Active Analogue CircuitsYear 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 2nd year lab.• “Art of Electronics” by Horowitz and Hill

October 2015 Todd Huffman

V0

I R1

R2

R3

-V0+IR1+IR2+IR3=0

0Vn+

+

+

+

–

–

–

–

–V0

+IR1

+IR2

+IR1

I1

I3

I2

I4

I1+I2–I3–I4=0

0In

Kirchoff’s laws

KCL

KVL

AC circuit theory

• Voltage represented by complex exponential

• Impedance relates current and voltage V=ZIin complex notation:

Resistance RInductance jLCapacitance 1/(jC)

and combinations thereof• Impedance has magnitude and phase

0V V cos t represented by real component of j t

0V V e

easily shown from dI

V Ldt

jZ Z e

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 ejwt 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!

j tj t0 0

j

V V e VI e

Z Z e Z

Op-amps Gain is very large (A)

Inputs draw no current (ZIN=)

Feedback v+=v–

VOUT+

–

v+

v–

VIN

R1R2

Non-Inverting Amplifier Circuit

+VOUT

VIN

R1

Inverting Amplifier Circuit

R2

–v–

v+i

i

First Non-ideal model

+

-

A(w)dV

+

dV

Instead of infinite gain, the devicehas finite, and frequency dep. Gain.

V0-

+

A(w) behaves like an RC filter.

Magnitude |A(w)|

Phase fA(w)

With a gain factor of over a million; and a roll-off around w = 1 rad/s

A(w) ≈ 106/(1+jw)

Model of this non-ideal gain curve

• Vx = A0V1

• KCL• (V2 – Vx)/R + jwCV2 = 0

– Substitute expression for Vx above and some algebra

• V2(1 + jwCR) = A0V1

• V2/V1 = A0/(1 + jwCR) ≡ A(w)

A0

R

CV1

V2

Vx

Note:Also Draw filter on Blackboard

How does this effect our negative feedback circuits?

• KVL• VR1 – dV – Vin = 0

• VR2 + dV + V0 = 0

• KCL• VR1/R1 = VR2/R2

• And also the Gain relationship

• dVA(w) = V0

Solve on board

+VOUT

VIN

R1

Inverting Amplifier CircuitR2

–v–

v+i

i

𝑉 0

𝑉 𝑖𝑛

=−𝑅2

𝑅1+(𝑅1+𝑅2 ) (1+ 𝑗 𝜔 )

106

The Transistor!Go to diode part of lecture

Take a step back to demonstrate a key

technique – Small Signal response

Current flows easily!Almost no current flow

𝐼=𝐼 0(𝑒𝑞𝑉 𝐴𝐶

𝑘𝑇 −1)

Diode (& Transistor) is nonlinear

• Norton and Thevenin equivalents• Superposition• Ohm’s law does not exist for a non-linear

device.

• Kirchoff’s laws• Power = VxI (but not V2/I or I2R)

This means some tricks do not work!

Other principles are still OK.

Small signal analysis

Current flows easily!

R

++

VACV0

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

vBE

Model works for npn and pnp(follow passive sign conv. on resistor)

CB

Cm

II

IkT

qg

+–

First Transistor Small Signal Model

gmvBEb/gm

base collector

emitter

Typical npn form shown

b is related to details of trans. Construction: = 100 b good start

How to use graphs?• Actually; start from Ebers-Moll equation

• A lot like Diode: if VBE ≈ 0.6 V or more IC starts to blow up. If IC changes by 10x, VBE still ~0.6 V

1. Assume VBE ≈ 0.6 V is true!

2. Assume VCE is ≥ 1 V (transistor is “active”)3. Assume b = 1004. Solve and rethink assumptions if inconsistency is

found.

10 e kT

qV

C

BE

II

Our First Transistor Circuit

VOUT

VIN

RB RC

+

+

How is it Biased?

What does it do?

The 741 op-amp’s actual diagram