Lossless Transmission Lines (Part-VI)

Voltage

(i)

Current

(ii)

Impedance

(iii)

Z_L = 0

Load Impedance of a Lossless Transmission Line (Lossless transmission line terminated with a short circuit)

Γ = -1

Reflection Coefficient for a Short Circuit (Voltage reflection coefficient of a short-circuited line)

S = ∞

Standing-Wave Ratio for a Short Circuit (VSWR of a short-circuited line)

Ṽ_sc(d) = 2j(V_0^+)sin(βd)

Voltage on a Short-Circuited Line (Voltage distribution along the line)

Ĩ_sc(d) = ((2(V_0^+))/(Z0))cos(βd)

Current on a Short-Circuited Line (Current distribution along the line)

Z_sc(d) = jZ0tan(βd)

Wave Impedance of a Short-Circuited Line (Position-dependent impedance)

zero

Voltage Behavior on a Short-Circuited Line: Voltage is ___ at the load and varies sinusoidally as sin(βd)

Current Behavior on a Short-Circuited Line: Current is _______ at the load and varies as cos(βd)

((Z_in)^sc) = jZ0tan(βl)

Input Impedance of a Short-Circuited Line: (Input impedance measured at distance l)

Input Resistance R_in

Real part of the input impedance Z_in = R_in + j(X_in)

Imaginary reactance X_in

Imaginary part of the input impedance Z_in = R_in + j(X_in)

R_in = 0

Purely Reactive Short-Circuited Line — A lossless short-circuited line has no real input resistance

tan(βl) ≥ 0

Inductive Short-Circuited Line Condition, line behaves inductively when

jω(L_eq) = jZ0tan(βl)

Equivalent Inductor Relation (Input impedance equivalent to an inductor)

L_eq = (Z0tan(βl))/ω

Equivalent Inductance (L_eq) (Inductance represented by a short-circuited line)

l = (1/β)arctan((ωL_eq)/Z0)

Minimum Line Length for Equivalent Inductor: Shortest line length producing inductance (L_eq)

tan(βl) ≤ 0

Capacitive Short-Circuited Line Condition: Line behaves capacitively when…

1/(jω(C_eq)) = jZ0tan(βl)

Equivalent Capacitor Relation: Input impedance equivalent to a capacitor

C_eq = 1/(Z0*ω*tan(βl))

Equivalent Capacitance: Capacitance represented by a short-circuited line

l = (1/β)[π - arctan(1/(ω*C_eq*Z0))]

Minimum Line Length for Equivalent Capacitor (Shortest line producing the capacitance C_eq)

l < λ/4

Quarter-Wavelength Behavior of a Short-Circuited Line, the line behaves inductively if…

λ/4 < l < λ/2

Quarter-Wavelength Behavior of a Short-Circuited Line, the line behaves capacitively if…

Reactive Transmission-Line Stub

Properly chosen short-circuited transmission lines can act as equivalent inductors or capacitors

Microwave Stub Application

Short-circuited transmission-line sections are commonly used in microwave and high-speed integrated circuits instead of physical inductors or capacitors.