Microwave parasitics: transmission lines

1
Microwave parasitics transmission lines

• transmission lines are "distributed" systems
• whenever the size of the circuit is large
  compared to the appropriate scale length (the
  electromagnetic wavelength here) the system
  cannot be represented by a single "lumped"
  circuit element
• generalized model in terms of infinitesimal
  circuit
• in time harmonic (ejwt) case get "telegraphist's
  equations"

2
Traveling wave solutions for T-lines

• propagation constant g
• characteristic impedance Zo
• for "low" loss or "high" frequency

3
"Input" impedance of a terminated T-line

• impedance is a function of both position and
  load
• reflection coefficient is a convenient number
• for a lossless line both are simple periodic
  functions

4
Behavior of reflection coefficient

• recall the reflection coefficient at a load is
  just

• how does rho behave as load varies?
• Zload 0 (short) ð rho -1
• Zload (open) ð rho 1

• Zload purely imaginary number

ð r 1
5
Relationships between rho and Z

• note relation between rho and the normalized
  impedance

• consider rho in complex plane

• so the relationship between the normalized
  impedance and the real and imaginary parts of rho
  is

6
Relationships between rho and Z

• from the real part

7
Plot of reflection coefficient in complex plane

• plot rho as a function of r
• these are circles!

• center

• radius

• r 0

• 0 lt r lt 1

• r 1

• r gt 1

curves of constant r Re(Z)
8
Plot of reflection coefficient

• from the Im part

• these are also circles!

• center

• radius

• plot rho as a function of x
• x

• x gt 0

• x lt 0

• x 0

curves of constant x Im(Z)
9
Plot of reflection coefficient in complex plane

• the Smith Chart is a plot of the reflection
  coefficient in the complex plane, with contours
  of constant load resistance and load reactance
  superimposed

10
Notes on rho

• for any non-negative value of Re(Z) (i.e., r 0)
  and any value of Im(Z) rho falls on or within the
  unit circle in the complex rho plane
• recalling that the reflection coefficient for a
  lossless transmission line of length l terminated
  by impedance ZL is

• so in the complex rho plane you trace out a
  circle of constant radius as l varies
• radius of circle is just

• recall the input impedance was

• you can read the values right off the Smith chart!

11
Plot of reflection coefficient in complex plane
curves of constant x Im(Z)
v, Im(reflection coef.)
u, Re(reflection coef.)
curves of constant r Re(Z)
12
Example 1 mm line, free space

• simple lossless T-line, Zo 377W, g jwvmoeo

13
For an RLC transmission line

• TEM line of length l,
• transverse dimensions of the conductors and their
  spacings much less than the wavelength and the
  length of the conductors
• terminated with an impedance

Rload
Xload
assume no dielectric loss G 0)
14
Low frequency behavior

• at low frequency (i.e., l ltlt 2p/Im(g), where g is
  the complex propagation constant for the line)
  the input impedance is approximately
• valid to terms up to order w (recall limit is for
  small w)
• Rt, Lt, and Ct are the total resistance,
  inductance, and capacitance of the T-line (i.e.,
  the per unit length values multiplied by the
  length l)

15
Special cases series R-L load

• Series R-L load

16
Special cases series R-C load

• series R-C load

17
Special case capacitive load
CL
Ct

• for RL 0, line capacitance much larger than
  load capacitance
• Ct gtgt CL,
• for RL 0, load capacitance much larger than
  line capacitance
• Ct ltlt CL,

18
For simple lumped models of T-line

• for one lump p model of T-line, pure capacitor
  load
• for RL 0, Ct gtgt CL,
• for one lump tee model of T-line, pure capacitor
  load
• for RL 0, Ct gtgt CL,
• both get capacitance right.
• correct answer
• for RL 0, Ct gtgt CL,
• for RL 0, Ct ltlt CL,

19
Example

• 1 mm thick copper, 10 mm wide, separation 1 mm, 1
  mm long
• Ctot 8.85x10-13 F
• Rtot 4x10-6 W (R out and back)
• Ltot 11.26x10-10 H
• termination pure capacitance Ctot 8.85x10-13
  F