[PPT] - Frequency Response Impact of Coupling Capacitors (1) v R = i i PowerPoint Presentation

SLIDE 1

Frequency Response

SLIDE 2

Impact of Coupling Capacitors (1)

F. Najmabadi, ECE65, Winter 2013, Discrete Amplifiers (31/42)

1 1 1 1 1

) ( 1 / 1 1 ] ) [( / 1 1 ) /( 1

c sig i p p sig i i sig i c sig i sig i i sig i c sig i i sig i

C R R j R R R v v C R R j R R R v v C j R R R v v + = − × + = + − × + = + + = ω ω ω ω ω High Pass filter with pole at ωp1

SLIDE 3

Impact of Coupling Capacitors (2)

F. Najmabadi, ECE65, Winter 2013, Discrete Amplifiers (32/42)

2 2 2 2

) ( 1 / 1 1 ) /( 1

c

L

p p

L

L v i

c
L

L i v

C

R R j R R R A v v C j R R R v A v + = − × + × = + + = ω ω ω ω High Pass filter with pole at ωp2

SLIDE 4

Impact of Coupling Capacitors (3)

F. Najmabadi, ECE65, Winter 2013, Discrete Amplifiers (33/42)

Low-frequency Response (included): ω ω ω ω ω ω ω ω ω ω ω ω / 1 1 / 1 1 / 1 1 / 1 1 / 1 1 / 1 1

2 1 2 1 2 1 p p v sig

p

p

L

L v sig i i sig

p
L

L v i

p

sig i i sig i

j j A v v j j R R R A R R R v v j R R R A v v j R R R v v − × − × = − × − × + × × + = − × + × = − × + =

Mid-band gain (caps are short):

L

L v sig i i v sig

R

R R A R R R A v v + × × + = =

Each coupling capacitor introduced a pole

SLIDE 5

Each Capacitor introduces a pole

F. Najmabadi, ECE65, Winter 2013, Discrete Amplifiers (34/42)

Each capacitor introduces a pole: 1) Coupling capacitor at the input: 2) Coupling capacitor at the output: 3) By-pass capacitor (CE wit RE, CS with RS, CB)*

1 1

) ( 2 1

c sig i p

C R R f + = π

pass by pass by p

C R f

− −

= 2 1

3

π

2 2

) ( 2 1

c L

p

C R R f + = π

Exact calculation of the lower cut-off

frequency can only be done numerically.

A surprisingly good approximation is

...

2 1

+ + ≈

p p p

f f f

* Formulas for Rby-pass are given in the Amp. summary sheets.

SLIDE 6

Multi-Stage Amplifiers

SLIDE 7

Gain of a multi-stage amplifier

F. Najmabadi, ECE65, Winter 2013, Discrete Amplifiers (36/42)

Example: A 3-stage amplifier

2 , 3 , 1 , 2 , 1 , 1 , 1 , 3 , 1 ,

i
i
i
v

v v v v v v v v v × × = = ...

3 2 1

× × × × + =

v v v sig i i sig

A

A A R R R v v

But we need to know RL,1, RL,2, RL,3, … in order to find AM s.

2 , 1 , i

v

v =

3 , 2 , i

v

v =

3 ,

v

v =

1 , 1 , 1 , 1 ,

,

i i sig i i sig i sig i

R R R R R v v v v = + = =

3 ,

R

R =

3 2 1 3 , 3 , 2 , 2 , 1 , 1 , 1 , v v v i

i
i
i
A

A A v v v v v v v v × × = × × =

2 1 i

v

v =

3 2 i

v

v =

SLIDE 8

What are RL and Rsig for each stage?

F. Najmabadi, ECE65, Winter 2013, Discrete Amplifiers (37/42)

Amp Model for Stage j-1

1 , , −

=

j

j

sig

R R

1 , , +

=

j i j L

R R

RL for each stage is the input resistance of the following stage.
Rsig for each stage is the output resistance of the previous stage.

Amp Model for Stage j+1

SLIDE 9

Procedure for Solving multi-stage Amplifiers

F. Najmabadi, ECE65, Winter 2013, Discrete Amplifiers (38/42)

Gain & Ri:

1.Start from the load side (nth stage),

Find the gain Av,n = (vo/vi)n and Ri,n .

2.For (n-1)th stage, set RL,n-1 = Ri,n

Find the gain Av,n-1 = (vo/vi)n-1 and Ri ,n-1 .
3. Repeat until reaching to the first stage. Then,

Ro:

1. Start from the source side (1st stage). Find Ro,1 .
2. Go to the second stage. Set Rsig,2 = Ro,1 . Find Ro,2

3.Continue to the last stage (nth stage). Then, Ro = Ro,n

...

3 2 1 1 ,

× × × × + = =

v v v sig i i sig

i

i

A A A R R R v v R R

SLIDE 10

What is “Buffer”

F. Najmabadi, ECE65, Winter 2013, Discrete Amplifiers (39/42)
Mid-band gain:
Sever loss in gain if RL is small compared to Ro
Main gain cells (CE or CS amps) have an Ro of several to

10s of kΩ.

This can be resolves by using a “buffer”:
Ideal Buffer: Avo = 1 , Ri = ∞ and Ro = 0.
Practical Buffer: Avo ≈ 1 , large Ri (several to 10s of

MΩ) and small Ro (10s or 100s of Ω).

CD amplifier (source follower) and CC amplifiers (emitter

follower) are examples of practical buffers

L

L v sig i i sig

R

R R A R R R v v + × × + =

* There is also loss in gain if Ri is small compared to Rsig . Buffer circuits can also solve this issues

SLIDE 11

Buffer circuit removes the impact of RL

F. Najmabadi, ECE65, Winter 2013, Discrete Amplifiers (40/42)

1 1

2 , 2 , 2 , 2 , 2 , 2 ,

= + × = + × =

L L i

L

L vo i

R

R v v R R R A v v

L

L v sig i i sig

R

R R A R R R v v + × × + =

1 , 1 , 2 , 2 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , vo

i

i vo i

L

L vo i

A

R R R A v v R R R A v v = + × = + × =

1

1 , 2 , 1 , 1 ,

× × + = × × =

vo sig i i sig

i
i
sig

i sig

A

R R R v v v v v v v v v v Without a buffer: With a buffer:

Buffer circuit has removed the dependence on Ro & RL.

∞ = = =

2 , 2 , 2 ,

1

i

vo

R R A

SLIDE 12

Buffer circuit can also be used to remove the impact of low Ri

F. Najmabadi, ECE65, Winter 2013, Discrete Amplifiers (41/42)
L

L vo i

L

L vo i

R

R R A v v R R R A v v + × = + × =

2 , 2 , 2 , 2 , 2 , 2 ,

L

L vo sig i i sig

R

R R A R R R v v + × × + =

1 1

1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 ,

= + × = + × =

L L i

L

L v i

R

R v v R R R A v v

L

L vo sig

i
i
sig

i sig

R

R R A v v v v v v v v v v + × = × × =

2 , 2 , 1 , 1 ,

Without a buffer: With a buffer:

Buffer circuit has removed the dependence on Ri & Rsig.

1

1 , 1 ,

= + =

sig i i sig i

R R R v v ∞ = = =

1 , 1 , 1 ,

1

i

vo

R R A

SLIDE 13

Cut-off frequency of a multi-stage amplifier

F. Najmabadi, ECE65, Winter 2013, Discrete Amplifiers (42/42)
The pole due to the Ccj , coupling

capacitor between stages j-1 and j, can be found by considering as the input capacitor to stage j (using formula from part 1) noting that Rsig, j = Ro, j-1 .

Similar to a single-stage amplifier, each capacitor introduces a pole: 1) Coupling capacitor at the input: 2) Coupling capacitor at the output: 3) Coupling capacitor between stages j-1 and j 4) By-pass capacitors for stage j (if exists) 5) Then:

1 1

) ( 2 1

c sig i p

C R R f + = π

pass by pass by bypass p

C R f

− −

= 2 1

,

π

co L

po

C R R f ) ( 2 1 + = π

cj j

j

i pj

C R R f ) ( 2 1

1 , , −

+ = π ...

2 1

+ + ≈

p p p