Electronics for physicists Lecture 2 Electronics Basics November 2018 Electronics for physicists

Electronics for physicists Current and Charge I: current in Ampere [A] = [C/s] Q: charge in Coulomb [C] t: time in seconds [s] Comments 1 C = 6.241 ∙ 1018 electrons 1 femtocoulomb = 1 fC ≈ 6000 electrons November 2018 Electronics for physicists

Charge deposition of a MIP in Silicon „Minimum Ionizing Particle“ (MIP) A MIP produces 80 electron-hole- pairs/μm in Silicon Thus the signal charge in 300 μm deep silicon is 4 fC November 2018 Electronics for physicists

Electronics for physicists Voltage U = ∫E ds U : voltage [V] E : electric field strength [V/m] s : distance [m] We do care for E in detectors, however less in electronic circuits. „Voltage is what makes the current flow“ will frequently do! November 2018 Electronics for physicists

Electronics for physicists Ohm‘s Law R : resistance in Ohm [Ω = V/A] Meaning of Ohm‘s Law „Voltage makes the current flow“ Flow of current through resistor causes voltage drop across resistor I 𝑆𝑙𝑜𝑝𝑒= 1 𝑅 U November 2018 Electronics for physicists

Electronics for physicists Power P : power in [W = VA] Comment In detector instrumentation the energy bill rarely matters Getting the heat out and the power in is more critical November 2018 Electronics for physicists

Electronics for physicists SCT power cables Example: 4 Ω cable resistance for 100 m Iong cables (one-way!) 1 A of current per each of 4000 modules P = 4 Ω x 1 A2 x 4000 = 16 kW ! This power is lost by heating cables November 2018 Electronics for physicists

Electronics for physicists How to reduce power ? The SVX2 chip helped find the top quark at FNAL near Chicago SVX’ 1990 SVX2 1996 SVX4 2002 SVX3 1998 Decreasing transistor size makes chips smaller Adding new functionality and channels makes them bigger Smaller feature size means smaller chip voltages and saves power! (Not true for technologies below 65 nm and with increasing subthreshold current) November 2018 Electronics for physicists

Electronics for physicists Specific resistance R : resistance in [Ω] : specific resistance in [Ωm] Resistance increases with length l and decreases with area A The same goes for other structures eg. FETs November 2018 Electronics for physicists

Electronics for physicists Material mass in R.L. Much of the detector „mass“ is cables Radiation length of copper is 14.4 mm, R.L. of aluminum is 88.9 mm Ideally it is much better to use aluminum cables November 2018 Electronics for physicists

Kirchhoff‘s Laws Kirchhoff‘s Current Law (KCL) Kirchhoff‘s Voltage Law (KVL) Charge conservation Conservation of energy Loop Consider „nodes“ Consider „loops“ Currents entering the node equal currents leaving the node The sum of all voltage drops around the loop is zero November 2018 Electronics for physicists

Resistors in series and in parallel Parallel resistors Same current through all series resistors: IR1 = IR2 Same voltage across all parallel resistors: VR1 = VR2 November 2018 Electronics for physicists

Electronics for physicists There are 2 types of power supplies: voltage sources and current sources Voltage source Ideally, the voltage is constant and does not depend on the load current. • RL is load resistance • Ri is „intenal“ resistance Power supply: Car battery: Mono-cell: Voltage sources are at its best at: „no load“ small load currents constant current I Slope = -1/Ri U RL >> Ri November 2018 Electronics for physicists

Electronics for physicists Current source The ideal current source supplies a constant current independently of the voltage at the load. Current sources are at its best at short-circuits. I Slope = -Ri U RL << Ri November 2018 Electronics for physicists

Example: ABCN Readout IC Current sources are much more frequent than you might think. Count them in this ASIC for a future ATLAS silicon detector (designer J. Kaplon, CERN). November 2018 Electronics for physicists

Electronics for physicists Voltage divider The voltage divider is one of the most frequent circuit elements. Assumptions: R₁ and R₂ are relatively large. the load current is small. U1 Loop U2 KVL: November 2018 Electronics for physicists

The capacitor Capacitor symbol Capacitor equations: Comment Sketch of a plate capacitor Comment Energy to charge and then discharge a capacitor: Power to charge and discharge with frequency f: November 2018 Electronics for physicists

Capacitance of a silicon sensor - + 20 mm 80 µm d εR = εSi Electrical field Calculate C for A = 80 μm x 20 mm, d = 300 μm, ε0 = 8.9 x 10-12 F/m and εR = 11.8 (silicon) Assuming full depletion and ignoring neigbouring strips, we get C ≈ 0.56 pF In reality; neighborings strips dominate and C = 1-2 pF/cm strip length November 2018 Electronics for physicists

Capacitors in series and in parallel Parallel capacitors Capacitors in series thus November 2018 Electronics for physicists

Series and parallel- summary Component Series Parallel Resistors Capacitors R Increase Decrease Decrease Increase November 2018 Electronics for physicists

Discharging a capacitor within 1 τ : discharge to 37% of U0 within 5 τ : discharge to 0.7% of U0 t = 0 U0 0.37 x U0 t = 0  Uc = U0 (Initial condition) t > 0  exponential drop of voltage t= (seconds) Slope =- 1/RC The current (Ic) in the discharge at that instant is therefore: Ic = IR with Ic = dq/dt= C dU/dt and U = IRR Therefore we have U = -CR dU/dt , rearranging and integrating gives: November 2018 Electronics for physicists

Charging a capacitor through a resistor Charging a larger capacitance C takes longer Charging through a larger resistance R takes longer too t = 0 t = 0  capacitor resembles “short circuit” t = ∞  capacitor resembles “open circuit” Uin 0.63 x Uin Slope =1/RC November 2018 Electronics for physicists

Charging a capacitor through a current source Charging C with a voltage sources vs. charging C with a current source November 2018 Electronics for physicists

Electronics for physicists Low-pass KCL: IC = IR KVL: Uein – UC – UR = 0 Loop 𝐼𝐶=𝐼𝑅→𝐶 𝑑𝑈𝐶 𝑑𝑡 = 𝑈𝑒𝑖𝑛 − 𝑈 𝐶 𝑅 → First approximation Uc small Low-pass filter is known as an integrator circuit November 2018 Electronics for physicists

Electronics for physicists High-pass KCL: IC = IR KVL: Uein – UC – UR = 0 Loop High-pass filter is known as a differentiator circuit November 2018 Electronics for physicists

Electronics for physicists Inductor Inductor equations: Air core solenoid vs. air core toroid The inductance L of a finite-length solenoid is given by with length l and solenoid cross section A November 2018 Electronics for physicists

Electronics for physicists Summary table Component Voltage Current Resistors 𝑈=𝑅𝐼 𝐼=𝐺𝑈 Capacitors 𝑈= 1 𝐶 𝐼 𝑑𝑡 𝐼=𝐶 𝑑𝑈 𝑑𝑡 Coils 𝑈=𝐿 𝑑𝐼 𝑑𝑡 𝐼= 1 𝐿 𝑈 𝑑𝑡 G = 1/R November 2018 Electronics for physicists

Coils in series and in parallel Parallel coils thus Same current through all series resistors: IL1 = IL2 Same voltage across all parallel resistors: VL1 = VL2 November 2018 Electronics for physicists

Series and parallel – summary Component Series Parallel Resistors Capacitors Coils R November 2018 Electronics for physicists

Electronics for physicists Inductor Magnet coil M1 for KATRIN B = 3.6 Tesla Stored Energy: 0.8 MJ Current: 300 A Inductivity L: 16.7 H Length: 3.3 m Wire bond of a few mm length Inductivity L: a few nH Substrate Wire bond November 2018 Electronics for physicists

Discharging an inductor Discharge of an inductor through a resistor t = 0  IL = I0 (Initial condition) t > 0  exponential drop of current Io (seconds) → Coil: exponential drop of current Capacitor: exponential drop of voltage November 2018 Electronics for physicists

Magnetizing an inductor Loop 0.63 x Uein/R (a) Electrical circuit with voltage source, inductor and resistor. KCL: Slope =R/L 𝐼𝐿=𝐼𝑅=𝐼𝑅𝐿 𝑈𝑒𝑖𝑛 =𝐿 𝑑𝐼 𝑑𝑡 +𝑅𝐼 KVL: 𝑈𝑒𝑖𝑛−𝑈𝐿 −𝑈𝑅=0 𝑑𝐼𝑅𝐿 𝑑𝑡 + 𝑅 𝐿 𝐼𝑅𝐿= 𝑈𝑒𝑖𝑛 𝐿 (b) Inductor current after switching on Uin. 𝐼𝑅𝐿= 𝑈𝑒𝑖𝑛 𝑅 (1− 𝑒 − 𝑅𝑡 𝐿 ) , t = 0  inductor resembles “open circuit” t = ∞ inductor resembles “short-circuit” November 2018 Electronics for physicists

Circuit response to a sequence of rectangular pulses RL Low-pass filter  integration circuit November 2018 Electronics for physicists

Time behaviour - summary table Component Energy stored t = 0 t  ∞ Resistors No memory time independent Capacitors Electric charge Q -> Voltage Q = 0  U = 0: short-circuit Q ≠ 0  U ≠ 0: voltage source Q is constant 𝑑𝑄 𝑑𝑡 =𝐼=0: open-circuit Coils Magnetic field B -> Current B = 0  I = 0: open-circuit B ≠ 0  I ≠ 0: current source M is constant 𝑑𝐼 𝑑𝑡 =𝑈=0: short-circuit November 2018 Electronics for physicists

Electronics for physicists RC and RL t Uein 𝑅 IRL IRL = Uein 𝑅 t = 0  capacitor resembles “short circuit” t = ∞  capacitor resembles “open-circuit” IRL = 0 t Uein 𝑅 IRL t = 0  inductor resembles “open circuit” t = ∞ inductor resembles “short-circuit” IRL = 0 IRL = Uein 𝑅 November 2018 Electronics for physicists

RL low-pass and high-pass filters Low-pass filter integrator circuit R UR L Uaus = UL High-pass filter differentiator circuit Homework November 2018 Electronics for physicists

Helmholtz theorem (Thévenin's theorem) Linear networks may be complicated and contain a large number of resistors, voltage sources and current sources. (See sketch at black board…) The good news is: Any net can be described as one ideal voltage source UTh and one (internal) resistance RTh . What is the voltage across and the current through a load resistor connected between any 2 nodes of the net ?+ (Only power consumption may not be predicted correctly.) November 2018 Electronics for physicists

Electronics for physicists Voltage divider How to calculate UTh and RTh ? Uth corresponds to the output voltage of voltage divider network without load (RL = ∞). All voltage sources are replaced by short circuits and all current sources by open circuits. The load resistor is removed. RTh is the resistance between the output terminals of the circuit thus modified. November 2018 Electronics for physicists

Electronics for physicists Norton's theorem Any linear circuit can be simplified to correspond to a single current source and internal parallel resistance. How to calculate INo and RNo ? The derivation is analogous to the Helmholtz-Thevenin equivalent circuit. 1. I No corresponds to the output current of the original circuit with a short circuit as a load. 2. All voltage sources are replaced by short circuits and all current sources by open circuits. RNo is the resistor between the output terminals of the thus modified circuit. (It turns out that RNo = RTh and INo = UTh /RNo ) Norton`s equivalent circuit November 2018 Electronics for physicists

The superposition principle (I) Circuit with 2 voltage sources and 3 resistors. Resistors are neither parallel or in series. The circuit above is a superpostion of the load currents of the circuits below with IL = IL1 + IL2 and UL = UL1 + UL2 . UL IL1 Output circuit for superposition principle IL2 IL November 2018 Electronics for physicists

The superposition principle (II) The superposition principle: equivalent circuits The voltage source U1 and U2 , respectively, are replaced by a short circuit November 2018 Electronics for physicists

Systems of linear equations What is the voltage U30 and current across the load RL ? Output circuit for linear system KCL: From ∑ Ii = 0 (all currents Ii flow into the node 3) follows KVL: From ∑ Uij = 0 follows and 3 unknown voltages U30 , U31 , U32 3 independent equations => This is soluble The solution is (see next slide) November 2018 Electronics for physicists

Electronics for physicists Algebra With we get: and November 2018 Electronics for physicists

Electronics for physicists Some waveforms harmonic oscillation square wave step function detector signal after pulse shaping by an RC-CR filter November 2018 Electronics for physicists

Electronics for physicists Harmonic signals Voltage Current Voltage and current are in phase! Voltage => Current Phase shift of 90°: voltage follows current! November 2018 Electronics for physicists

Electronics for physicists „Harmonic signals Voltage Current with => Phase shift of 90°: current follows voltage! November 2018 Electronics for physicists

Time to complex plane - summary 𝑑𝐹 𝑑𝑡 →𝑗𝜔 𝐹 𝐹 𝑑𝑡 → 1 𝑗𝜔 𝐹 𝜔=2𝜋𝑓 Time-domain differentiation becomes multiplication by frequency 𝑗𝜔 Time-domain integration becomes division by frequency 𝑗𝜔 Component Time domain Complex plane Resistors U=𝑅𝐼 𝑈 =𝑅 𝐼 Capacitors 𝑈 =𝑗𝜔𝐿 𝐼 Coils 𝐼 =𝑗𝜔𝐶 𝑈 November 2018 Electronics for physicists

Electronics for physicists Generalized Ohm's law Resistor: Capacitor: Inductor: Impedances Z1 and Z2 in series: Impedances Z1 and Z2 in parallel: Z : complex impedance November 2018 Electronics for physicists

Representation of Z in the complex plane November 2018 Electronics for physicists

Frequency behaviour - summary Component 𝜔 ≈ 0 𝜔  ∞ Resistors 𝑈 =𝑅 𝐼 Frequency independent Capacitors 𝐼 =𝑗𝜔𝐶 𝑈 𝐼 = 0 open circuit 𝑈  0 short circuit Coils 𝑈 =𝑗𝜔𝐿 𝐼 U = 0 𝐼  0 November 2018 Electronics for physicists

RC and RL filters - summary Circuit Low-pass filter High-pass filter RC RL November 2018 Electronics for physicists

Resistance and inductance in series Homework Use: and Calculate: and for November 2018 Electronics for physicists

Electronics for physicists Solution and November 2018 Electronics for physicists

„Real“ resistors Maximum ratings Precision Temperature stability Types and sizes HF properties Costs Wired resistors SMD resistors with ruler November 2018 Electronics for physicists

Electronics for physicists „Real“ resistors Equivalent circuit November 2018 Electronics for physicists

„Real“ capacitors Multi-layer ceramic capacitor (MLCC) November 2018 Electronics for physicists

„Real“ capacitors Examples of wired capacitors November 2018 Electronics for physicists

Cylindrical capacitors November 2018 Electronics for physicists

Electrolytic capacitors November 2018 Electronics for physicists

Equivalent circuit of a capacitor ESR: equivalent series resistance ESL: equivalent series inductance For small ω, the capacitive term 1 ω𝐶 dominates. For large ω, the term ωL dominates. The frequency response of a capacitor is often more important than the exact value of its capacitance. November 2018 Electronics for physicists

Electronics for physicists Low-pass filter Integrator or low-pass filter November 2018 Electronics for physicists

Effect of a low-pass filter arbitrary vertical position no phase information shown Low-frequency signals pass without change High-frequency are signals attenuated Output signal is delayed w.r.t. input at high frequencies November 2018 Electronics for physicists

Electronics for physicists Low-pass filter Amplitude of output signal is frequency dependent Phase is frequency dependent Frequency of output equals frequency of input November 2018 Electronics for physicists

Electronics for physicists Decibel log stands for logarithm to base 10 Definitions: for power for voltage and current So far so good, but what about: dbm or dbmA ? Note: November 2018 Electronics for physicists

Electronics for physicists Bode plot (low pass) tan φ = -ωRC Bode plot: amplitude of transfer function Z in db vs. log f phase of Z vs. log f November 2018 Electronics for physicists

Electronics for physicists Low pass at f-3db Z is reduced to - the slope of Z is -20 db/decade November 2018 Electronics for physicists

Compensated voltage divider November 2018 Electronics for physicists

Electronics for physicists Bode plot (high pass) November 2018 Electronics for physicists

RLC Oscillator Note the current source which drives the circuit November 2018 Electronics for physicists

Electronics for physicists RLC Oscillator small output voltages for very small and large ω maximum output voltages at November 2018 Electronics for physicists

Electronics for physicists RLC Oscillator Bandwidth: At ω1 , ω2 , amplitude is of maximum Q factor is related to loss of energy per oscillation: November 2018 Electronics for physicists

Electronics for physicists Inductors November 2018 Electronics for physicists

Electronics for physicists Inductors Equivalent circuit November 2018 Electronics for physicists

Capacitors and inductors in a nutshell November 2018 Electronics for physicists