# Electronic Circuits Summary

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1 Electronic Circuits Summary Andreas Biri, D-ITET Constants ε 0 = F m m 0 = kg k = J K = ev/k kt q = V, q kt = 38.6, kt = 5.9 mev V Small Signal Equivalent Circuit BJT Consider only small oscillations around operation point Linearize as approximation, V CC = 0 = V EE as const. i C = β v E = d I C β + R E d V BE v out ( v in v E ) = V BE v E R E R L + R E v in = d I C = I C, g d V BE V π = d I B = T d V BE β Small Signal Equivalent Circuit MOSFET g 0 = d I D d V DS = I D = d I D d V GS K W L I D λ λ I + λ V D, r 0 = DS g 0 λ I D ev = J q = A s. Transistor Characteristics Resistor: V R = R I R g 0 = d I C d V CE = r π = g π = β, I C V A + V CE I C V A r 0 = g 0 = V A + V CE I C V A I C. Single-Transistor Amplifiers Capacitor: I C = C d dt V C Inductor: V L = L d dt I L Bipolar Junction Transistor ( BJT ) V BE I C = I S e V T ( + V CE ), V V T = kt A q 6 mv I B = I C β, I E = ( + β) I β, V A Early voltage Biasing of a BJT ( Setting the operationg point ) Voltage divider R B, R B sets the bias voltage Transistor in active region : V BE 0.7 V MOSFET Impedances: Gains: Z in = v in i in, Z out = v out i out ( v S = 0 ) A V = v out v S, A I = i out i s ( R L = 0 ) I D = K W L ( V GS V t ) ( + λ V DS ), V DS > V GS V t Millers theorem K : Intrinsic transconduct. coeff. V t : Threshold voltage W / L : Gate width / Gate length λ : Characteristic length Z in = Z + A V, Z out = Z + A V

2 Common-Emitter / Source Amplifier R in = v in i in = r π R out = v out = r i 0 L v in =0 Common-Base / Gate Amplifier Common-Collector / Drain Amplifier Also known as emitter / source follower Rout i L = r 0 r 0 + R L v in A V = v out v S r π r π + R S R L, v out v in Inverting Amplifier 80 phase shift A I = i out = R S β, i S R v out =0 S + r π = i L R L v in R L i out = g i m r π = β in v out =0 MOSFET: instead of BJT, no current into the gate R in = v in i in R out = v out i out R S r π ((r π R S ) + ) r 0 β r 0 A V = v out v S R L + R S non inverting amp. A I = i out = for R i in S r π v out =0 + g R S r m π MOSFET: R S = 0 voltage source ; R S = current source Usually: R L r 0 R L r 0 R L R in = v in = r i π + ( + β) R L, R out = in + r π A V = v out v in + R L, A I = + β MOSFET: no current flowing into the gate ( A I = ) v out R L v in R in = R out = No current flow into the gate: R out, v in = v S R out = r 0, A V R L A V = v out R L, A v S + R I = i out = S i S R in = v in i in, R out ( + R S ) r 0 A V, + R L i out = v out

3 Comparison of the three basic amplifiers Multi-Stage Amplifier 4. Differential Amplifiers Transmitting information with two complementary signals Information is contained in the difference, same DC value Small signal model: all constant voltage supplies become ground After voltage buffer: lower output resistance (better V-source) After current buffer: larger output resistance (better C-source) Impedance Matching P L is maximized when R L = R S 3. Frequency Response of Amplifiers Change of charge vs. voltage across pn-junctions between BJTs can be represented by a parasitic capacitance Stage : Stage : C C large, shorts, C gd often negligible Differential amplifier: In order to filter out DC component before the amplification, we use a fixed tail current I E, which also enables DC coupling of stages (current splitted) C π capacitance between B and E, C μ capacitance between B and C, C gs C gd v out = v Z R + sc L R v, Z A V (s) = v out(s) v (s) = R + s C L R Cut-off frequencies: defined by poles R + sc L R A V (s) = v out(s) v in (s) = R R ( + s R C gs )( + s R C L ) Bandwidth-Broadening Additional shunt/feedback resistor R f up to doubles bandwidth! ω = p = R C L ω = p = R C gs Time Domain representation: v out (t) = A e p t + B e p t + C V E ( emitter-node potential) remains constant v E = 0 Symmetry between left and right branch split circuit into two independent parts and analyze separately (once) Differential amplification: A vd = v od v id = R C Common-mode amplification: A vcm = v 0 = v ocm = R C v i v icm R E 3

4 Common Mode Rejection Ratio Indicates how strong a common mode signal is attenuated compared to a differential signal G = A vd A vcm = R C R C /R E = R E CMRR = G db = 0 log 0 G GBP = A 0 ω C Operational amplifiers Non-inverting amplifier Integrator 5. Instrumentation Amplifier Precise amplification of weak, distorted sensor signals High input impedance, internal feedback loop Basic Instrumentation Amplifier Amplifies voltage difference with a precise gain Differential gain must be equal for both input branches Ideal: Z in, Z out 0, A vd, CMMR Non-ideal Small Signal Equivalent Differentiator Voltage Comparator Voltage follower ( Buffer ) V out = sign(v in ) V CC V 0 = R R + R 4 / + R /R V i+ R 4 V i Set R =, R = R 4 to equally load both input branches: V 0 = G V i+ G V i, G = R R = R 4 Buffered Instrumental Amplifier Obtain ideally high input impedance by input buffering Inverting amplifier V 0 = V icm ( R ( + R 4 ) (R + R ) R 4 ) + V id (R ( + R 4 ) (R + R ) + R 4 ) CMMR = A d = V 0/V id = ( + R 4 )R + (R + R )R 4 A cm V 0 /V icm (R R 4 R ) 4

5 Input stage gain Differential & common mode gain of input stage: A B = V Bd V id = V B+ V B V i+ V i = + R 5 + R 6 R 7 6. Voltage Regulators, Logarithmic & Anti-Logarithmic Amplifiers Linear voltage regulators Small Signal Equivalent: Z out = v out = s + ωp 0 i o A 0 ( + s ) ( + s C L ) ωp 0 A 0 Logarithmic & Anti-Logarithmic Amplifiers Non-linear circuit whose output voltage is proportional to the logarithm / exponential of the input voltage A cm,b = V B+ + V B V i+ + V i = no current through R 5, R 6, R 7 Differential & common mode gain in total: A d = V 0 V id = A B ( R ( + R 4 ) (R + R ) + R 4 ) Logarithmic Amplifier: Rely on logarithmic relationship of I C & V BE CMMR = A d R =, R = R 4 A d = R R A B A cm = A cm,b ( R ( + R 4 ) (R + R ) R 4 ) A cm = A B A d A cm increased by factor A B I in = V in = I R C = I S e V out V T, V out = V T ln ( I in ) = V I T ln ( V in ) S R I S Anti-Logarithmic Amplifier Voltage offset: Offset voltage in combination with a small input signal is highly undesired. The output signal then reaches the saturation level even for small values of V i and is therefore distorted. If this is not appropriate, chopper amplifiers can be used. V out = ( + R F R F ) V ref Choose R F, R F R L I L I E / I C V BE = V in, I C = I S e V in V T V out = I C R = I S R e V in V T 5

6 7. Active RC Filters First order active filters Filters and amplifies signal Resonance frequency: ω 0 = / L C Filter is a frequency-selective circuit that passes a specified band of frequencies and blocks frequencies outside of it. Quality factor: Q 0 = L R C Passive Filters: based on passive elements such as R / L / C Active Filters: based on op-amps in addition to R / L / C Energy vs. freq. The higher Q, the narrower and sharper the peak. Comparison of first order filters Low-pass filter Cutoff frequency Poles define cut-off: ω n = p n, First order passive filters T(jω c ) = max ( T(jω) ) BW 3dB = ω 0 Q0 Second order passive filters nd order passive filters can be synthesized from st order T(s) = High-pass filter Band-pass filter LC s + R L s + = LC T(s) = s ω 0 s + ω 0 Q 0 s + ω 0 ω 0 s + ω 0 Q 0 s + ω 0 T(s) = R C s + 3 RC s +, ω 0 = RC, Q 0 = 3 R C Denominator: D(s) = s + ω 0 Q o s + ω 0 T(s) = s R L s + R L s + LC Poles: p, = ω 0 Q 0 ± ω 0 4 Q 0 6

7 Sallen-Key amplifier Allow sharp gains without using inductors (expensive) Low-pass filter Band-pass filter 8. Switched capacitor filters Motivation: some systems require an active RC low-pass filter with very low f cutoff We need a large resistor in a highly integrated chip, whereby it is also inaccurate. Concept of switched capacitor devices High-pass filter Tow-Thomas Biquad filter Less sensitive to tolerance difference; combine Sallen-Keys Transfer of charge Q from potential V to potential V at a fixed rate f c = T C. Transferred charge per T C : Q = C ( V V ) Average current: Equivalent resistor: I,avg = Q T C = C ( V V ) T C R eq = T C C = f c C 7

8 Inverting Integrator using SC Non-inverting Integrator using SC Replace all reistors by switched capacitors Phase : Φ on, charge accumulates on C and C Phase : Φ on, C is discharged C V out [nt C ] = C V out [(n )T C ] C V in [n T C ] Same circuit as before, only change of switching schedule Charge on C is inverted compared to before Phase : C is charged to V in Design equations: C R4 C R3 = k, C R C = C R3 C = ω 0 T C, C R3 C R = Q Phase : Charge is transferred to C C V out [n T C ] = C V out [ (n )T C ] + C V in [n T C ] C V in (z) = C ( z ) V Out (z) Z-transform extended V out (z) V in (z) = C C z Switched capacitor Tow-Thomas biquad seems continuous for small enough T C V out [n T C ] = C n T C V T C C in (t) dt Ratio of capacitors can be realized more accurate than absolute values of R and C. Z-transform 0 8

9 9. Appendix Transimpedance amplifiers Sensing an input current & converting it to output voltage Frequency Response of Transimpedance Amplifiers Step Response of Second-Order Systems r m = dv 0 di in, Z in 0, Z out 0 9

10 Switched Capacitors Examples 0

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