Lecture 1: Course Overview and Introduction to Phasors

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Lecture 1 Course Overview and Introduction to
Phasors

• Prof. Niknejad

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EECS 105 Course Overview

• Phasors and Frequency Domain (2 weeks)
• Integrated Passives (R, C, L) (2 weeks)
• MOSFET Physics/Model (1 week)
• PN Junction / BJT Physics/Model (1.5 weeks)
• Single Stage Amplifiers (2 weeks)
• Feedback and Diff Amps (1 week)
• Freq Resp of Single Stage Amps (1 week)
• Multistage Amps (2.5 weeks)
• Freq Resp of Multistage Amps (1 week)

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EECS 105 in the Grand Scheme

• Example Cell Phone

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Transistors are Bricks

• Transistors are the building blocks (bricks) of
  the modern electronic world
• Focus of course
• Understand device physics
• Build analog circuits
• Learn electronic prototyping and measurement
• Learn simulations tools such as SPICE

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SPICE
Example netlist Q1 1 2 0 npnmod R1 1 3 1k Vdd 3
0 3v .tran 1u 100u
SPICE
stimulus
netlist
response

• SPICE Simulation Program with IC Emphasis
• Invented at Berkeley (released in 1972)
• .DC Find the DC operating point of a circuit
• .TRAN Solve the transient response of a circuit
  (solve a system of generally non-linear ordinary
  differential equations via adaptive time-step
  solver)
• .AC Find steady-state response of circuit to a
  sinusoidal excitation

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BSIM

• Transistors are complicated. Accurate sim
  requires 2D or 3D numerical sim (TCAD) to solve
  coupled PDEs (quantum effects, electromagnetics,
  etc)
• This is slow a circuit with one transistor will
  take hours to simulation
• How do you simulate large circuits (100s-1000s of
  transistors)?
• Use compact models. In EECS 105 we will derive
  the so called level 1 model for a MOSFET.
• The BSIM family of models are the industry
  standard models for circuit simulation of
  advanced process transistors.
• BSIM Berkeley Short Channel IGFET Model

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Berkeley

• A great place to study circuits, devices, and CAD!

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Review of LTI Systems

• Since most periodic (non-periodic) signals can be
  decomposed into a summation (integration) of
  sinusoids via Fourier Series (Transform), the
  response of a LTI system to virtually any input
  is characterized by the frequency response of the
  system

Phase Shift
Any linear circuit With L,C,R,M and dep. sources

Amp Scale
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Example Low Pass Filter (LPF)

• Input signal
• We know that

Phase shift
Amp shift
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LPF the hard way (cont.)

• Plug the known form of the output into the
  equation and see if it can satisfy KVL and KCL
• Since sine and cosine are linearly independent
  functions

IFF
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LPF Solving for response

• Applying linear independence

Phase Response
Amplitude Response
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LPF Magnitude Response
Passband of filter
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LPF Phase Response
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dB Honor the inventor of the phone

• The LPF response quickly decays to zero
• We can expand range by taking the log of the
  magnitude response
• dB deciBel (deci 10)

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Why 20? Power!

• Why multiply log by 20 rather than 10?
• Power is proportional to voltage squared
• At breakpoint
• Observe slope of signal attenuation is 20
  dB/decade in frequency

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Why introduce complex numbers?

• They actually make things easier
• One insightful derivation of
• Consider a second order homogeneous DE
• Since sine and cosine are linearly independent,
  any solution is a linear combination of the
  fundamental solutions

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Insight into Complex Exponential

• But note that is also a solution!
• That means
• To find the constants of prop, take derivative of
  this equation
• Now solve for the constants using both equations

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The Rotating Complex Exponential

• So the complex exponential is nothing but a point
  tracing out a unit circle on the complex plane

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Magic Turn Diff Eq into Algebraic Eq

• Integration and differentiation are trivial with
  complex numbers
• Any ODE is now trivial algebraic manipulations
  in fact, well show that you dont even need to
  directly derive the ODE by using phasors
• The key is to observe that the current/voltage
  relation for any element can be derived for
  complex exponential excitation

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Complex Exponential is Powerful

• To find steady state response we can excite the
  system with a complex exponential
• At any frequency, the system response is
  characterized by a single complex number H
• This is not surprising since a sinusoid is a sum
  of complex exponentials (and because of
  linearity!)
• From this perspective, the complex exponential is
  even more fundamental

Mag Response
Phase Response
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LPF Example The soft way

• Lets excite the system with a complex exp

use j to avoid confusion
complex
real
Easy!!!
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Magnitude and Phase Response

• The system is characterized by the complex
  function
• The magnitude and phase response match our
  previous calculation

?
?
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Why did it work?

• The system is linear
• If we excite system with a sinusoid
• If we push the complex exp through the system
  first and take the real part of the output, then
  thats the real sinusoidal response

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And yet another perspective

• Again, the system is linear
• To find the response to a sinusoid, we can find
  the response to and and sum the
  results

LTI System H
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Another persepctive (cont.)

• Since the input is real, the output has to be
  real
• That means the second term is the conjugate of
  the first
• Therefore the output is

?
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Proof for Linear Systems

• For an arbitrary linear circuit (L,C,R,M, and
  dependent sources), decompose it into linear
  sub-operators, like multiplication by constants,
  time derivatives, or integrals
• For a complex exponential input x this simplifies
  to

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Proof (cont.)

• Notice that the output is also a complex exp
  times a complex number
• The amplitude of the output is the magnitude of
  the complex number and the phase of the output is
  the phase of the complex number

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Phasors

• With our new confidence in complex numbers, we go
  full steam ahead and work directly with them we
  can even drop the time factor since it will
  cancel out of the equations.
• Excite system with a phasor
• Response will also be phasor
• For those with a Linear System background, were
  going to work in the frequency domain
• This is the Laplace domain with

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Capacitor I-V Phasor Relation

• Find the Phasor relation for current and voltage
  in a cap

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Inductor I-V Phasor Relation

• Find the Phasor relation for current and voltage
  in an inductor