12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
E1.1 Analysis of Circuits (2016-8284)
12: Resonance
Resonance: 12 – 1 / 11
Quadratic Factors
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
2
A quadratic factor in a transfer function is: F (jω) = a (jω) + b (jω) + c.
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 2 / 11
Quadratic Factors
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
2
A quadratic factor in a transfer function is: F (jω) = a (jω) + b (jω) + c.
Case 1: If b2 ≥ 4ac then we can factorize it:
F (jω) = a(jω − p1 )(jω − p2 )
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 2 / 11
Quadratic Factors
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
2
A quadratic factor in a transfer function is: F (jω) = a (jω) + b (jω) + c.
Case 1: If b2 ≥ 4ac then we can factorize it:
F (jω) = a(jω − p1 )(jω − p2 )
where pi =
√
−b± b2 −4ac
.
2a
filter
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 2 / 11
Quadratic Factors
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
2
A quadratic factor in a transfer function is: F (jω) = a (jω) + b (jω) + c.
Case 1: If b2 ≥ 4ac then we can factorize it:
F (jω) = a(jω − p1 )(jω − p2 )
where pi =
√
−b± b2 −4ac
.
2a
filter
• Summary
E1.1 Analysis of Circuits (2016-8284)
Y
X (jω)
=
1
6R2 C 2 (jω)2 +7RCjω+1
Resonance: 12 – 2 / 11
Quadratic Factors
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
2
A quadratic factor in a transfer function is: F (jω) = a (jω) + b (jω) + c.
Case 1: If b2 ≥ 4ac then we can factorize it:
F (jω) = a(jω − p1 )(jω − p2 )
where pi =
√
−b± b2 −4ac
.
2a
filter
• Summary
Y
X (jω)
=
=
E1.1 Analysis of Circuits (2016-8284)
1
6R2 C 2 (jω)2 +7RCjω+1
1
(6jωRC+1)(jωRC+1)
Resonance: 12 – 2 / 11
Quadratic Factors
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
2
A quadratic factor in a transfer function is: F (jω) = a (jω) + b (jω) + c.
Case 1: If b2 ≥ 4ac then we can factorize it:
F (jω) = a(jω − p1 )(jω − p2 )
where pi =
√
−b± b2 −4ac
.
2a
filter
• Summary
Y
X (jω)
=
=
1
6R2 C 2 (jω)2 +7RCjω+1
1
(6jωRC+1)(jωRC+1)
ωc =
E1.1 Analysis of Circuits (2016-8284)
1
0.17
,
RC
RC
Resonance: 12 – 2 / 11
Quadratic Factors
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
2
A quadratic factor in a transfer function is: F (jω) = a (jω) + b (jω) + c.
Case 1: If b2 ≥ 4ac then we can factorize it:
F (jω) = a(jω − p1 )(jω − p2 )
Resonance
where pi =
• Low Pass Filter
• Resonance Peak for LP
√
−b± b2 −4ac
.
2a
filter
• Summary
Y
X (jω)
0
-20
=
=
1
6R2 C 2 (jω)2 +7RCjω+1
1
(6jωRC+1)(jωRC+1)
-40
0.1/RC
E1.1 Analysis of Circuits (2016-8284)
0.3/RC
ω
1/RC
3/RC
ωc =
1
0.17
,
RC
RC
Resonance: 12 – 2 / 11
Quadratic Factors
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
2
A quadratic factor in a transfer function is: F (jω) = a (jω) + b (jω) + c.
Case 1: If b2 ≥ 4ac then we can factorize it:
F (jω) = a(jω − p1 )(jω − p2 )
Resonance
where pi =
• Low Pass Filter
• Resonance Peak for LP
√
−b± b2 −4ac
.
2a
filter
• Summary
Y
X (jω)
0
-20
=
=
1
6R2 C 2 (jω)2 +7RCjω+1
1
(6jωRC+1)(jωRC+1)
-40
0.1/RC
E1.1 Analysis of Circuits (2016-8284)
0.3/RC
ω
1/RC
3/RC
ωc =
1
0.17
,
RC
RC
= |p1 | , |p2 |
Resonance: 12 – 2 / 11
Quadratic Factors
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
2
A quadratic factor in a transfer function is: F (jω) = a (jω) + b (jω) + c.
Case 1: If b2 ≥ 4ac then we can factorize it:
F (jω) = a(jω − p1 )(jω − p2 )
Resonance
where pi =
• Low Pass Filter
• Resonance Peak for LP
√
−b± b2 −4ac
.
2a
filter
• Summary
Y
X (jω)
0
-20
=
=
1
6R2 C 2 (jω)2 +7RCjω+1
1
(6jωRC+1)(jωRC+1)
-40
0.1/RC
0.3/RC
ω
1/RC
3/RC
ωc =
1
0.17
,
RC
RC
= |p1 | , |p2 |
Case 2: If b2 < 4ac, we cannot factorize with real coefficients so we leave
it as a quadratic.
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 2 / 11
Quadratic Factors
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
2
A quadratic factor in a transfer function is: F (jω) = a (jω) + b (jω) + c.
Case 1: If b2 ≥ 4ac then we can factorize it:
F (jω) = a(jω − p1 )(jω − p2 )
Resonance
where pi =
• Low Pass Filter
• Resonance Peak for LP
√
−b± b2 −4ac
.
2a
filter
• Summary
Y
X (jω)
0
-20
=
=
1
6R2 C 2 (jω)2 +7RCjω+1
1
(6jωRC+1)(jωRC+1)
-40
0.1/RC
0.3/RC
ω
1/RC
3/RC
ωc =
1
0.17
,
RC
RC
= |p1 | , |p2 |
Case 2: If b2 < 4ac, we cannot factorize with real coefficients so we leave
it as a quadratic. Sometimes called a quadratic resonance.
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 2 / 11
Quadratic Factors
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
2
A quadratic factor in a transfer function is: F (jω) = a (jω) + b (jω) + c.
Case 1: If b2 ≥ 4ac then we can factorize it:
F (jω) = a(jω − p1 )(jω − p2 )
Resonance
where pi =
• Low Pass Filter
• Resonance Peak for LP
√
−b± b2 −4ac
.
2a
filter
• Summary
Y
X (jω)
0
-20
=
=
1
6R2 C 2 (jω)2 +7RCjω+1
1
(6jωRC+1)(jωRC+1)
-40
0.1/RC
0.3/RC
ω
1/RC
3/RC
ωc =
1
0.17
,
RC
RC
= |p1 | , |p2 |
Case 2: If b2 < 4ac, we cannot factorize with real coefficients so we leave
it as a quadratic. Sometimes called a quadratic resonance.
Any polynomial with real coefficients can be factored into linear and
quadratic factors
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 2 / 11
Quadratic Factors
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
2
A quadratic factor in a transfer function is: F (jω) = a (jω) + b (jω) + c.
Case 1: If b2 ≥ 4ac then we can factorize it:
F (jω) = a(jω − p1 )(jω − p2 )
Resonance
where pi =
• Low Pass Filter
• Resonance Peak for LP
√
−b± b2 −4ac
.
2a
filter
• Summary
Y
X (jω)
0
-20
=
=
1
6R2 C 2 (jω)2 +7RCjω+1
1
(6jωRC+1)(jωRC+1)
-40
0.1/RC
0.3/RC
ω
1/RC
3/RC
ωc =
1
0.17
,
RC
RC
= |p1 | , |p2 |
Case 2: If b2 < 4ac, we cannot factorize with real coefficients so we leave
it as a quadratic. Sometimes called a quadratic resonance.
Any polynomial with real coefficients can be factored into linear and
quadratic factors ⇒ a quadratic factor is as complicated as it gets.
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 2 / 11
Damping Factor and Q
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
2
Suppose b2 < 4ac in F (jω) = a (jω) + b (jω) + c.
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 3 / 11
Damping Factor and Q
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
2
Suppose b2 < 4ac in F (jω) = a (jω) + b (jω) + c.
Low/High freq asymptotes:
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 3 / 11
Damping Factor and Q
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
2
Suppose b2 < 4ac in F (jω) = a (jω) + b (jω) + c.
Low/High freq asymptotes: FLF (jω) = c
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 3 / 11
Damping Factor and Q
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
2
Suppose b2 < 4ac in F (jω) = a (jω) + b (jω) + c.
Low/High freq asymptotes: FLF (jω) = c,
2
FHF (jω) = a (jω)
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 3 / 11
Damping Factor and Q
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
2
Suppose b2 < 4ac in F (jω) = a (jω) + b (jω) + c.
Low/High freq asymptotes: FLF (jω) = c,
2
FHF (jω) = a (jω)
The asymptote magnitudes cross at the corner frequency:
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 3 / 11
Damping Factor and Q
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
2
Suppose b2 < 4ac in F (jω) = a (jω) + b (jω) + c.
Low/High freq asymptotes: FLF (jω) = c,
2
FHF (jω) = a (jω)
The asymptote
magnitudes
cross at the corner frequency:
filter
• Summary
E1.1 Analysis of Circuits (2016-8284)
pc
2
a (jωc ) = |c| ⇒ ωc = a .
Resonance: 12 – 3 / 11
Damping Factor and Q
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
2
Suppose b2 < 4ac in F (jω) = a (jω) + b (jω) + c.
Low/High freq asymptotes: FLF (jω) = c,
2
FHF (jω) = a (jω)
The asymptote
magnitudes
cross at the corner frequency:
pc
2
a (jωc ) = |c| ⇒ ωc = a .
We define the damping factor , “zeta”, to be ζ = √ b
E1.1 Analysis of Circuits (2016-8284)
4ac
Resonance: 12 – 3 / 11
Damping Factor and Q
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
2
Suppose b2 < 4ac in F (jω) = a (jω) + b (jω) + c.
Low/High freq asymptotes: FLF (jω) = c,
2
FHF (jω) = a (jω)
The asymptote
magnitudes
cross at the corner frequency:
pc
2
a (jωc ) = |c| ⇒ ωc = a .
We define the damping factor , “zeta”, to be ζ = √ b
E1.1 Analysis of Circuits (2016-8284)
4ac
=
bωc
2c
=
b
2aωc
Resonance: 12 – 3 / 11
Damping Factor and Q
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
2
Suppose b2 < 4ac in F (jω) = a (jω) + b (jω) + c.
Low/High freq asymptotes: FLF (jω) = c,
2
FHF (jω) = a (jω)
The asymptote
magnitudes
cross at the corner frequency:
pc
2
a (jωc ) = |c| ⇒ ωc = a .
We define the damping factor , “zeta”, to be ζ = √ b
E1.1 Analysis of Circuits (2016-8284)
=
4ac
2
ω
ω
⇒ F (jω) = c
j ωc + 2ζ j ωc + 1
bωc
2c
=
b
2aωc
Resonance: 12 – 3 / 11
Damping Factor and Q
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
2
Suppose b2 < 4ac in F (jω) = a (jω) + b (jω) + c.
Low/High freq asymptotes: FLF (jω) = c,
2
FHF (jω) = a (jω)
The asymptote
magnitudes
cross at the corner frequency:
pc
2
a (jωc ) = |c| ⇒ ωc = a .
We define the damping factor , “zeta”, to be ζ = √ b
=
4ac
2
ω
ω
⇒ F (jω) = c
j ωc + 2ζ j ωc + 1
bωc
2c
=
b
2aωc
Properties to notice in this expression:
(a) c is just an overall scale factor.
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 3 / 11
Damping Factor and Q
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
2
Suppose b2 < 4ac in F (jω) = a (jω) + b (jω) + c.
Low/High freq asymptotes: FLF (jω) = c,
2
FHF (jω) = a (jω)
The asymptote
magnitudes
cross at the corner frequency:
pc
2
a (jωc ) = |c| ⇒ ωc = a .
We define the damping factor , “zeta”, to be ζ = √ b
=
4ac
2
ω
ω
⇒ F (jω) = c
j ωc + 2ζ j ωc + 1
bωc
2c
=
b
2aωc
Properties to notice in this expression:
(a) c is just an overall scale factor.
(b) ωc just scales the frequency axis since F (jω) is a function of ωω .
c
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 3 / 11
Damping Factor and Q
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
2
Suppose b2 < 4ac in F (jω) = a (jω) + b (jω) + c.
Low/High freq asymptotes: FLF (jω) = c,
2
FHF (jω) = a (jω)
The asymptote
magnitudes
cross at the corner frequency:
pc
2
a (jωc ) = |c| ⇒ ωc = a .
We define the damping factor , “zeta”, to be ζ = √ b
=
4ac
2
ω
ω
⇒ F (jω) = c
j ωc + 2ζ j ωc + 1
bωc
2c
=
b
2aωc
Properties to notice in this expression:
(a) c is just an overall scale factor.
(b) ωc just scales the frequency axis since F (jω) is a function of ωω .
c
(c) The shape of the F (jω) graphs is determined entirely by ζ .
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 3 / 11
Damping Factor and Q
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
2
Suppose b2 < 4ac in F (jω) = a (jω) + b (jω) + c.
Low/High freq asymptotes: FLF (jω) = c,
2
FHF (jω) = a (jω)
The asymptote
magnitudes
cross at the corner frequency:
pc
2
a (jωc ) = |c| ⇒ ωc = a .
We define the damping factor , “zeta”, to be ζ = √ b
=
4ac
2
ω
ω
⇒ F (jω) = c
j ωc + 2ζ j ωc + 1
bωc
2c
=
b
2aωc
Properties to notice in this expression:
(a) c is just an overall scale factor.
(b) ωc just scales the frequency axis since F (jω) is a function of ωω .
c
(c) The shape of the F (jω) graphs is determined entirely by ζ .
(d) The quadratic cannot be factorized ⇔ b2 < 4ac ⇔ |ζ| < 1.
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 3 / 11
Damping Factor and Q
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
2
Suppose b2 < 4ac in F (jω) = a (jω) + b (jω) + c.
Low/High freq asymptotes: FLF (jω) = c,
2
FHF (jω) = a (jω)
The asymptote
magnitudes
cross at the corner frequency:
pc
2
a (jωc ) = |c| ⇒ ωc = a .
We define the damping factor , “zeta”, to be ζ = √ b
=
4ac
2
ω
ω
⇒ F (jω) = c
j ωc + 2ζ j ωc + 1
bωc
2c
=
b
2aωc
Properties to notice in this expression:
(a) c is just an overall scale factor.
(b) ωc just scales the frequency axis since F (jω) is a function of ωω .
c
(c) The shape of the F (jω) graphs is determined entirely by ζ .
(d) The quadratic cannot be factorized ⇔ b2 < 4ac ⇔ |ζ| < 1.
(e) At ω = ωc , asymptote gain = c but F (jω) = c × 2jζ .
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 3 / 11
Damping Factor and Q
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
2
Suppose b2 < 4ac in F (jω) = a (jω) + b (jω) + c.
Low/High freq asymptotes: FLF (jω) = c,
2
FHF (jω) = a (jω)
The asymptote
magnitudes
cross at the corner frequency:
pc
2
a (jωc ) = |c| ⇒ ωc = a .
We define the damping factor , “zeta”, to be ζ = √ b
=
4ac
2
ω
ω
⇒ F (jω) = c
j ωc + 2ζ j ωc + 1
bωc
2c
=
b
2aωc
Properties to notice in this expression:
(a) c is just an overall scale factor.
(b) ωc just scales the frequency axis since F (jω) is a function of ωω .
c
(c) The shape of the F (jω) graphs is determined entirely by ζ .
(d) The quadratic cannot be factorized ⇔ b2 < 4ac ⇔ |ζ| < 1.
(e) At ω = ωc , asymptote gain = c but F (jω) = c × 2jζ .
Alternatively, we sometimes use the quality factor , Q ≈
E1.1 Analysis of Circuits (2016-8284)
1
2ζ
=
√
ac
b .
Resonance: 12 – 3 / 11
Parallel RLC
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Y
I
=
1
1
1
R + jωL +jωC
=
jωL
L
LC(jω)2 + R
jω+1
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 4 / 11
Parallel RLC
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Y
I
1
=
ωc =
1
1
R + jωL +jωC
pc
a
=
= 1000, ζ
jωL
L
LC(jω)2 + R
jω+1
b
= 0.083
= √4ac
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 4 / 11
Parallel RLC
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Y
I
1
=
ωc =
1
1
R + jωL +jωC
pc
a
=
= 1000, ζ
jωL
L
LC(jω)2 + R
jω+1
b
= 0.083
= √4ac
1
.
Asymptotes: jωL and jωC
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 4 / 11
Parallel RLC
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Y
I
1
=
ωc =
1
1
R + jωL +jωC
pc
a
=
= 1000, ζ
jωL
L
LC(jω)2 + R
jω+1
b
= 0.083
= √4ac
1
.
Asymptotes: jωL and jωC
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
Power absorbed by resistor ∝ Y 2 . It peaks quite
sharply at ω = 1000.
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 4 / 11
Parallel RLC
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Y
I
1
=
ωc =
1
1
R + jωL +jωC
pc
a
=
= 1000, ζ
jωL
L
LC(jω)2 + R
jω+1
b
= 0.083
= √4ac
1
.
Asymptotes: jωL and jωC
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
Power absorbed by resistor ∝ Y 2 . It peaks quite
sharply at ω = 1000. The resonant frequency,
ωr , is when the impedance is purely real:
at ωr = 1000, ZRLC = YI = R.
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 4 / 11
Parallel RLC
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Y
I
1
=
ωc =
1
1
R + jωL +jωC
pc
a
=
= 1000, ζ
jωL
L
LC(jω)2 + R
jω+1
b
= 0.083
= √4ac
1
.
Asymptotes: jωL and jωC
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
Power absorbed by resistor ∝ Y 2 . It peaks quite
sharply at ω = 1000. The resonant frequency,
ωr , is when the impedance is purely real:
at ωr = 1000, ZRLC = YI = R.
A system with a strong peak in power absorption
is a resonant system.
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 4 / 11
Parallel RLC
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Y
I
1
=
ωc =
1
1
R + jωL +jωC
pc
a
=
= 1000, ζ
jωL
L
LC(jω)2 + R
jω+1
b
= 0.083
= √4ac
1
.
Asymptotes: jωL and jωC
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
Power absorbed by resistor ∝ Y 2 . It peaks quite
sharply at ω = 1000. The resonant frequency,
ωr , is when the impedance is purely real:
at ωr = 1000, ZRLC = YI = R.
A system with a strong peak in power absorption
is a resonant system.
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 4 / 11
Behaviour at Resonance
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
ω = 1000 ⇒ ZL = 100j, ZC = −100j .
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 5 / 11
Behaviour at Resonance
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
ω = 1000 ⇒ ZL = 100j, ZC = −100j .
ZL = −ZC ⇒ IL = −IC
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 5 / 11
Behaviour at Resonance
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
ω = 1000 ⇒ ZL = 100j, ZC = −100j .
ZL = −ZC ⇒ IL = −IC
⇒ I = IR + IL + IC = IR = 1
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 5 / 11
Behaviour at Resonance
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
ω = 1000 ⇒ ZL = 100j, ZC = −100j .
ZL = −ZC ⇒ IL = −IC
⇒ I = IR + IL + IC = IR = 1
⇒ Y = IR R = 600∠0◦ = 56 dBV
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 5 / 11
Behaviour at Resonance
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
ω = 1000 ⇒ ZL = 100j, ZC = −100j .
ZL = −ZC ⇒ IL = −IC
⇒ I = IR + IL + IC = IR = 1
⇒ Y = IR R = 600∠0◦ = 56 dBV
600
⇒ IL = ZYL = 100j
= −6j
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 5 / 11
Behaviour at Resonance
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
ω = 1000 ⇒ ZL = 100j, ZC = −100j .
ZL = −ZC ⇒ IL = −IC
⇒ I = IR + IL + IC = IR = 1
⇒ Y = IR R = 600∠0◦ = 56 dBV
600
⇒ IL = ZYL = 100j
= −6j
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 5 / 11
Behaviour at Resonance
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
ω = 1000 ⇒ ZL = 100j, ZC = −100j .
ZL = −ZC ⇒ IL = −IC
⇒ I = IR + IL + IC = IR = 1
⇒ Y = IR R = 600∠0◦ = 56 dBV
600
⇒ IL = ZYL = 100j
= −6j
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 5 / 11
Behaviour at Resonance
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
ω = 1000 ⇒ ZL = 100j, ZC = −100j .
ZL = −ZC ⇒ IL = −IC
⇒ I = IR + IL + IC = IR = 1
⇒ Y = IR R = 600∠0◦ = 56 dBV
600
⇒ IL = ZYL = 100j
= −6j
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 5 / 11
Behaviour at Resonance
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
ω = 1000 ⇒ ZL = 100j, ZC = −100j .
ZL = −ZC ⇒ IL = −IC
⇒ I = IR + IL + IC = IR = 1
⇒ Y = IR R = 600∠0◦ = 56 dBV
600
⇒ IL = ZYL = 100j
= −6j
Large currents in L and C exactly cancel out ⇒ IR = I and Z = R (real)
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 5 / 11
Away from resonance
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
ω = 2000 ⇒ ZL = 200j, ZC = −50j
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 6 / 11
Away from resonance
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
ω = 2000 ⇒ ZL = 200j, ZC = −50j
−1
Z = R1 + Z1L + Z1C
= 66∠ − 84◦
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 6 / 11
Away from resonance
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
ω = 2000 ⇒ ZL = 200j, ZC = −50j
−1
Z = R1 + Z1L + Z1C
= 66∠ − 84◦
Y = I × Z = 66∠ − 84◦ = 36 dBV
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 6 / 11
Away from resonance
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
ω = 2000 ⇒ ZL = 200j, ZC = −50j
−1
Z = R1 + Z1L + Z1C
= 66∠ − 84◦
Y = I × Z = 66∠ − 84◦ = 36 dBV
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 6 / 11
Away from resonance
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
ω = 2000 ⇒ ZL = 200j, ZC = −50j
−1
Z = R1 + Z1L + Z1C
= 66∠ − 84◦
Y = I × Z = 66∠ − 84◦ = 36 dBV
IR = YR = 0.11∠ − 84◦
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 6 / 11
Away from resonance
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
ω = 2000 ⇒ ZL = 200j, ZC = −50j
−1
Z = R1 + Z1L + Z1C
= 66∠ − 84◦
Y = I × Z = 66∠ − 84◦ = 36 dBV
IR = YR = 0.11∠ − 84◦
IL = ZYL = 0.33∠ − 174◦
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 6 / 11
Away from resonance
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
ω = 2000 ⇒ ZL = 200j, ZC = −50j
−1
Z = R1 + Z1L + Z1C
= 66∠ − 84◦
Y = I × Z = 66∠ − 84◦ = 36 dBV
IR = YR = 0.11∠ − 84◦
IL = ZYL = 0.33∠ − 174◦ , IC = 1.33∠ + 6◦
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 6 / 11
Away from resonance
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
ω = 2000 ⇒ ZL = 200j, ZC = −50j
−1
Z = R1 + Z1L + Z1C
= 66∠ − 84◦
Y = I × Z = 66∠ − 84◦ = 36 dBV
IR = YR = 0.11∠ − 84◦
IL = ZYL = 0.33∠ − 174◦ , IC = 1.33∠ + 6◦
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 6 / 11
Away from resonance
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
ω = 2000 ⇒ ZL = 200j, ZC = −50j
−1
Z = R1 + Z1L + Z1C
= 66∠ − 84◦
Y = I × Z = 66∠ − 84◦ = 36 dBV
IR = YR = 0.11∠ − 84◦
IL = ZYL = 0.33∠ − 174◦ , IC = 1.33∠ + 6◦
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 6 / 11
Away from resonance
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
ω = 2000 ⇒ ZL = 200j, ZC = −50j
−1
Z = R1 + Z1L + Z1C
= 66∠ − 84◦
Y = I × Z = 66∠ − 84◦ = 36 dBV
IR = YR = 0.11∠ − 84◦
IL = ZYL = 0.33∠ − 174◦ , IC = 1.33∠ + 6◦
Most current now flows through C , only 0.11 through R.
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 6 / 11
Bandwidth and Q
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Y
I
=
1
1/R+j(ωC−1/ωL)
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 7 / 11
Bandwidth and Q
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
Y
I
=
1
1/R+j(ωC−1/ωL)
Bandwidth is the range of frequencies for
Y 2
which is greater than half its peak.
I
Also called half-power bandwidth or 3dB
bandwidth.
filter
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 7 / 11
Bandwidth and Q
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
Y
I
=
1
1/R+j(ωC−1/ωL)
Bandwidth is the range of frequencies for
Y 2
which is greater than half its peak.
I
Also called half-power bandwidth or 3dB
bandwidth.
filter
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 7 / 11
Bandwidth and Q
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
Y
I
=
1
1/R+j(ωC−1/ωL)
Bandwidth is the range of frequencies for
Y 2
which is greater than half its peak.
I
Also called half-power bandwidth or 3dB
bandwidth.
Y 2
=
I
E1.1 Analysis of Circuits (2016-8284)
1
(1/R) +(ωC−1/ωL)2
2
Resonance: 12 – 7 / 11
Bandwidth and Q
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
Y
I
=
1
1/R+j(ωC−1/ωL)
Bandwidth is the range of frequencies for
Y 2
which is greater than half its peak.
I
Also called half-power bandwidth or 3dB
bandwidth.
Y 2
=
I
1
(1/R) +(ωC−1/ωL)2
2
Y
2
Peak is I (ω0 ) = R2 @ ω0 = 1000
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 7 / 11
Bandwidth and Q
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
Y
I
=
1
1/R+j(ωC−1/ωL)
Bandwidth is the range of frequencies for
Y 2
which is greater than half its peak.
I
Also called half-power bandwidth or 3dB
bandwidth.
Y 2
=
I
1
(1/R) +(ωC−1/ωL)2
2
Y
2
Peak is I (ω0 ) = R2 @ ω0 = 1000
2
2
Y
1 Y
At ω3dB : I (ω3dB ) = 2 I (ω0 )
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 7 / 11
Bandwidth and Q
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
Y
I
=
1
1/R+j(ωC−1/ωL)
Bandwidth is the range of frequencies for
Y 2
which is greater than half its peak.
I
Also called half-power bandwidth or 3dB
bandwidth.
Y 2
=
I
1
(1/R) +(ωC−1/ωL)2
2
Y
2
Peak is I (ω0 ) = R2 @ ω0 = 1000
2
2
Y
1 Y
At ω3dB : I (ω3dB ) = 2 I (ω0 )
1
2
1
( /R) +(ω3dB C−1/ω3dB L)2
E1.1 Analysis of Circuits (2016-8284)
=
R2
2
Resonance: 12 – 7 / 11
Bandwidth and Q
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
Y
I
=
1
1/R+j(ωC−1/ωL)
Bandwidth is the range of frequencies for
Y 2
which is greater than half its peak.
I
Also called half-power bandwidth or 3dB
bandwidth.
Y 2
=
I
1
(1/R) +(ωC−1/ωL)2
2
Y
2
Peak is I (ω0 ) = R2 @ ω0 = 1000
2
2
Y
1 Y
At ω3dB : I (ω3dB ) = 2 I (ω0 )
1
(1/R)2 +(ω3dB C−1/ω
E1.1 Analysis of Circuits (2016-8284)
3dB L)
2
=
2
R
2
⇒ 1 + ω3dB RC −
R
ω3dB L
2
=2
Resonance: 12 – 7 / 11
Bandwidth and Q
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
Y
I
=
1
1/R+j(ωC−1/ωL)
Bandwidth is the range of frequencies for
Y 2
which is greater than half its peak.
I
Also called half-power bandwidth or 3dB
bandwidth.
Y 2
=
I
1
(1/R) +(ωC−1/ωL)2
2
Y
2
Peak is I (ω0 ) = R2 @ ω0 = 1000
2
2
Y
1 Y
At ω3dB : I (ω3dB ) = 2 I (ω0 )
1
(1/R)2 +(ω3dB C−1/ω
3dB L)
2
=
2
R
2
⇒ 1 + ω3dB RC −
R
ω3dB L
2
=2
ω3dB RC − R/ω3dB L = ±1
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 7 / 11
Bandwidth and Q
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
Y
I
=
1
1/R+j(ωC−1/ωL)
Bandwidth is the range of frequencies for
Y 2
which is greater than half its peak.
I
Also called half-power bandwidth or 3dB
bandwidth.
Y 2
=
I
1
(1/R) +(ωC−1/ωL)2
2
Y
2
Peak is I (ω0 ) = R2 @ ω0 = 1000
2
2
Y
1 Y
At ω3dB : I (ω3dB ) = 2 I (ω0 )
1
(1/R)2 +(ω3dB C−1/ω
3dB L)
2
=
2
R
2
⇒ 1 + ω3dB RC −
ω3dB RC − R/ω3dB L = ±1 ⇒
E1.1 Analysis of Circuits (2016-8284)
R
ω3dB L
2
=2
2
ω3dB
RLC ± ω3dB L − R = 0
Resonance: 12 – 7 / 11
Bandwidth and Q
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
Y
I
=
1
1/R+j(ωC−1/ωL)
Bandwidth is the range of frequencies for
Y 2
which is greater than half its peak.
I
Also called half-power bandwidth or 3dB
bandwidth.
Y 2
=
I
1
(1/R) +(ωC−1/ωL)2
2
Y
2
Peak is I (ω0 ) = R2 @ ω0 = 1000
2
2
Y
1 Y
At ω3dB : I (ω3dB ) = 2 I (ω0 )
1
(1/R)2 +(ω3dB C−1/ω
3dB L)
2
=
2
R
2
⇒ 1 + ω3dB RC −
ω3dB RC − R/ω3dB L = ±1 ⇒
Positive roots: ω3dB =
E1.1 Analysis of Circuits (2016-8284)
R
ω3dB L
2
=2
2
ω3dB
RLC ± ω3dB L − R = 0
√
±L+ L2 +4R2 LC
2RLC
= {920, 1086} rad/s
Resonance: 12 – 7 / 11
Bandwidth and Q
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
Y
I
=
1
1/R+j(ωC−1/ωL)
Bandwidth is the range of frequencies for
Y 2
which is greater than half its peak.
I
Also called half-power bandwidth or 3dB
bandwidth.
Y 2
=
I
1
(1/R) +(ωC−1/ωL)2
2
Y
2
Peak is I (ω0 ) = R2 @ ω0 = 1000
2
2
Y
1 Y
At ω3dB : I (ω3dB ) = 2 I (ω0 )
1
(1/R)2 +(ω3dB C−1/ω
3dB L)
2
=
2
R
2
⇒ 1 + ω3dB RC −
ω3dB RC − R/ω3dB L = ±1 ⇒
Positive roots: ω3dB =
R
ω3dB L
2
=2
2
ω3dB
RLC ± ω3dB L − R = 0
√
±L+ L2 +4R2 LC
2RLC
= {920, 1086} rad/s
Bandwidth: B = 1086 − 920 = 167 rad/s.
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 7 / 11
Bandwidth and Q
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
Y
I
=
1
1/R+j(ωC−1/ωL)
Bandwidth is the range of frequencies for
Y 2
which is greater than half its peak.
I
Also called half-power bandwidth or 3dB
bandwidth.
Y 2
=
I
1
(1/R) +(ωC−1/ωL)2
2
Y
2
Peak is I (ω0 ) = R2 @ ω0 = 1000
2
2
Y
1 Y
At ω3dB : I (ω3dB ) = 2 I (ω0 )
1
(1/R)2 +(ω3dB C−1/ω
3dB L)
2
=
2
R
2
⇒ 1 + ω3dB RC −
ω3dB RC − R/ω3dB L = ±1 ⇒
Positive roots: ω3dB =
R
ω3dB L
2
=2
2
ω3dB
RLC ± ω3dB L − R = 0
√
±L+ L2 +4R2 LC
2RLC
= {920, 1086} rad/s
Bandwidth: B = 1086 − 920 = 167 rad/s.
E1.1 Analysis of Circuits (2016-8284)
Q factor ≈
ω0
B
=
1
2ζ
= 6. (Q = “Quality”)
Resonance: 12 – 7 / 11
Power and Energy at Resonance
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 8 / 11
Power and Energy at Resonance
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Absorbed Power =v(t)i(t):
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 8 / 11
Power and Energy at Resonance
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Absorbed Power =v(t)i(t):
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 8 / 11
Power and Energy at Resonance
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Absorbed Power =v(t)i(t):
PL and PC opposite and ≫ PR .
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 8 / 11
Power and Energy at Resonance
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Absorbed Power =v(t)i(t):
PL and PC opposite and ≫ PR .
Stored Energy = 12 Li2L + 21 Cy 2 :
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 8 / 11
Power and Energy at Resonance
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Absorbed Power =v(t)i(t):
PL and PC opposite and ≫ PR .
Stored Energy = 12 Li2L + 21 Cy 2 :
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 8 / 11
Power and Energy at Resonance
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Absorbed Power =v(t)i(t):
PL and PC opposite and ≫ PR .
Stored Energy = 12 Li2L + 21 Cy 2 :
sloshes between L and C .
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 8 / 11
Power and Energy at Resonance
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
Absorbed Power =v(t)i(t):
PL and PC opposite and ≫ PR .
Stored Energy = 12 Li2L + 21 Cy 2 :
sloshes between L and C .
Q , ω × Wstored ÷ P R
filter
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 8 / 11
Power and Energy at Resonance
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
Absorbed Power =v(t)i(t):
PL and PC opposite and ≫ PR .
Stored Energy = 12 Li2L + 21 Cy 2 :
sloshes between L and C .
Q , ω × Wstored ÷ P R
2
2
= ω × 21 C |IR| ÷ 21 |I| R= ωRC
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 8 / 11
Power and Energy at Resonance
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
Absorbed Power =v(t)i(t):
PL and PC opposite and ≫ PR .
Stored Energy = 12 Li2L + 21 Cy 2 :
sloshes between L and C .
Q , ω × Wstored ÷ P R
2
2
= ω × 21 C |IR| ÷ 21 |I| R= ωRC
• Summary
Q , ω× peak stored energy ÷ average power loss.
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 8 / 11
Low Pass Filter
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Y
X
=
1/jωC
1
R+jωL+ jωC
=
1
LC(jω) +RCjω+1
2
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 9 / 11
Low Pass Filter
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Y
X
=
1/jωC
1
R+jωL+ jωC
=
1
LC(jω) +RCjω+1
2
1
(jω)−2 .
Asymptotes: 1 and LC
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 9 / 11
Low Pass Filter
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
Y
X
1/jωC
=
1
R+jωL+ jωC
=
1
LC(jω) +RCjω+1
2
1
(jω)−2 .
Asymptotes: 1 and LC
ωc =
pc
a
= 1000, ζ =
√b
4ac
=
R
200
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 9 / 11
Low Pass Filter
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
Y
X
1/jωC
=
1
R+jωL+ jωC
=
1
LC(jω) +RCjω+1
2
1
(jω)−2 .
Asymptotes: 1 and LC
ωc =
pc
a
= 1000, ζ =
√b
4ac
=
R
200
@ωc : ZL = −ZC = 100j , I = X
R
• Summary
C
L
R
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 9 / 11
Low Pass Filter
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
Y
X
1/jωC
=
1
R+jωL+ jωC
=
1
LC(jω) +RCjω+1
2
1
(jω)−2 .
Asymptotes: 1 and LC
pc
√b
4ac
=
R
200
@ωc : ZL = −ZC = 100j , I =
X
R,
ωc =
a
• Summary
C
= 1000, ζ =
Y =
X
1
RCω
=
1
2ζ ,
Y
∠X
= − π2
L
R
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 9 / 11
Low Pass Filter
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Y
X
1/jωC
=
1
R+jωL+ jωC
=
1
LC(jω) +RCjω+1
2
1
(jω)−2 .
Asymptotes: 1 and LC
pc
√b
4ac
=
R
200
filter
@ωc : ZL = −ZC = 100j , I =
X
R,
• Summary
Magntitude Plot:
Resonance
• Low Pass Filter
• Resonance Peak for LP
ωc =
a
= 1000, ζ =
Y =
X
1
RCω
=
1
2ζ ,
Y
∠X
= − π2
R=20, ζ =0.1
20
0
C
L
-20
R
E1.1 Analysis of Circuits (2016-8284)
-40
100
1k
ω (rad/s)
10k
Resonance: 12 – 9 / 11
Low Pass Filter
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
Y
X
1/jωC
=
1
R+jωL+ jωC
=
1
LC(jω) +RCjω+1
2
1
(jω)−2 .
Asymptotes: 1 and LC
pc
√b
4ac
=
R
200
@ωc : ZL = −ZC = 100j , I =
X
R,
ωc =
a
= 1000, ζ =
Y =
X
1
RCω
=
1
2ζ ,
Y
∠X
= − π2
Magntitude Plot:
Small ζ ⇒ less loss, higher peak, smaller bandwidth.
R=5, ζ =0.03
R=20, ζ =0.1
20
0
C
L
-20
R
E1.1 Analysis of Circuits (2016-8284)
-40
100
1k
ω (rad/s)
10k
Resonance: 12 – 9 / 11
Low Pass Filter
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
Y
X
1/jωC
=
1
R+jωL+ jωC
=
1
LC(jω) +RCjω+1
2
1
(jω)−2 .
Asymptotes: 1 and LC
pc
√b
4ac
=
R
200
@ωc : ZL = −ZC = 100j , I =
X
R,
ωc =
a
= 1000, ζ =
Y =
X
1
RCω
=
1
2ζ ,
Y
∠X
= − π2
Magntitude Plot:
Small ζ ⇒ less loss, higher peak, smaller bandwidth.
Large ζ more loss, smaller peak at a lower ω , larger bandwidth.
R=5, ζ =0.03
R=20, ζ =0.1
R=60, ζ =0.3
20
0
C
L
-20
R
E1.1 Analysis of Circuits (2016-8284)
-40
100
1k
ω (rad/s)
10k
Resonance: 12 – 9 / 11
Low Pass Filter
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
Y
X
1/jωC
=
1
R+jωL+ jωC
=
1
LC(jω) +RCjω+1
2
1
(jω)−2 .
Asymptotes: 1 and LC
pc
√b
4ac
=
R
200
@ωc : ZL = −ZC = 100j , I =
X
R,
ωc =
a
= 1000, ζ =
Y =
X
1
RCω
=
1
2ζ ,
Y
∠X
= − π2
Magntitude Plot:
Small ζ ⇒ less loss, higher peak, smaller bandwidth.
Large ζ more loss, smaller peak at a lower ω , larger bandwidth.
R=5, ζ =0.03
R=20, ζ =0.1
R=60, ζ =0.3
R=120, ζ =0.6
20
0
C
L
-20
R
E1.1 Analysis of Circuits (2016-8284)
-40
100
1k
ω (rad/s)
10k
Resonance: 12 – 9 / 11
Low Pass Filter
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
Y
X
1/jωC
=
1
R+jωL+ jωC
=
1
LC(jω) +RCjω+1
2
1
(jω)−2 .
Asymptotes: 1 and LC
pc
√b
4ac
=
R
200
@ωc : ZL = −ZC = 100j , I =
X
R,
ωc =
a
= 1000, ζ =
Y =
X
1
RCω
=
1
2ζ ,
Y
∠X
= − π2
Magntitude Plot:
Small ζ ⇒ less loss, higher peak, smaller bandwidth.
Large ζ more loss, smaller peak at a lower ω , larger bandwidth.
Phase Plot:
0
C
L
0
R=20, ζ =0.1
π
R=5, ζ =0.03
R=20, ζ =0.1
R=60, ζ =0.3
R=120, ζ =0.6
20
-0.5
-20
R
E1.1 Analysis of Circuits (2016-8284)
-40
100
1k
ω (rad/s)
10k
-1
100
1k
ω (rad/s)
10k
Resonance: 12 – 9 / 11
Low Pass Filter
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
Y
X
1/jωC
=
1
R+jωL+ jωC
=
1
LC(jω) +RCjω+1
2
1
(jω)−2 .
Asymptotes: 1 and LC
pc
√b
4ac
=
R
200
@ωc : ZL = −ZC = 100j , I =
X
R,
ωc =
a
= 1000, ζ =
Y =
X
1
RCω
=
1
2ζ ,
Y
∠X
= − π2
Magntitude Plot:
Small ζ ⇒ less loss, higher peak, smaller bandwidth.
Large ζ more loss, smaller peak at a lower ω , larger bandwidth.
Phase Plot:
Small ζ ⇒ fast phase change: π over 2ζ decades.
0
C
L
0
π
R=5, ζ =0.03
R=20, ζ =0.1
R=60, ζ =0.3
R=120, ζ =0.6
20
R=5, ζ =0.03
R=20, ζ =0.1
-0.5
-20
R
E1.1 Analysis of Circuits (2016-8284)
-40
100
1k
ω (rad/s)
10k
-1
100
1k
ω (rad/s)
10k
Resonance: 12 – 9 / 11
Low Pass Filter
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
Y
X
1/jωC
=
1
R+jωL+ jωC
=
1
LC(jω) +RCjω+1
2
1
(jω)−2 .
Asymptotes: 1 and LC
pc
√b
4ac
=
R
200
@ωc : ZL = −ZC = 100j , I =
X
R,
ωc =
a
= 1000, ζ =
Y =
X
1
RCω
=
1
2ζ ,
Y
∠X
= − π2
Magntitude Plot:
Small ζ ⇒ less loss, higher peak, smaller bandwidth.
Large ζ more loss, smaller peak at a lower ω , larger bandwidth.
Phase Plot:
Small ζ ⇒ fast phase change: π over 2ζ decades.
0
C
L
0
π
R=5, ζ =0.03
R=20, ζ =0.1
R=60, ζ =0.3
R=120, ζ =0.6
20
R=5, ζ =0.03
R=20, ζ =0.1
R=60, ζ =0.3
-0.5
-20
R
E1.1 Analysis of Circuits (2016-8284)
-40
100
1k
ω (rad/s)
10k
-1
100
501 1k 2.00k
ω (rad/s)
10k
Resonance: 12 – 9 / 11
Low Pass Filter
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
Y
X
1/jωC
=
1
R+jωL+ jωC
=
1
LC(jω) +RCjω+1
2
1
(jω)−2 .
Asymptotes: 1 and LC
pc
√b
4ac
=
R
200
@ωc : ZL = −ZC = 100j , I =
X
R,
ωc =
a
= 1000, ζ =
Y =
X
1
RCω
=
1
2ζ ,
Y
∠X
= − π2
Magntitude Plot:
Small ζ ⇒ less loss, higher peak, smaller bandwidth.
Large ζ more loss, smaller peak at a lower ω , larger bandwidth.
Phase Plot:
Small ζ ⇒ fast phase change: π over 2ζ decades.
0
C
L
0
R=5, ζ =0.03
R=20, ζ =0.1
R=60, ζ =0.3
R=120, ζ =0.6
π
R=5, ζ =0.03
R=20, ζ =0.1
R=60, ζ =0.3
R=120, ζ =0.6
20
-0.5
-20
R
E1.1 Analysis of Circuits (2016-8284)
-40
100
1k
ω (rad/s)
10k
-1
100
251
1k
ω (rad/s)
3.98k
10k
Resonance: 12 – 9 / 11
Low Pass Filter
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
Y
X
1/jωC
=
1
R+jωL+ jωC
=
1
LC(jω) +RCjω+1
2
1
(jω)−2 .
Asymptotes: 1 and LC
pc
√b
4ac
=
R
200
@ωc : ZL = −ZC = 100j , I =
X
R,
ωc =
a
= 1000, ζ =
Y =
X
1
RCω
=
1
2ζ ,
Y
∠X
= − π2
Magntitude Plot:
Small ζ ⇒ less loss, higher peak, smaller bandwidth.
Large ζ more loss, smaller peak at a lower ω , larger bandwidth.
Phase Plot:
Small ζ ⇒ fast
π over 2ζ decades.
phase change:
Y
≈ −π
1 + ζ1 log10 ωω for 10−ζ < ωω < 10+ζ
∠X
2
c
0
L
0
R=5, ζ =0.03
R=20, ζ =0.1
R=60, ζ =0.3
R=120, ζ =0.6
π
R=5, ζ =0.03
R=20, ζ =0.1
R=60, ζ =0.3
R=120, ζ =0.6
20
C
c
-0.5
-20
R
E1.1 Analysis of Circuits (2016-8284)
-40
100
1k
ω (rad/s)
10k
-1
100
251
1k
ω (rad/s)
3.98k
10k
Resonance: 12 – 9 / 11
Resonance Peak for LP filter
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Y
X
=
1
LC(jω) +RCjω+1
2
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 10 / 11
Resonance Peak for LP filter
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Y
X
1
LC(jω) +RCjω+1
=
ωc =
2
pc
a
= 1000, ζ =
√b
4ac
=
bωc
2c
=
R
200
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 10 / 11
Resonance Peak for LP filter
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Y
X
1
LC(jω) +RCjω+1
=
ωc =
2
pc
a
= 1000, ζ =
=
(j
√b
4ac
ω
ωc
=
)
2
1
+2ζj ωωc +1
bωc
2c
=
R
200
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 10 / 11
Resonance Peak for LP filter
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
Y
X
ωc =
Y
X
1
LC(jω) +RCjω+1
=
2
pc
a
= 1000, ζ =
=
(j
√b
4ac
ω
ωc
=
)
2
1
+2ζj ωωc +1
bωc
2c
=
R
200
is a function of ωω so ωc just scales frequency axis (= shift on log axis).
c
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 10 / 11
Resonance Peak for LP filter
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
Y
X
ωc =
Y
X
1
LC(jω) +RCjω+1
=
2
pc
a
= 1000, ζ =
=
(j
√b
4ac
ω
ωc
=
)
2
1
+2ζj ωωc +1
bωc
2c
=
R
200
is a function of ωω so ωc just scales frequency axis (= shift on log axis).
c
The damping factor , ζ , (“zeta”) determines the shape of the peak.
filter
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 10 / 11
Resonance Peak for LP filter
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
Y
X
ωc =
Y
X
1
LC(jω) +RCjω+1
=
2
pc
a
= 1000, ζ =
=
(j
√b
4ac
ω
ωc
=
)
2
1
+2ζj ωωc +1
bωc
2c
=
R
200
is a function of ωω so ωc just scales frequency axis (= shift on log axis).
c
The damping factor , ζ , (“zeta”) determines the shape of the peak.
Peak frequency: p
ωp = ωc
1 − 2ζ 2
30
20
R=20, ζ =0.1
990, 14dB
10
0
-10
E1.1 Analysis of Circuits (2016-8284)
0.7
0.8
0.9 1
ω (krad/s)
1.2
1.4
Resonance: 12 – 10 / 11
Resonance Peak for LP filter
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
Y
X
ωc =
Y
X
1
LC(jω) +RCjω+1
=
2
pc
a
= 1000, ζ =
=
(j
√b
4ac
ω
ωc
=
)
2
1
+2ζj ωωc +1
bωc
2c
=
R
200
is a function of ωω so ωc just scales frequency axis (= shift on log axis).
c
The damping factor , ζ , (“zeta”) determines the shape of the peak.
Peak frequency: p
ωp = ωc
1 − 2ζ 2
ζ ≥ 0.71 ⇒ no peak,
E1.1 Analysis of Circuits (2016-8284)
30
20
R=20, ζ =0.1
990, 14dB
10
0
-10
0.7
0.8
0.9 1
ω (krad/s)
1.2
1.4
Resonance: 12 – 10 / 11
Resonance Peak for LP filter
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
Y
X
ωc =
Y
X
1
LC(jω) +RCjω+1
=
2
pc
a
= 1000, ζ =
=
(j
√b
4ac
ω
ωc
=
)
2
1
+2ζj ωωc +1
bωc
2c
=
R
200
is a function of ωω so ωc just scales frequency axis (= shift on log axis).
c
The damping factor , ζ , (“zeta”) determines the shape of the peak.
Peak frequency: p
ωp = ωc
1 − 2ζ 2
ζ ≥ 0.71 ⇒ no peak,
ζ ≥ 1 ⇒ can factorize
E1.1 Analysis of Circuits (2016-8284)
30
20
R=20, ζ =0.1
990, 14dB
10
0
-10
0.7
0.8
0.9 1
ω (krad/s)
1.2
1.4
Resonance: 12 – 10 / 11
Resonance Peak for LP filter
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
Y
X
ωc =
Y
X
1
LC(jω) +RCjω+1
=
=
2
pc
a
= 1000, ζ =
(j
√b
4ac
ω
ωc
=
)
2
1
+2ζj ωωc +1
bωc
2c
=
R
200
is a function of ωω so ωc just scales frequency axis (= shift on log axis).
c
The damping factor , ζ , (“zeta”) determines the shape of the peak.
Peak frequency: p
ωp = ωc
30
1 − 2ζ 2
10
ζ ≥ 0.71 ⇒ no peak,
ζ ≥ 1 ⇒ can factorize
Gain relative to asymptote:
E1.1 Analysis of Circuits (2016-8284)
R=20, ζ =0.1
990, 14dB
20
0
-10
@ ωp :
2ζ
√1
0.7
1−ζ 2
0.8
0.9 1
ω (krad/s)
1.2
1.4
1
@ ωc : 2ζ
≈Q
Resonance: 12 – 10 / 11
Resonance Peak for LP filter
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
Y
X
ωc =
Y
X
1
LC(jω) +RCjω+1
=
=
2
pc
a
= 1000, ζ =
(j
√b
4ac
ω
ωc
=
)
2
1
+2ζj ωωc +1
bωc
2c
=
R
200
is a function of ωω so ωc just scales frequency axis (= shift on log axis).
c
The damping factor , ζ , (“zeta”) determines the shape of the peak.
Peak frequency: p
ωp = ωc
1−
30
2ζ 2
20
10
ζ ≥ 0.71 ⇒ no peak,
ζ ≥ 1 ⇒ can factorize
Gain relative to asymptote:
E1.1 Analysis of Circuits (2016-8284)
R=5, ζ =0.03
R=20, ζ =0.1
999, 26dB
990, 14dB
0
-10
@ ωp :
2ζ
√1
0.7
1−ζ 2
0.8
0.9 1
ω (krad/s)
1.2
1.4
1
@ ωc : 2ζ
≈Q
Resonance: 12 – 10 / 11
Resonance Peak for LP filter
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
Y
X
ωc =
Y
X
1
LC(jω) +RCjω+1
=
=
2
pc
a
= 1000, ζ =
(j
√b
4ac
ω
ωc
=
)
2
1
+2ζj ωωc +1
bωc
2c
=
R
200
is a function of ωω so ωc just scales frequency axis (= shift on log axis).
c
The damping factor , ζ , (“zeta”) determines the shape of the peak.
Peak frequency: p
ωp = ωc
1−
30
2ζ 2
20
10
ζ ≥ 0.71 ⇒ no peak,
ζ ≥ 1 ⇒ can factorize
Gain relative to asymptote:
E1.1 Analysis of Circuits (2016-8284)
R=5, ζ =0.03
R=20, ζ =0.1
R=60, ζ =0.3
999, 26dB
990, 14dB
906, 5dB
0
-10
@ ωp :
2ζ
√1
0.7
1−ζ 2
0.8
0.9 1
ω (krad/s)
1.2
1.4
1
@ ωc : 2ζ
≈Q
Resonance: 12 – 10 / 11
Resonance Peak for LP filter
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
Y
X
ωc =
Y
X
1
LC(jω) +RCjω+1
=
=
2
pc
a
= 1000, ζ =
(j
√b
4ac
)
ω
ωc
=
2
1
+2ζj ωωc +1
bωc
2c
=
R
200
is a function of ωω so ωc just scales frequency axis (= shift on log axis).
c
The damping factor , ζ , (“zeta”) determines the shape of the peak.
Peak frequency: p
30
2ζ 2
ωp = ωc 1 −
ζ ≥ 0.5 ⇒ passes under corner,
ζ ≥ 0.71 ⇒ no peak,
ζ ≥ 1 ⇒ can factorize
Gain relative to asymptote:
E1.1 Analysis of Circuits (2016-8284)
@ ωp :
R=5, ζ =0.03
R=20, ζ =0.1
R=60, ζ =0.3
R=120, ζ =0.6
999, 26dB
990, 14dB
906, 5dB
529, 4dB
20
10
0
-10
2ζ
√1
0.7
1−ζ 2
0.8
0.9 1
ω (krad/s)
1.2
1.4
1
@ ωc : 2ζ
≈Q
Resonance: 12 – 10 / 11
Resonance Peak for LP filter
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
Y
X
ωc =
Y
X
1
LC(jω) +RCjω+1
=
=
2
pc
a
= 1000, ζ =
(j
√b
4ac
)
ω
ωc
=
2
1
+2ζj ωωc +1
bωc
2c
=
R
200
is a function of ωω so ωc just scales frequency axis (= shift on log axis).
c
The damping factor , ζ , (“zeta”) determines the shape of the peak.
Peak frequency: p
30
2ζ 2
ωp = ωc 1 −
ζ ≥ 0.5 ⇒ passes under corner,
ζ ≥ 0.71 ⇒ no peak,
ζ ≥ 1 ⇒ can factorize
Gain relative to asymptote:
@ ωp :
R=5, ζ =0.03
R=20, ζ =0.1
R=60, ζ =0.3
R=120, ζ =0.6
999, 26dB
990, 14dB
906, 5dB
529, 4dB
20
10
0
-10
2ζ
√1
0.7
1−ζ 2
0.8
0.9 1
ω (krad/s)
1.2
1.4
1
@ ωc : 2ζ
≈Q
Three frequencies: ωp = peak, ωc = asymptotes cross, ωr = real impedance
For ζ < 0.3, ωp ≈ ωc ≈ ωr . All get called the resonant frequency.
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 10 / 11
Resonance Peak for LP filter
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
Y
X
ωc =
Y
X
1
LC(jω) +RCjω+1
=
=
2
pc
a
= 1000, ζ =
(j
√b
4ac
)
ω
ωc
=
2
1
+2ζj ωωc +1
bωc
2c
=
R
200
is a function of ωω so ωc just scales frequency axis (= shift on log axis).
c
The damping factor , ζ , (“zeta”) determines the shape of the peak.
Peak frequency: p
30
2ζ 2
ωp = ωc 1 −
ζ ≥ 0.5 ⇒ passes under corner,
ζ ≥ 0.71 ⇒ no peak,
ζ ≥ 1 ⇒ can factorize
Gain relative to asymptote:
@ ωp :
R=5, ζ =0.03
R=20, ζ =0.1
R=60, ζ =0.3
R=120, ζ =0.6
999, 26dB
990, 14dB
906, 5dB
529, 4dB
20
10
0
-10
2ζ
√1
0.7
1−ζ 2
0.8
0.9 1
ω (krad/s)
1.2
1.4
1
@ ωc : 2ζ
≈Q
Three frequencies: ωp = peak, ωc = asymptotes cross, ωr = real impedance
For ζ < 0.3, ωp ≈ ωc ≈ ωr . All get called the resonant frequency.
The exact relationship between ωp , ωc and ωr and the gain at these
frequencies is affected by any other corner frequencies in the response.
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 10 / 11
Summary
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
• Resonance is a peak in energy absorption
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 11 / 11
Summary
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
• Resonance is a peak in energy absorption
◦ Parallel or series circuit has a real impedance at ωr
⊲ peak response may be at a slightly different frequency
Resonance
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 11 / 11
Summary
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Resonance is a peak in energy absorption
◦ Parallel or series circuit has a real impedance at ωr
⊲ peak response may be at a slightly different frequency
◦ The quality factor, Q, of the resonance is
• Low Pass Filter
• Resonance Peak for LP
filter
Q,
ω0 ×stored
energy
ω0
1
≈
≈
2ζ
power in R
3 dB bandwidth
• Summary
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 11 / 11
Summary
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Resonance is a peak in energy absorption
◦ Parallel or series circuit has a real impedance at ωr
⊲ peak response may be at a slightly different frequency
◦ The quality factor, Q, of the resonance is
Q,
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
ω0 ×stored
energy
ω0
1
≈
≈
2ζ
power in R
3 dB bandwidth
◦ 3 dB bandwidth is where power falls by
E1.1 Analysis of Circuits (2016-8284)
1
2
or voltage by √1
2
Resonance: 12 – 11 / 11
Summary
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Resonance is a peak in energy absorption
◦ Parallel or series circuit has a real impedance at ωr
⊲ peak response may be at a slightly different frequency
◦ The quality factor, Q, of the resonance is
Q,
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
ω0 ×stored
energy
ω0
1
≈
≈
2ζ
power in R
3 dB bandwidth
◦ 3 dB bandwidth is where power falls by
1
2
or voltage by √1
2
◦ The stored energy sloshes between L and C
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 11 / 11
Summary
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Resonance is a peak in energy absorption
◦ Parallel or series circuit has a real impedance at ωr
⊲ peak response may be at a slightly different frequency
◦ The quality factor, Q, of the resonance is
Q,
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
ω0 ×stored
energy
ω0
1
≈
≈
2ζ
power in R
3 dB bandwidth
◦ 3 dB bandwidth is where power falls by
1
2
or voltage by √1
2
◦ The stored energy sloshes between L and C
2
jω
• Quadratic factor: ωc + 2ζ jω
ωc + 1
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 11 / 11
Summary
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Resonance is a peak in energy absorption
◦ Parallel or series circuit has a real impedance at ωr
⊲ peak response may be at a slightly different frequency
◦ The quality factor, Q, of the resonance is
Q,
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
ω0 ×stored
energy
ω0
1
≈
≈
2ζ
power in R
3 dB bandwidth
◦ 3 dB bandwidth is where power falls by
1
2
or voltage by √1
◦ The stored energy sloshes between L and C
2
jω
• Quadratic factor: ωc + 2ζ jω
ωc + 1
pc
2
◦ a (jω) + b (jω) + c ⇒ ωc = a and ζ =
E1.1 Analysis of Circuits (2016-8284)
2
√b
4ac
=
b
2aωc
Resonance: 12 – 11 / 11
Summary
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Resonance is a peak in energy absorption
◦ Parallel or series circuit has a real impedance at ωr
⊲ peak response may be at a slightly different frequency
◦ The quality factor, Q, of the resonance is
Q,
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
ω0 ×stored
energy
ω0
1
≈
≈
2ζ
power in R
3 dB bandwidth
◦ 3 dB bandwidth is where power falls by
1
2
or voltage by √1
2
◦ The stored energy sloshes between L and C
2
jω
• Quadratic factor: ωc + 2ζ jω
ωc + 1
pc
2
b
=
◦ a (jω) + b (jω) + c ⇒ ωc = a and ζ = √4ac
◦ ±40 dB/decade slope change in magnitude response
E1.1 Analysis of Circuits (2016-8284)
b
2aωc
Resonance: 12 – 11 / 11
Summary
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Resonance is a peak in energy absorption
◦ Parallel or series circuit has a real impedance at ωr
⊲ peak response may be at a slightly different frequency
◦ The quality factor, Q, of the resonance is
Q,
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
ω0 ×stored
energy
ω0
1
≈
≈
2ζ
power in R
3 dB bandwidth
◦ 3 dB bandwidth is where power falls by
1
2
or voltage by √1
2
◦ The stored energy sloshes between L and C
2
jω
• Quadratic factor: ωc + 2ζ jω
ωc + 1
pc
2
b
=
◦ a (jω) + b (jω) + c ⇒ ωc = a and ζ = √4ac
◦ ±40 dB/decade slope change in magnitude response
◦ phase changes rapidly by 180◦ over ω = 10∓ζ ωc
E1.1 Analysis of Circuits (2016-8284)
b
2aωc
Resonance: 12 – 11 / 11
Summary
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Resonance is a peak in energy absorption
◦ Parallel or series circuit has a real impedance at ωr
⊲ peak response may be at a slightly different frequency
◦ The quality factor, Q, of the resonance is
Q,
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
ω0 ×stored
energy
ω0
1
≈
≈
2ζ
power in R
3 dB bandwidth
◦ 3 dB bandwidth is where power falls by
1
2
or voltage by √1
2
◦ The stored energy sloshes between L and C
2
jω
• Quadratic factor: ωc + 2ζ jω
ωc + 1
pc
2
b
=
◦ a (jω) + b (jω) + c ⇒ ωc = a and ζ = √4ac
◦ ±40 dB/decade slope change in magnitude response
◦ phase changes rapidly by 180◦ over ω = 10∓ζ ωc
1
◦ Gain error in asymptote is 2ζ
≈ Q at ω0
E1.1 Analysis of Circuits (2016-8284)
b
2aωc
Resonance: 12 – 11 / 11
Summary
12: Resonance
• Quadratic Factors
• Damping Factor and Q
• Parallel RLC
• Behaviour at Resonance
• Away from resonance
• Bandwidth and Q
• Power and Energy at
Resonance
• Resonance is a peak in energy absorption
◦ Parallel or series circuit has a real impedance at ωr
⊲ peak response may be at a slightly different frequency
◦ The quality factor, Q, of the resonance is
Q,
• Low Pass Filter
• Resonance Peak for LP
filter
• Summary
ω0 ×stored
energy
ω0
1
≈
≈
2ζ
power in R
3 dB bandwidth
◦ 3 dB bandwidth is where power falls by
1
2
or voltage by √1
2
◦ The stored energy sloshes between L and C
2
jω
• Quadratic factor: ωc + 2ζ jω
ωc + 1
pc
2
b
=
◦ a (jω) + b (jω) + c ⇒ ωc = a and ζ = √4ac
◦ ±40 dB/decade slope change in magnitude response
◦ phase changes rapidly by 180◦ over ω = 10∓ζ ωc
1
◦ Gain error in asymptote is 2ζ
≈ Q at ω0
b
2aωc
For further details see Hayt et al. Chapter 16.
E1.1 Analysis of Circuits (2016-8284)
Resonance: 12 – 11 / 11