Lecture 1: Course Overview and Introduction to Phasors

1. Lecture 1: Course Overview and Introduction to Phasors Prof. Niknejad

2. 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) University of California, Berkeley

3. EECS 105 in the Grand Scheme • Example: Cell Phone University of California, Berkeley

4. MOS Cap Digital Gate Analog “Amp” Variable Capacitor PN Junction 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 University of California, Berkeley

5. SPICE • SPICE = Simulation Program with ICEmphasis • 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 Q1 1 2 0 npnmod R1 1 3 1k Vdd 3 0 3v .tran 1u 100u SPICE stimulus response netlist University of California, Berkeley

6. 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 University of California, Berkeley

7. Berkeley… • A great place to study circuits, devices, and CAD! University of California, Berkeley

8. 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 University of California, Berkeley

9. Example: Low Pass Filter (LPF) • Input signal: • We know that: Phase shift Amp shift University of California, Berkeley

10. 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 University of California, Berkeley

11. LPF: Solving for response… • Applying linear independence Phase Response: Amplitude Response: University of California, Berkeley

12. LPF Magnitude Response Passband of filter University of California, Berkeley

13. LPF Phase Response University of California, Berkeley

14. 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) University of California, Berkeley

15. 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 University of California, Berkeley

16. 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 University of California, Berkeley

17. 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: University of California, Berkeley

18. The Rotating Complex Exponential • So the complex exponential is nothing but a point tracing out a unit circle on the complex plane: University of California, Berkeley

19. 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 University of California, Berkeley

20. LTI System H 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 University of California, Berkeley

21. LPF Example: The “soft way” • Let’s excite the system with a complex exp: use j to avoid confusion complex real Easy!!! University of California, Berkeley

22. Magnitude and Phase Response • The system is characterized by the complex function • The magnitude and phase response match our previous calculation:   University of California, Berkeley

23. 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 University of California, Berkeley

24. LTI System H LTI System H 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 University of California, Berkeley

25. 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:  University of California, Berkeley

26. “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: University of California, Berkeley

27. “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 University of California, Berkeley

28. 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 University of California, Berkeley

29. + _ Capacitor I-V Phasor Relation • Find the Phasor relation for current and voltage in a cap: University of California, Berkeley

30. + _ Inductor I-V Phasor Relation • Find the Phasor relation for current and voltage in an inductor: University of California, Berkeley