Agenda:

• Passive RLC networks (review)

• Impedance transformation

REC: Lee, pp 74-87

• Power matching networks

RLC networks:

“Low frequencies”

⇒ inductor dominates admittance

Iin L

“High frequencies”

⇒ capacitor dominates admittance

Iin C

What separates “high” and “low” frequencies? → Resonance Frequency

Admittance: 𝑌(𝜔) =1

𝑅+ 𝑗 (𝜔𝐶 −

1

𝜔𝐿)

At resonance: 𝜔0𝐶 =1

𝜔0𝐿 ⇒ 𝜔0 =

1

√𝐿𝐶

Equivalent circuit @ resonance:

Purely resistive

Define 𝑍(𝜔) = impedance of paralleled

RLC tank

At resonance

Magnitude |𝑍(𝜔0)| = 𝑅

Phase ∡𝑍(𝜔0) = 0°

Quality Factor Q:

Measure of how well a system stores energy

Analogy of pendulum:

At position ②: pendulum has only kinetic energy, and no

potential energy

At position ① and ③: pendulum has only potential energy,

and no kinetic energy

For a lossless pendulum: KE + PE = constant for ever

→ no dissipation

Lossy pendulum: Energy dissipates until pendulum stops oscillating

⇒ Quality is high if energy is dissipated very slowly:

For RLC circuit (or any circuit or system for that matter)

𝑄 = 𝜔𝐸𝑛𝑒𝑟𝑔𝑦 𝑠𝑡𝑜𝑟𝑒𝑑

𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑝𝑜𝑤𝑒𝑟 𝑑𝑖𝑠𝑠𝑖𝑝𝑎𝑡𝑒𝑑

RLC tank → energy sloshes between L and C with a constant sum at resonance

CIin RLIin

Ipk

When current reaches 𝐼𝑝𝑘, voltage across capacitor (at resonance)

𝑉𝑐 = 𝐼𝑝𝑘 ∙ 𝑅

⇒ Energy in capacitor reaches a peak at this instant & the energy in the inductor at this instant is

zero.

⇒ Peak energy stored 𝐸𝑝𝑘 =1

2 𝐶𝑉𝑝𝑘

2 =1

2 𝐶(𝐼𝑝𝑘𝑅)2

Also, at resonance, all the current from the current source flows into the resistor

⇒ Average power dissipated 𝑃𝑎𝑣𝑔 =1

2𝐼𝑝𝑘

2 𝑅

⇒ 𝑄 =𝜔0∙𝐸𝑝𝑘

𝑃𝑎𝑣𝑔=

1

√𝐿𝐶∙

1

2𝐶(𝐼𝑝𝑘𝑅)

2∙

11

2𝐼𝑝𝑘

2 𝑅=

𝑅

√𝐿 𝐶⁄

√𝐿

𝐶 → called characteristic impedance (analogy with transmission lines)

𝑄 =𝑅

√𝐿 𝐶⁄ → As 𝑅 ⟶ ∞, 𝑄 ⟶ ∞

⇒ intuitively correct

At resonance, |𝑍𝑐| =1

𝜔0𝐶=

√𝐿𝐶

𝐶= √

𝐿

𝐶

⇒ Characteristic impedance

|𝑍𝐿| = 𝜔0𝐿 =𝐿

√𝐿𝐶= √

𝐿

𝐶

Three forms of Q for parallel RLC networks:

𝑄 =𝑅

√𝐿 𝐶⁄

𝑄 =𝑅

|𝑍𝐿|=

𝑅

𝜔0𝐿

𝑄 =𝑅

|𝑍𝐿|= 𝜔0𝑅𝐶

Branch currents at resonance:

|𝐼𝐿| =|𝐼𝑖𝑛|𝑅

𝜔0𝐿= 𝑄|𝐼𝑖𝑛| Beware → for very high Q,

|𝐼𝐶| = |𝐼𝑖𝑛|𝑅 ∙ 𝜔0𝐶 = 𝑄|𝐼𝑖𝑛| we have very large branch currents.

→ To say that the inductor and capacitor “cancel” at resonance dangerously incorrect!

Relationship between bandwidth and Q

→ What happens @ frequencies near resonance?

𝜔 = 𝜔0 + ∆𝜔

𝑌(𝜔) = 𝐺 + 𝑗 (𝜔𝐶 −1

𝜔𝐿) = 𝐺 + 𝑗 [(𝜔0 + ∆𝜔)𝐶 −

1

(𝜔0+∆𝜔)𝐿]

= 𝐺 +𝑗

(𝜔0+∆𝜔)𝐿[(𝜔0 + ∆𝜔)2𝐶𝐿 − 1]

= 𝐺 +𝑗

(1+∆𝜔

𝜔0)𝜔0𝐿

{[𝜔02 + 2𝜔0∆𝜔 + (∆𝜔)2]𝐿𝐶 − 1}

𝜔02𝐿𝐶 = 1

= 𝐺 +𝑗

(1+∆𝜔

𝜔0)𝜔0𝐿

[2𝜔0∆𝜔𝐿𝐶 + (∆𝜔)2𝐿𝐶]

= 𝐺 +𝑗∆𝜔𝐿𝐶

(1+∆𝜔

𝜔0)𝜔0𝐿

[2𝜔0 + ∆𝜔]

= 𝐺 +𝑗∆𝜔∙𝐿𝐶

(1+∆𝜔

𝜔0)𝜔0𝐿

[2 +∆𝜔

𝜔0] ∙ 𝜔0

Use 1

1+𝑥≃ 1 − 𝑥 for 𝑥 ≪ 1,

= 𝐺 + 𝑗∆𝜔𝐶 (1 −∆𝜔

𝜔0) (2 +

∆𝜔

𝜔0)

= 𝐺 + 𝑗∆𝜔𝐶 [2 −∆𝜔

𝜔0− (

∆𝜔

𝜔0)

2

]

neglect ≪ 1

⇒ 𝑌(𝜔) ≃ 𝐺 + 𝑗2∆𝜔𝐶

⇒ Equivalent circuit @ resonance →

2CR

→ Near resonance, we can replace the original circuit with this equivalent circuit, and replace

absolute frequency by frequency offset with respect to 𝜔0.

⇒ 𝑍(𝜔) →

-3dB

ω0

Bandwidth of RLC network

2RC

1

Δω

⇒ BW of RLC network = 21

2𝑅𝐶=

1

𝑅𝐶

→ Normalize w.r.t 𝜔0 → ∆𝜔

𝜔0=

1 𝑅𝐶⁄

1 √𝐿𝐶⁄=

1

𝑅∙ √

𝐿

𝐶

⇒𝜔0

∆𝜔=

𝑅

√𝐿 𝐶⁄= 𝑄 ⇒ Higher the Q, narrower the frequency.

Series RLC network:

R

L

C

Vin

Vin L

Vin C

Vin R

LF

HF

Resonance

Capacitance dominates

the impedance

Inductance dominates

the impedance

Purely resistance

At resonance 𝑉𝐶 + 𝑉𝐿 = 0, 𝑍(𝜔) = 𝑅 + 𝑗 (𝜔𝐿 −1

𝜔𝐶) → resonance @ 𝜔0 =

1

√𝐿𝐶

+20dB/dec

inductive

|Z(ω)|

log scale

ω0

-20dB/dec

capactive

log scale ωZ(ω)

+90º

0º

-90ºω0

R

log scale ω

𝑄 = 𝜔 ∙𝐸𝑛𝑒𝑟𝑔𝑦 𝑠𝑡𝑜𝑟𝑒𝑑

𝐴𝑣𝑔.𝑝𝑜𝑤𝑒𝑟 𝑑𝑖𝑠𝑠𝑖𝑝𝑎𝑡𝑒𝑑

At resonance, when voltage reaches its peak, the current is 𝐼𝑝𝑘 =𝑉𝑝𝑘

𝑅

⇒ peak magnetic energy in the inductor at this instant & zero capacitive energy at that instant

⇒ 𝐸𝑝𝑘 =1

2𝐿𝐼𝑝𝑘

2 =1

2𝐿 (

𝑉𝑝𝑘

𝑅)

2

Average power dissipated 𝑃𝐴𝑣𝑔 =𝐼𝑝𝑘

2 𝑅

2=

𝑉𝑝𝑘2

2𝑅

⇒ 𝑄 = 𝜔0 ∙

1

2(

𝑉𝑝𝑘

𝑅)

2

𝐿

1

2

𝑉𝑝𝑘2

𝑅

=𝜔0𝐿

𝑅=

√𝐿 𝐶⁄

𝑅 → characteristic impedance

𝑄 =√𝐿 𝐶⁄

𝑅=

𝜔0𝐿

𝑅=

1

𝜔0𝑅𝐶 𝑄 → ∞ 𝑎𝑠 𝑅 → 0 intuitively correct

Branch voltages at resonance:

|𝑉𝐿| = |𝐼𝑖𝑛| ∙ 𝜔0𝐿 =|𝑉𝑖𝑛|𝜔0𝐿

𝑅= 𝑄|𝑉𝑖𝑛|

|𝑉𝐶| = |𝐼𝑖𝑛| ∙1

𝜔0𝐶=

|𝑉𝑖𝑛|

𝜔0𝑅𝐶= 𝑄|𝑉𝑖𝑛|

⇒ High branch voltage at resonance → breakdown

→ useful for voltage amplification in LNA’s

Bandwidth near resonance

For 𝜔 = 𝜔0 + ∆𝜔 for ∆𝜔 ≪ 𝜔0: 𝑍(𝜔) ≃ 𝑅 + 𝑗𝑍∆𝜔𝐿 (PROVE THIS!)

→ Equivalent circuit near resonance (replace absolute frequency by

offset frequency w.r.t 𝜔0)

3dB Bandwidth of equivalent circuit =𝑅

2𝐿

𝐵𝑊 = 2𝜔3𝑑𝐵 = 𝑅 𝐿⁄

Normalized BW =𝐵𝑊

𝜔0=

𝑅 𝐿⁄

1 √𝐿𝐶⁄=

𝑅

√𝐿 𝐶⁄=

1

𝑄

⇒ 𝑄 =𝜔0

𝐵𝑊 or 𝐵𝑊 =

𝜔0

𝑄

Series-parallel transformations:

• Ubiquitous in RF design

• Can we use what we know about pure-series or pure-parallel RLC networks to analyze networks

that are not pure series or parallel?

• On-chip RF circuit → inductors are a lot more lossy than capacitors

LC tank with a lossy inductor:

Convert to a pure parallel RLC

Rs

Ls

C C RpLp

Equate impedances at resonance:

𝑅𝑠 + 𝑗𝜔0𝐿𝑠 = 𝑅𝑝 ∥ 𝑗𝜔0𝐿𝑝 =𝑅𝑝𝑗𝜔0𝐿𝑝

𝑅𝑝 + 𝑗𝜔0𝐿𝑝∙

𝑅𝑝 − 𝑗𝜔0𝐿𝑝

𝑅𝑝 − 𝑗𝜔0𝐿𝑝=

(𝜔0𝐿𝑝)2

𝑅𝑝 + 𝑗𝜔0𝐿𝑝𝑅𝑝2

𝑅𝑝2 + (𝜔0𝐿𝑝)

2

Equate real and imaginary parts:

𝑅𝑠 =(𝑗𝜔0𝐿𝑝)

2𝑅𝑝

𝑅𝑝2+(𝜔0𝐿𝑝)

2 𝜔0𝐿𝑠 =𝜔0𝐿𝑝𝑅𝑝

2

𝑅𝑝2+(𝜔0𝐿𝑝)

2

Note 𝑄 =𝜔0𝐿𝑠

𝑅𝑠=

𝑅𝑝

𝜔0𝐿𝑝 (Q’s must be the same after transformation if we want the circuit to be

identical)

𝑅𝑠 =𝑅𝑝

1+(𝑅𝑝

𝜔0𝐿𝑝)2

𝜔0𝐿𝑠 =[𝑅𝑝

2 (𝜔0𝐿𝑝)2

]∙(𝜔0𝐿𝑝)⁄

1+(𝑅𝑝

𝜔0𝐿𝑝)2

⇒ 𝑅𝑠 =𝑅𝑝

𝑄2+1 ⇒ 𝐿𝑠 =

𝐿𝑝∙𝑄2

𝑄2+1

⇒ 𝑅𝑝 = (𝑄2 + 1)𝑅𝑠 ⇒ 𝐿𝑝 = (𝑄2+1

𝑄2)𝐿𝑠

Similarly, to transform series-RC to parallel-RC:

Rs

Cs

L RpL Cp

𝑅𝑝 = 𝑅𝑠(𝑄2 + 1) 𝐶𝑝 =𝑄2

𝑄2+1𝐶𝑠

Generally:

𝑅𝑝 = 𝑅𝑠(𝑄2 + 1), 𝑋𝑝 = (𝑄2+1

𝑄2 ) 𝑋𝑠

𝑄 =𝑋𝑠

𝑅𝑠=

𝑅𝑝

𝑋𝑝

Note that these transformations hold only in a narrow band centered about 𝜔0.

The maximum power transfer theorem:

Suppose that we have a source with a source impedance 𝑍𝑠, at

what value of load impedance will maximum (real) power be

delivered to the load?

→ 𝑍𝐿 = 𝑍𝑠 ∗ (complex conjugate)

𝑅𝐿 = 𝑅𝑠 → Called a conjugate match.

or → Other types of “matcher” are

𝑋𝐿 = −𝑋𝑠 → reflection matching and noise matching.

Derivation (good refresher in basics!)

Power delivered to load 𝑃𝐿 =1

2𝑅𝑒[𝑉𝐿𝐼𝐿

∗] (peak phasors!)

𝑉𝐿 =𝑍𝐿

𝑍𝐿+𝑍𝑆𝑉𝑠, 𝐼𝐿 =

𝑉𝑠

𝑍𝐿+𝑍𝑆

∴ 𝑃𝐿 =1

2𝑅𝑒 [

𝑍𝐿

𝑍𝐿+𝑍𝑆𝑉𝑠 ∙

𝑉𝑠∗

(𝑍𝐿+𝑍𝑆)∗] =

|𝑉𝑠|2

2|𝑍𝐿+𝑍𝑆|2∙ 𝑅𝑒(𝑍𝐿)

𝑃𝐿 =|𝑉𝑠|2𝑅𝐿

2|𝑍𝐿+𝑍𝑆|2 =|𝑉𝑠|2𝑅𝐿

2[(𝑅𝐿+𝑅𝑠)2+(𝑋𝐿+𝑋𝑠)2]

Note that the source impedance is fixed, and we are allowed to vary only the load impedance.

For maximum power, 𝜕𝑃𝐿

𝜕𝑋𝐿= 0 and

𝜕𝑃𝐿

𝜕𝑅𝐿= 0

𝜕𝑃𝐿

𝜕𝑋𝐿=

|𝑉𝑠|2

2∙

(𝑅𝐿)[−2(𝑋𝐿+𝑋𝑠)]

[(𝑅𝐿+𝑅𝑠)2+(𝑋𝐿+𝑋𝑠)2]= 0 ⇒ 𝑋𝐿 + 𝑋𝑠 = 0 or 𝑋𝐿 = −𝑋𝑠

𝜕𝑃𝐿

𝜕𝑅𝐿=

|𝑉𝑠|2

2{

1

(𝑅𝐿 + 𝑅𝑠)2 + (𝑋𝐿 + 𝑋𝑠)2−

2𝑅𝐿(𝑅𝐿 + 𝑅𝑠)

[(𝑅𝐿 + 𝑅𝑠)2 + (𝑋𝐿 + 𝑋𝑠)2]2} = 0

⇒ (𝑅𝐿 + 𝑅𝑠)2 + (𝑋𝐿 + 𝑋𝑠)2 = 2𝑅𝐿(𝑅𝐿 + 𝑅𝑠)

= 0

Since 𝑋𝐿 = −𝑋𝑠

∴ 𝑅𝐿 + 𝑅𝑠 = 2𝑅𝐿 or 𝑅𝐿 = 𝑅𝑠

The power delivered to the load when 𝑍𝐿 = 𝑍𝑠 ∗ is defined as the “available” power from a source.

𝑃𝑎𝑣𝑠 = 𝑃𝐿𝑚𝑎𝑥=

|𝑉𝑠|2

8𝑅𝑠

RLC networks as impedance transformers:

1) Upward transformation:

Find 𝐿𝑠 and C to match the load to the source at

5GHz

Series-to-parallel transformation:

(A)

Suppose 𝐿𝑝 & C resonate @ 5GHz,

then 𝑅𝑝 = 200Ω @ 5GHz.

𝑅𝑝 = 𝑅𝐿(𝑄2 + 1)

𝑅𝐿 = 50Ω, 𝑅𝑝 = 200Ω ⇒ 𝑄 = √𝑅𝑝

𝑅𝐿− 1

or 𝑄 = √3 = 1.732

But 𝑄 =𝜔0𝐿𝑠

𝑅𝐿=

𝑅𝑝

𝜔0𝐿𝑝 ⇒ 𝐿𝑠 =

𝑄∙𝑅𝐿

𝜔0=

√3∙50

2𝜋∙5𝐺𝐻𝑧 or 𝐿𝑠 = 2.75𝑛𝐻,

∴ 𝐿𝑝 =𝑅𝑝

𝜔0∙𝑄=

200

2𝜋∙5𝐺𝐻𝑧√3= 3.66𝑛𝐻

(Alternatively, we can use the formula 𝐿𝑝 =𝐿𝑠(𝑄2+1)

𝑄2)

Finally, 𝐿𝑝 & C must resonate at 5GHz

⇒ 𝜔0 =1

√𝐿𝑝𝐶=

1

√3.66𝑛𝐻∙𝐶= 2𝜋 ∙ 5𝐺𝐻𝑧 or 𝐶 = 276.83𝑓𝐹.

This network has a low pass response from the source to the load. We can design another network

with a high pass response:

(B)

C

RL=50ΩL

200Ω

Rp=200Ω @5GHz

L Cp

200Ω

Series – to –

parallel

𝑅𝑝 = 𝑅𝐿(𝑄2 + 1) ⇒ 𝑄 = √𝑅𝑝

𝑅𝐿− 1 = √3

𝑄 =1

𝜔0𝑅𝐿𝐶⇒ 𝐶 =

1

𝜔0𝑅𝐿𝑄=

1

2𝜋∙5𝐺𝐻𝑧∙50∙√3 or 𝐶 = 367.55𝑓𝐹

𝐶𝑝 = 𝐶𝑄2

𝑄2+1= 275.66𝑓𝐹

We want L and 𝐶𝑝 to reach resonance @ 5GHz

⇒ 𝐿 =1

𝜔02𝐶𝑝

= 3.67𝑛𝐻

Which network to choose, (A) or (B)?

-- High-pass vs Low pass response

-- Usually want the network with the smallest inductor (larger inductors have more loss and

area)

-- Parasitics

-- Bandwidth: Match exact only at 5GHz. At other frequencies, we will have an impedance

mismatch.

Basic microwave theory: an impedance mismatch causes power to be “reflected” back to

the source.

Mismatch quantified by reflection coefficient (=S11 for a 1-port network)

𝑆11 = Γ =𝑍𝐿 − 𝑍0

𝑍𝐿 + 𝑍0

Rule-of-Thumb: |Γ| ≈ −10𝑑𝐵 is

generally a good number.

Freq

|Γ|

0dB

10dB

5GHz

High passLow pass

Note: For a mismatch |Γ| between the load & source impedance, the power delivered to

the load is 𝑃𝐿 = 𝑃𝑎𝑣(1 − |Γ|2) when source impedance is real.

“available power” from source.

2) Downward transformation Match @ 5GHz

50Ω 50Ω

Paralle – to

– seriesC RL=200Ω

L

Rs

Cs

L

If we resonance L & 𝐶𝑠 @5GHz, then 𝑅𝑠 = 50Ω @ 5GHz

𝑅𝑠 =𝑅𝑝

𝑄2+1⇒ 𝑄 = √

𝑅𝑝

𝑅𝐿− 1 = √3, 𝑄 =

1

𝜔0𝑅𝑠𝐶𝑠 ⇒ 𝐶𝑠 =

1

2𝜋∙5𝐺𝐻𝑧∙50∙√3= 367.55𝑓𝐹

𝐶𝑆 = 𝐶 (𝑄2+1

𝑄2) ⇒ 𝐶 = 𝐶𝑆 (

𝑄2

𝑄2+1) =

(√3)2

(√3)2+1∙ 367.5𝑓𝐹 or 𝐶 = 275.5𝑓𝐹

Since L and 𝐶𝑆 resonate @ 5GHz,

𝜔02 =

1

𝐿𝐶𝑠⇒ 𝐿 =

1

(2𝜋×5𝐺𝐻𝑧)2(367.55𝑓𝐹)= 2.75𝑛𝐻 or 𝐿 = 2.75𝑛𝐻

(Not surprisingly, the values are the same as the upward transformation case looking

backwards)

Mnemonic: How to find if an L-match network gives an upward or downward

transformation?

Since we add L in series, we expect 𝑅𝑖𝑛 > 𝑅𝐿 ⇒ upward

tx.

Add “something” (ie, L here) in parallel ⇒ 𝑅𝑖𝑛 < 𝑅𝐿 ⇒

downward transformation