ωL1 −

1ωC1

)

R1

]

Magnitude of I ⇒ I = A

R12 + (ωL1 −

1ωC1

)2

Angle of I ⇒∠I = ∠− tan−1[(ωL1 −

1ωC1

)

R1

]

•  Plotting the magnitude and phase versus frequency.

0

0.2

0.4

0.6

0.8

1

1.E-02 1.E+00 1.E+02 1.E+04 1.E+06Radians

Magnitude

-2-1.5-1

-0.50

0.51

1.52

1.E-02 1.E+00 1.E+02 1.E+04 1.E+06

Radians

Angle

Bode Plots

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•  Easy way to plot the magnitude. I = A

R1 + jωL1 +1jωC1

First evaluate at the initial condition ω=0

I |ω=0 = A

R1 + jωL1 +1jωC1

|ω=0→A

R1 + j0L1 +1j0C1

→Determine the dominate terms

A1j0C1

→ j A∞→ 0∠π

2

Next evaluate at the final condition ω →∞

I |ω→∞ = A

R1 + jωL1 +1jωC1

|ω→∞→A

R1 + j∞L1 +1j∞C1

→Determine the dominate terms

Aj∞L1

→− j A∞→ 0∠− π

2

Find an interesting point; for this example choose ω = ω0 =1

LC since this is when the imaginary part

of the deminator is zero.

I |ω0 =

1

LC

= A

R1 + jωL1 +1jωC1

|ω0=

1

L1C1

= A

R1 + j(1L1C1

L1 −1

1L1C1

C1

)= A

R1 + j(L1

C1

− 1C1

L1

)

= AR1

= AR1

∠0

Bode Plots

•  With the three point, plots can be made

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ω

Magnitude

0

AR1

0

ω0ω

Phase π2

0

−π2

ω0

35

Homework R31k

R11k a

b

Find the total resistance

R2R45k5k

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Homework Find the total resistance Rab where R1 = 3Ω, R2 = 6Ω, R3 = 12Ω, R4 = 4Ω, R5 = 2Ω, R6 = 2Ω, R7 = 4Ω, R8 = 4Ω

R1

R7

R2

R8

R3

R4

R6

R5

a

b

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Homework Find the total resistance Rab where R1 = 2Ω, R2 = 4Ω, R3 = 2Ω, R4 = 2Ω, R5 = 2Ω, R6 = 4Ω,

R6

R1

R2

R3

R4

R5

a

b

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Homework Find the total resistance Rab for this infinite resistive network

R

R

R

R

R

R

a

b

R

R

R

R

R

R

R

R

R

R

R

continues to infinity

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Homework C3

1f

C1

1f a

b

Find the total capacitance

C2C45f5f

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Homework

Find and plot the impedance Zab(jω) as a function of frequency. Use Matlab to calculate the Bode plot.

R=1 C=1

L=1

b

a

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Homework

R=1 C=1 L=1

b

a

Find and plot the impedance Zab(jω) as a function of frequency. Use Matlab to calculate the Bode plot.

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Homework R25k

R11k

C11n

C25n

a

b

Find and plot the impedance Zab(jω) as function of ω. Use Matlab to calculate the Bode plot.

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