Introduction to

RF DesignRF Electronics Spring, 2018

Robert R. Krchnavek Rowan University

Objectives

• Understand why RF design is different from low-frequency design.

• Develop RF models of passive components.

• Learn how to measure passive components (lab exercise).

Generic RF System

Circuit Diagram - 2 GHz Power Amp

?

?

?

?

?

??

?

Circuit Diagram - 2 GHz Power Amp

Printed Circuit Board - Power Amp

Why RF Design?

• DC Circuit Analysis (and low-frequency)

• Uses KCL and KVL.

• Assumes “lumped” components.

• As the frequency increases, or more precisely, when the circuit dimensions reach an “appreciable” percentage of the wavelength,

• KCL and KVL no longer apply and the wave nature must be considered.

• Components are no longer lumped but are in reality “distributed” networks.

Electromagnetic Waves

Recall the Engineering Electromagnetics, the equations which describe a plane wave (TEM):

What do these equations mean?

E = Exx = E0xe−jβzx ⇒ E0x cos(ωt − βz)x

H = Hy y = H0ye−jβz y ⇒ H0y cos(ωt − βz)y

Important Relationships

Propagation Constant:

Wavelength:

Phase Velocity:

β =2π

λ= ω

√

µϵ

vp =ω

β=

1√

µϵ

λ =2π

β=

2πvp

ω=

vp

f

Conclusion #1

• Because electromagnetic energy travels as waves, the electric (and magnetic) field will vary as a function of position in the material.Therefore, the voltage (and current) will vary as a function of position.

• When a wire’s length is a significant fraction of the wavelength of the EM waves in the conducting system, the voltage will not be constant along the wire. This contradicts KVL.

• In a similar fashion, the current will not be constant along the length of wire. This contradicts KCL.

• This leads to the concept of a distributed parameter network and will be considered next.

Components

• All components (passive, active, and even interconnects) need to be viewed as distributed parameter networks. We will consider resistors, capacitors, inductors, and the skin effect in conductors.

• Recall some fundamental principles:

• Resistance - occurs in any conducting medium (except superconductors) and limits the flow of current.

• Capacitance - occurs whenever two conductors are separated by a dielectric.

Components - continued

• Inductance - occurs whenever magnetic flux links a conductor.

• The physical dimensions and material properties of a component determines the equivalent distributed parameter network and we model the component as a network of discrete components.

ResistorWhat does a resistor look like at very high frequencies?

CapacitorWhat does a capacitor look like at very high frequencies?

Inductor

What does an inductor look like at very high frequencies?

Skin Effect

As frequency increases, the current density is greatest near the outer edges of the conductor. Only at DC is the current uniformly dense in the conductor.

Skin Depth:

AC Resistance:

δ =1

√

πfµσcond

R = RDC

a

2δ

Conclusion #2

• All components have resistance, capacitance, and inductance.

• At low frequencies, the unintended device components are insignificant.

• At high frequencies, the unintended device components become significant.

• The unintended components are distributed throughout the device.

• We model these devices as consisting of a network of discrete devices.

Transmission Lines

Objectives• Understand the distributed parameter model for a

transmission line.

• Be able to DERIVE the general transmission line equations from the distributed parameter model for a transmission line.

• Be able to determine when transmission line modeling may be necessary in the analysis of an electronic circuit.

• Know how to calculate the characteristic impedance for the common transmission line structures.

• Know how to calculate the reflection coefficient, standing wave ratio, and input impedance for a terminated, lossless transmission line.

• Know how to do simple impedance matching using transmission lines.

Transmission Lines

• Most of what you need to know was covered in EEMAG.

• From earlier, we know passive components are more complex at RF frequencies than at low frequencies. The complexity is not only restricted to resistors, capacitors, and inductors, but even the interconnects on PWBs. PWBs are networks consisting of R, L, C, and G.

• Transmission line analysis provides a key method for designing and analyzing RF circuits.

Network Model of a Transmission Line

R, L, G, and C are “distributed” parameters. In other words, their units are: Ω/m, H/m, mhos/m, and F/m respectively.

When do we have to use the distributed parameter network model?

• Use the wavelength in the medium, not the free-space wavelength.

• If the average size of a component is given by lA, then the frequency at which we should consider using transmission line theory is given by:

Rule of Thumb:

When the average size of a discrete component is more than a tenth of the wavelength, the distributed parameter network model, i.e., transmission line theory, should be used.

f =vp

10lA

Common Transmission LinesTwo-wire

Microstrip

CoaxialTriplate

Common Transmission LinesR, L, G, and C depend on the particular transmission line structure and the material properties. R, L, G, and C can be calculated using fundamental EEMAG techniques.

Parameter Two-Wire Line Coaxial Line Parallel-Plate Line

Unit

R

L

G

Cπϵ

arc cosh!

D

2a

"

πσd

arc cosh!

D

2a

"

µ

πarc cosh

!

D

2a

"

1

πaσcδ

1

2πσcδ

!

1

a+

1

b

"

µ

2πln

!

b

a

"

2πσd

ln!

b

a

"

2πϵ

ln!

b

a

"

2

wσcδ

µd

w

σd

w

d

ϵw

d

Ω/m

H/m

S/m

F/m

The Transmission Line Equations

V (z) − I(z)R∆z − ȷωL∆zI(z) − V (z + ∆z) = 0

−

V (z + ∆z) − V (z)

∆z= (R + ȷωL)I(z)

−

dV (z)

dz= (R + ȷωL)I(z)

Using KVL:

The Transmission Line Equations

Using KCL:

I(z + ∆z) − I(z) + V (z + ∆z)(G + ȷωC)∆z = 0

I(z + ∆z) − I(z)

∆z= V (z + ∆z)(G + ȷωC)

dI(z)

dz= (G + ȷωC)V (z)

Solution

• k is the complex propagation constant.

• V+ and I+ are wavefronts propagating in the +z direction.

• V- and I- are wavefronts propagation in the -z direction.

V (z) = V+e−kz + V

−

e+kz

I(z) = I+e−kz

− I−

e+kz

k = kr + ȷki =!

(R + ȷωL)(G + ȷωC)

Characteristic Impedance

Consider a semi-infinite transmission line

The voltage and current on this line (no reflections) is given by

The voltage and current on the semi-infinite line are related by the “characteristic impedance”

V (z) = V+e−kz

I(z) = I+e−kz

Z0 =V (z)

I(z)=

V +

I+

Characteristic ImpedanceRecall

So, Z0 is a function of the line parameters R, L, G, and C and the frequency. In a similar fashion, we can show

−

dV (z)

dz= (R + ȷωL)I(z)

V (z) = V+e−kz

I(z) = I+e−kz

kV +e−kz = (R + ȷωL)I+e−kz

Z0 =V +

I+=

R + ȷωL

k=

!

R + ȷωL

G + ȷωC

Z0 = −

V −

I−

Terminated, Lossless Transmission Line

The voltage on this line is given by

Define the voltage reflection coefficient as

V (z) = V+e−kz + V

−

e+kz

V (z) = V+

!

e−kz +

V −

V +e+kz

"

Γ0 =V −

V +

Terminated, Lossless Transmission Line

Similarly,

Then,

The impedance at the load end, ZL, is given by

The impedance anywhere along the line is given by

V (z) = V+

!

e−kz + Γ0e

+kz"

I(z) =V +

Z0

!

e−kz

− Γ0e+kz

"

Z(z) =V (z)

I(z)= Z0

e−kz + Γ0e+kz

e−kz− Γ0e

+kz

Z(0) = ZL = Z0

1 + Γ0

1 − Γ0

Terminated, Lossless Transmission LineThen,

CONCLUSION: The reflection coefficient is a function of the load impedance and the characteristic impedance.

Recall

This is often written as

For the lossless case, , and

Then,

Γ0 =ZL − Z0

ZL + Z0

k = kr + ȷki =!

(R + ȷωL)(G + ȷωC)

k = α + ȷβ

β = ω√

LC = kiα = 0

V (z) = V+

!

e−ȷβz + Γ0e

+ȷβz"

I(z) =V +

Z0

!

e−ȷβz

− Γ0e+ȷβz

"

Terminated, Lossless Transmission Line

Rearranging,

It is customary to change to a new coordinate system, d, at this point.

Rewriting the expressions for voltage and current, we have

V (d) = V+

!

e+ȷβd + Γ0e

−ȷβd"

I(d) =V +

Z0

!

e+ȷβd

− Γ0e−ȷβd

"

V (d) = V+e+ȷβd

!

1 + Γ0e−2ȷβd

"

I(d) =V +

Z0

e+ȷβd

!

1 − Γ0e−2ȷβd

"

ImpedanceThe impedance anywhere along the line is given by

The reflection coefficient can be modified as follows

Then, the impedance can be written as

After some algebra, an alternative expression for the impedance is given by

CONCLUSION: The load impedance is “transformed” as we move away from the load.

Z(d) =V (d)

I(d)= Z0

1 + Γ0e−2ȷβd

1 − Γ0e−2ȷβd

Γ(d) = Γ0e−2ȷβd

Z(d) = Z0

1 + Γ(d)

1 − Γ(d)

Z(d) = Z0

ZL + ȷZ0 tanβd

Z0 + ȷZL tanβd

Miscellaneous - But Important!The (Voltage) Standing Wave Ratio - SWR (or VSWR) is defined as

SWR =|Vmax|

|Vmin|=

|Imax|

|Imin|

SWR =1 + |Γ0|

1 − |Γ0|

Closing Comments

• RF circuit design requires impedance transformations/matching to maximize the transfer of power.

• Components (passive, active, PWB interconnects) do not have the idealized impedances seen at low frequency.

• Techniques are needed that determine the impedance of a component and then how to transform its impedance as necessary.