active analogue circuits year 2 - university of oxford ...huffman/mphys/circuits/aady2... · •...
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Active Analogue Circuits Year 2
B. Todd Huffman
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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 2016 Todd Huffman
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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
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AC circuit theory
• Voltage represented by complex exponential
• Impedance relates current and voltage V=ZI in complex notation:
Resistance R Inductance jL Capacitance 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
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• 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 ejt 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
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Op-amps Gain is very large (A)
Inputs draw no current (ZIN=)
Feedback v+=v–
VOUT +
–
v+
v–
VIN
R1 R2
Non-Inverting Amplifier Circuit
+ VOUT
VIN R1
Inverting Amplifier Circuit
R2
– v–
v+ i
i
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First Non-ideal model
+
-
A()dV
+
dV
Instead of infinite gain, the device
has finite, and frequency dep. Gain.
V0 -
+
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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() ≈ 106/(1+j)
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Model of this non-ideal gain curve
• Vx = A0V1
• KCL • (V2 – Vx)/R + jCV2 = 0
– Substitute expression for Vx above and some algebra
• V2(1 + jCR) = A0V1
• V2/V1 = A0/(1 + jCR) ≡ A()
A0
R
C V1
V2
Vx
Note:
Also Draw filter
on Blackboard
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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() = V0
Solve on board
+ VOUT
VIN R1
Inverting Amplifier Circuit
R2
– v–
v+ i
i
𝑉0𝑉𝑖𝑛
=−𝑅2
𝑅1 +𝑅1 + 𝑅2 1 + 𝑗𝜔
106
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The Transistor!
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• Silicon (Si) is a semiconductor
– Also Ge
• Atoms in diamond lattice
• Doping
• P-type
• N-type
N N P
base
emmitter
collector Coll.
emmitter
base
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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
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Bipolar Junction Transistor curves
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BJT – How to approach this?!
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
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How to use graphs?
• Actually; start from Ebers-Moll equation
• 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 = 100
4. Solve and rethink assumptions if inconsistency is found.
CmVq
kTII e kT
qV
C
BE
0.27@2510
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Biasing the BJT –
“active” operation • Will use npn transistors
(but pnp are simple complements of npn)
• If “Active” the Base-Emitter voltage drop is a diode drop ~0.6 V.
– Assume this is the case
• Use KVL and KCL and Ohm’s law on rest of circuit to find the DC collector current, IC.
N N P
base
emmitter
collector
0.6v
+
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vBE
Model works for npn and pnp (follow passive sign conv. on resistor)
CB
Cm
II
IkT
qg
b
+ –
First Transistor Small Signal Model
gmvBE b/gm
base collector
emitter
Typical npn form shown
b is related to details of trans.
Construction: b 100 good start
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Our First Transistor Circuit
VOUT
VIN
RB RC
+
+
How is it Biased?
What does it do?
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The 741 op-amp’s actual diagram