Department of EECS University of California, Berkeley EECS 105 Fall 2003, Lecture 1 Lecture 1: Course Overview and Introduction to Phasors Prof. Niknejad

EECS 105 Fall 2003, Lecture 1Prof. A. Niknejad Department of EECS University of California, Berkeley 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)

EECS 105 Fall 2003, Lecture 1Prof. A. Niknejad Department of EECS University of California, Berkeley EECS 105 in the Grand Scheme Example: Cell Phone

EECS 105 Fall 2003, Lecture 1Prof. A. Niknejad Department of EECS University of California, Berkeley 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 Analog “Amp” Digital Gate MOS Cap PN Junction Variable Capacitor

EECS 105 Fall 2003, Lecture 1Prof. A. Niknejad Department of EECS University of California, Berkeley SPICE 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 * Example netlist Q npnmod R k Vdd 3 0 3v.tran 1u 100u SPICE stimulus netlist response

EECS 105 Fall 2003, Lecture 1Prof. A. Niknejad Department of EECS University of California, Berkeley 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

EECS 105 Fall 2003, Lecture 1Prof. A. Niknejad Department of EECS University of California, Berkeley Berkeley… A great place to study circuits, devices, and CAD!

EECS 105 Fall 2003, Lecture 1Prof. A. Niknejad Department of EECS University of California, Berkeley 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: Any linear circuit With L,C,R,M and dep. sources Amp Scale Phase Shift

EECS 105 Fall 2003, Lecture 1Prof. A. Niknejad Department of EECS University of California, Berkeley Example: Low Pass Filter (LPF) Input signal: We know that: Amp shift Phase shift

EECS 105 Fall 2003, Lecture 1Prof. A. Niknejad Department of EECS University of California, Berkeley 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

EECS 105 Fall 2003, Lecture 1Prof. A. Niknejad Department of EECS University of California, Berkeley LPF: Solving for response… Applying linear independence Phase Response: Amplitude Response:

EECS 105 Fall 2003, Lecture 1Prof. A. Niknejad Department of EECS University of California, Berkeley LPF Magnitude Response Passband of filter

EECS 105 Fall 2003, Lecture 1Prof. A. Niknejad Department of EECS University of California, Berkeley LPF Phase Response

EECS 105 Fall 2003, Lecture 1Prof. A. Niknejad Department of EECS University of California, Berkeley 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)

EECS 105 Fall 2003, Lecture 1Prof. A. Niknejad Department of EECS University of California, Berkeley 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

EECS 105 Fall 2003, Lecture 1Prof. A. Niknejad Department of EECS University of California, Berkeley 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

EECS 105 Fall 2003, Lecture 1Prof. A. Niknejad Department of EECS University of California, Berkeley 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:

EECS 105 Fall 2003, Lecture 1Prof. A. Niknejad Department of EECS University of California, Berkeley The Rotating Complex Exponential So the complex exponential is nothing but a point tracing out a unit circle on the complex plane:

EECS 105 Fall 2003, Lecture 1Prof. A. Niknejad Department of EECS University of California, Berkeley Magic: Turn Diff Eq into Algebraic Eq Integration and differentiation are trivial with complex numbers: Any ODE is now trivial algebraic manipulations … in fact, we’ll show that you don’t 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

EECS 105 Fall 2003, Lecture 1Prof. A. Niknejad Department of EECS University of California, Berkeley 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 LTI System H Mag Response Phase Response

EECS 105 Fall 2003, Lecture 1Prof. A. Niknejad Department of EECS University of California, Berkeley LPF Example: The “soft way” Let’s excite the system with a complex exp: real complex use j to avoid confusion Easy!!!

EECS 105 Fall 2003, Lecture 1Prof. A. Niknejad Department of EECS University of California, Berkeley Magnitude and Phase Response The system is characterized by the complex function The magnitude and phase response match our previous calculation:

EECS 105 Fall 2003, Lecture 1Prof. A. Niknejad Department of EECS University of California, Berkeley 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 that’s the “real” sinusoidal response

EECS 105 Fall 2003, Lecture 1Prof. A. Niknejad Department of EECS University of California, Berkeley 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 LTI System H LTI System H

EECS 105 Fall 2003, Lecture 1Prof. A. Niknejad Department of EECS University of California, Berkeley 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:

EECS 105 Fall 2003, Lecture 1Prof. A. Niknejad Department of EECS University of California, Berkeley “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:

EECS 105 Fall 2003, Lecture 1Prof. A. Niknejad Department of EECS University of California, Berkeley “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

EECS 105 Fall 2003, Lecture 1Prof. A. Niknejad Department of EECS University of California, Berkeley 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, we’re going to work in the frequency domain – This is the Laplace domain with

EECS 105 Fall 2003, Lecture 1Prof. A. Niknejad Department of EECS University of California, Berkeley Capacitor I-V Phasor Relation Find the Phasor relation for current and voltage in a cap: +_+_

EECS 105 Fall 2003, Lecture 1Prof. A. Niknejad Department of EECS University of California, Berkeley Inductor I-V Phasor Relation Find the Phasor relation for current and voltage in an inductor: +_+_