Active Analogue Circuits Year 2 B. Todd Huffman

Circuit Theory Reminders Basics, Kirchoff’s laws, Thevenin and Norton’s theorem, Capacitors, Inductors AC theory, complex notation, LCR circuits Passive Sign Convention What is “Passive Sign Convention”? Good Texts: Electronics Course Manual for 2 nd year lab. “Art of Electronics” by Horowitz and Hill October 2015 Todd Huffman

V0V0 I R1R1 R2R2 R3R3 -V 0 +IR 1 +IR 2 +IR 3 = – – – – –V 0 +IR 1 +IR 2 +IR 1 I1I1 I3I3 I2I2 I4I4 I 1 +I 2 –I 3 –I 4 =0 Kirchoff’s laws KCL KVL

AC circuit theory Voltage represented by complex exponential Impedance relates current and voltage V=ZI in complex notation: Resistance  R Inductance  jL Capacitance 1/(jC) and combinations thereof Impedance has magnitude and phase represented by real component of easily shown from Q=VC

Current is given by So |Z| gives the ratio of magnitudes of V and I, and  give the phase difference by which current lags voltage Notice that the time dependent part is a common factor – So e j  t can be removed and is “understood” to be present when returning to the time domain. – WARNING!!! This is only true for circuits with Linear behaviour!

Op-amps Gain is very large (A) Inputs draw no current (Z IN =) Feedback  v + =v – V OUT + – v+v+ v–v– V IN R1R1 R2R2 Non-Inverting Amplifier Circuit + V OUT V IN R1R1 Inverting Amplifier Circuit R2R2 – v–v– v+v+ i i

First Non-ideal model + - A(  )  V + VV Instead of infinite gain, the device has finite, and frequency dep. Gain. V0V0 - +

A(  ) behaves like an RC filter. Magnitude |A(  )| Phase  A (  ) With a gain factor of over a million; and a roll-off around  = 1 rad/s A(  ) ≈ 10 6 /(1+j  )

Model of this non-ideal gain curve V x = A 0 V 1 KCL (V 2 – V x )/R + j  CV 2 = 0 – Substitute expression for Vx above and some algebra V 2 (1 + j  CR) = A 0 V 1 V 2 /V 1 = A 0 /(1 + j  CR) ≡ A(  ) A0A0 R CV1V1 V2V2 VxVx Note: Also Draw filter on Blackboard

How does this effect our negative feedback circuits? KVL V R1 –  V – V in = 0 V R2 +  V + V 0 = 0 KCL V R1 /R 1 = V R2 /R 2 And also the Gain relationship  VA(  ) = V 0 Solve on board + V OUT V IN R1R1 Inverting Amplifier Circuit R2R2 – v–v– v+v+ i i

The Transistor!

Take a step back to demonstrate a key technique – Small Signal response Current flows easily!Almost no current flow

Diode (& Transistor) is nonlinear Norton and Thevenin equivalents Superposition Ohm’s law does not exist for a non-linear device. Kirchoff’s laws Power = Vx I (but not V 2 /I or I 2 R) This means some tricks do not work! Other principles are still OK.

Small signal analysis Current flows easily! R + + V AC V0V0

Simple Transistor Model It can be a “switch” – Flow is “on” one way – Flow is “off” the other way It can be an amplifier – The flow is proportional to the amount you turn the valve. – If you turn the valve fast enough you can communicate in Morse- Code-litres

Bipolar Junction Transistor curves On black board!

BJT – How to approach this?! Technique just like for diode Assume it is working as expected Find an “operating point” using DC parameters (check assumptions!) Use some kind of “equivalent circuit” which is linear Solve linear circuit for “small signals” Check consistency

v BE Model works for npn and pnp (follow passive sign conv. on resistor) + – First Transistor Small Signal Model g m v BE  /g m base collector emitter Typical npn form shown  is related to details of trans. Construction:  good start

How to use graphs? Actually; start from Ebers-Moll equation A lot like Diode: if V BE ≈ 0.6 V or more I C starts to blow up. If I C changes by 10x, V BE still ~0.6 V 1.Assume V BE ≈ 0.6 V is true! 2.Assume VCE is ≥ 1 V (transistor is “active”) 3.Assume  = Solve and rethink assumptions if inconsistency is found.

Our First Transistor Circuit V OUT V IN RBRB RCRC + + How is it Biased? What does it do?

The 741 op-amp’s actual diagram