Lecture 1:
Feedback and Gain/Bandwidth
Prof. Ali M. Niknejad
Department of EECS
University of California, Berkeley
EE 105 Fall 2016
Prof. A. M. Niknejad
Feedback Control
l
Feedback is a universal way to design systems
–
–
–
l
Speed control example:
–
–
–
–
–
2
Thermostat in your house
Cruise control / keep-in-lane technology (driver assist)
Walking / standing in humans
Set desired speed of car: vdes
Measure the current speed of car: vcar
Find the difference: vdiff = vdes – vcar (this is also called the error signal)
Proportional scheme: Adjust car speed (accelerate) based on Kp vdiff
PID control with dynamics: Use proportional, integral, and derivative of vcar to
control speed
EE 105 Fall 2016
Prof. A. M. Niknejad
Negative Feedback Block Diagram
l
To find the transfer function, note that the error signal is a function of
the input and output:
s = s − f ⋅sout
err in
3
sout = G ⋅serr = G ⋅(sin − fsout )
EE 105 Fall 2016
Prof. A. M. Niknejad
Closed-Loop Transfer Function
l
Solve :
l
For very large gain G, such that Gf >>1, we have
sout (1+ Gf ) = G ⋅sin
sout
G
Gclosed =
=
sin 1+ Gf
Gclosed
4
G 1
≈
=
Gf f
EE 105 Fall 2016
Prof. A. M. Niknejad
Electronic Feedback
l
l
We have already seen electronic feedback in op-amps
Example: Non-inverting amplifier
–
–
–
l
Resistor divider samples output voltage
Error signal formed at input of op-amp
Op-amp output is a gained version of the error signal
Strangeness:
–
–
Op-amp gain is very large (ideally infinity)
Error signal is driven to zero
l
5
All practical op-amps have finite gain, so error signal is nearly zero
EE 105 Fall 2016
Prof. A. M. Niknejad
History
l
l
l
6
Invention of electronic negative
feedback
Inventor Harold Black was on a ferry
ride to Bell Labs (1927) thinking
about how to increase the gain and
improve linearity of of vacuum tube
amplifiers.
Realized that positive feedback can
be used to boost the gain and
negative feedback could improve the
linearity…
EE 105 Fall 2016
Prof. A. M. Niknejad
Precision Analog
l
Feedback allows us to use a really “crappy” open-loop op-amp and
get good performance
–
–
7
When gain is sufficiently high, the gain is determined by the feedback
network, not the open-loop gain
Open loop gain can vary over temperature, process, it can age, it can be highly
non-linear ... In the end, if it’s high enough, it does not play a role !
EE 105 Fall 2016
Prof. A. M. Niknejad
Benefits of Feedback
l
Closed-loop gain depends on passive feedback network
–
l
Linearization
–
l
l
8
Can set gain precisely
Op-amp transfer function is almost perfectly linear, despite using a very nonlinear core amplifier
Bandwidth enhancement (see next section)
Interference rejection (loop can correct for unwanted signals that are
injected into the signal path)
EE 105 Fall 2016
Prof. A. M. Niknejad
Loop Gain
l
Recall that the closed-loop transfer function is given by
G
G
Gclosed =
=
1+ Gf 1+T
l
For a precise transfer function, the key to feedback is to realize
sufficiently high loop gain
T = Gf ≫ 1
9
EE 105 Fall 2016
Prof. A. M. Niknejad
Noise Rejection
l
Another benefit of high loop gain is
interference rejection. Imagine an unwanted
signal couples into the loop as show
s = Gserr + snoise = G(sin − fsout )+ snoise
out
l
10
The noise is rejected like 1/(1+T). Any
unwanted signal, including distortion, is
rejected by the loop
sout
G
1
=
sin +
snoise
1+T
1+T
EE 105 Fall 2016
Prof. A. M. Niknejad
Positive Feedback
l
l
l
11
Up to now we have been considering a negative feedback system
whereby the output is subtracted from the input
If we were to add the output to the input, the system would be a
positive feedback system, which is also useful but not what we
intended.
Positive feedback systems tend to “rail out”, in other words they are
regenerative
EE 105 Fall 2016
Prof. A. M. Niknejad
Stability
l
l
l
12
Any real amplifier will introduce some
phase shift when the input frequency
1
increases. For example, a single-pole
G( jω ) =
1+ jωτ
system has the following form
As the frequency increases, the phase of
the output signal lags the input,
asymptotically up to 90°.
∠[G( jω )]= −arctan( jωτ )
When the system has more poles, the
phase shift can reach 180°. What
happens to the negative feedback system?
EE 105 Fall 2016
Prof. A. M. Niknejad
Instability
l
If the loop gain has a phase shift of 180°, then our negative feedback
system appears like a positive feedback system at frequency
T = G( jω x ) f = −1
l
l
l
13
Since there is always noise and disturbances in the system at this
frequency in the system, this noise is regenerated and potentially can
cause problems if |T| > 1.
The condition for stability is then to ensure that when loop gain is
unity, the phase of T should be less than 180°
The phase margin is a measure of stability. A good design should
have 60° phase margin or more.
EE 105 Fall 2016
Prof. A. M. Niknejad
Oscillation
l
l
l
The condition T(jω) = -1 is in fact how we build oscillators, which are
inherently unstable
If the circuit has T = -1 at a particular frequency, then the gain at that
frequency is theoretically infinite
Any noise or disturbance can leads to a strong oscillation at this
particular frequency
–
14
Take EECS 142 to understand this in more detail
EE 105 Fall 2016
Prof. A. M. Niknejad
Feedback Types: Series-Shunt
l
l
So far we have looked at feedback from the perspective of ideal
signals. In a real circuit, the feedback signals can be currents and/or
voltages
A system that samples the output voltage and feeds back an input
voltage is known as series-shunt feedback
–
–
15
A shunt connection is used to sample the output voltage
A series connection is used to subtract or add the input voltage
EE 105 Fall 2016
Prof. A. M. Niknejad
Shunt-Series Feedback
l
A system that samples the output current and feeds back an input
current is known as shunt-series feedback
–
–
16
A series connection is used to sample the output current
A shunt connection is used to subtract or add the input current
EE 105 Fall 2016
Prof. A. M. Niknejad
Series Feedback
l
A system that samples the output current and feeds back an input
voltage is known as series-series feedback, or simply series feedback
–
–
17
A series connection is used to sample the output current
A series connection is used to subtract or add the input voltage
EE 105 Fall 2016
Prof. A. M. Niknejad
Shunt Feedback
l
A system that samples the output voltage and feeds back an input
current is known as shunt-shunt feedback, or simply shunt feedback
–
–
18
A shunt connection is used to sample the output voltage
A shunt connection is used to subtract or add the input current
EE 105 Fall 2016
Prof. A. M. Niknejad
Effect on Input Impedance: Series at Input
l
l
If there is series negative feedback at the input, the action of the
feedback tends to raise the input impedance.
Note that error signal is very small if the loop gain is large
verr = vin − fvout
⎛
fG ⎞
= vin ⎜ 1 ⎝ 1+ Gf ⎟⎠
verr
l
l
19
vin
=
1+ T
This reduces the input current by (1+T), or effectively increases the
input impedance by the same amount
For a shunt connection at the input, the opposite is true, the input
impedance drops by (1+T)
EE 105 Fall 2016
Prof. A. M. Niknejad
Effect on Output Impedance: Shunt at Output
l
l
l
l
l
20
Without feedback the output resistance is given by Ro
When there’s shunt feedback at the output, the current flowing the
input of the amplifier through the feedback resistance “loads” the
output because the action of the feedback activates the amplifier
which then draws a large current, making the output impedance much
small than Ro
We can show the output impedance actually drops by (1+T)
This means that a relatively high output impedance amplifier (such as
an OTA) can be used to drive a voltage if sufficient loop gain is used
For a series connection at the output, the opposite is true, the output
impedance increases by (1+T)
EE 105 Fall 2016
Prof. A. M. Niknejad
Gain/Bandwidth Trade-off
21
EE 105 Fall 2016
Prof. A. M. Niknejad
Bandwidth Extension
l
Suppose the core amplifier is single pole with bandwidth:
G0
G( jω ) =
1+ jωτ
22
l
When used feedback, the overall transfer function is given by
l
The bandwidth has expanded by (1 + T)
G0
G( jω )
1+ jωτ
G fb ( jω ) =
=
1+ G( jω ) f 1+ G0 f
1+ jωτ
G0
G fb ( jω ) =
1+ jωτ + G0 f
G0
Gclosed−loop (0)
1+ G0 f
G fb (s) =
=
τ
τ
1+ jω
1+ jω
1+ G0 f
1+ T
EE 105 Fall 2016
Prof. A. M. Niknejad
Circuit Model
l
23
Here’s the equivalent circuit for an amplifier with feedback
EE 105 Fall 2016
Prof. A. M. Niknejad
Circuit Interpretation
l
24
Here we see the action of the feedback is to lower the impedance seen
by the Gm by the loop gain, which expands the bandwidth by the
same factor
EE 105 Fall 2016
Prof. A. M. Niknejad
Gain / Bandwidth Product in Feedback
l
l
Even though the bandwidth expanded by (1+T), the gain drops by the
same factor. So overall the gain-bandwidth (GBW) product is
constant
The GBW product depends only the the Gm of the op-amp and the Cx
internal capacitance (or load in the case of an OTA)
G0
1
GBW = G × BW =
× (1+ T )
1+ T
RoC x
1
GBW = G0 × RoC x = Gm Ro ×
RoC x
Gm
GBW =
Cx
25
EE 105 Fall 2016
Prof. A. M. Niknejad
Unity Gain Frequency
l
l
The GBW product is also known as the unity gain frequency.
To see this, consider the frequency at which the gain drops to unity
G0
| G |=
=1
1+ jω uτ
G02 = 1+ ω u2τ 2
G0
1+ ω τ
2
u
=1
ω u2 = (G02 − 1) / τ 2 ≈ G02 / τ 2
Gm Ro Gm
ω u = G0 / τ =
=
C x Ro C x
26
2
EE 105 Fall 2016
Prof. A. M. Niknejad
Unity Gain Feedback Amplifier
l
27
An amplifier that has a feedback factor f=1, such as a unity gain
buffer, has the full GBW product frequency range
EE 105 Fall 2016
Prof. A. M. Niknejad
Non-Dominant Poles
l
l
l
28
As we have seen, poles in the system tend to make an amplifier less
stable. A single pole cannot do harm since it has a maximum phase
shift of 90°
A second pole in the system is not affected by feedback (prove this)
and it will add phase shift as the frequency approaches this second
pole
For this reason, non-dominant poles should be at a much higher
frequency than the unity-gain frequency
EE 105 Fall 2016
Prof. A. M. Niknejad
Positive Feedback
l
l
l
l
Positive Feedback is also useful
We can create a comparator circuit with hysteresis
Also, as long as T < 1, we can get stable gain … instead of reducing
the gain (negative feedback), positive feedback enhances the gain.
In theory we can boost the gain to any desired level simply by making
T close to unity:
T = 1− ε
l
ε is a very small number
–
–
29
In practice if the gain varies over process / temperature / voltage, then the
circuit can go stable and oscillate
Positive feedback also has a narrow-banding effect