Question

The following circuit(s) representing a lumped element equivalent of an infinitesimal section of a transmission line is/are 

• A. Series branch: $L A z$ then $R A z$; shunt to ground at the left: $C A z/2$; shunt to ground at the right: $G A z/2$.
• B. Shunt to ground at the input: $G A z$ in parallel with $C A z$; then series branch: $L A z$ followed by $R A z$.
• C. $\pi$-form: shunt to ground at input: $G A z/2$ in parallel with $C A z/2$; middle series branch: $L A z$ in series with $R A z$; shunt to ground at output: $G A z/2$ in parallel with $C A z/2$.
• D. $T$-form (symmetric): input series $R A z/2$ then $L A z/2$; shunt to ground at center: $G A z$ in parallel with $C A z$; output series $L A z/2$ then $R A z/2$.

Hint:

For an infinitesimal line section, any arrangement that (to $O(\Delta z)$) gives series $Z_s=(R+sL)\Delta z$ and shunt $Y_s=(G+sC)\Delta z$ is acceptable. Symmetric $T$ or $\pi$ always works; avoid separating $G$ and $C$ to different nodes.

Answer 1

The Correct Option is B

For a transmission line, per–unit–length parameters are $R,\ L,\ G,\ C$. For an infinitesimal section of length $\Delta z$, a first-order lumped equivalent must realize
\[ Z_s=(R+sL)\,\Delta z \text{in series, and} Y_s=(G+sC)\,\Delta z \ \text{in shunt}, \] so that the ABCD matrix is $A=D=1$, $B=Z_s$, $C=Y_s$ up to $O(\Delta z^2)$.
Check (D): Symmetric $T$ with $Z_s/2$–$Y_s$–$Z_s/2$ gives $A=D=1$, $B=Z_s$, $C=Y_s$ to first order. \(\Rightarrow\) Valid.
Check (C): Symmetric $\pi$ with $Y_s/2$–$Z_s$–$Y_s/2$ also yields $A=D=1$, $B=Z_s$, $C=Y_s$ to first order. \(\Rightarrow\) Valid.
Check (B): Placing the entire shunt $Y_s$ at one end and the entire series $Z_s$ along the line gives $A=D=1$, $B=Z_s$, $C=Y_s$ ignoring the product $Z_sY_s$ (which is $O(\Delta z^2)$). For an infinitesimal section, this is first-order equivalent. \(\Rightarrow\) Valid.
Check (A): Splitting unequally the shunt elements (capacitive on one end, conductive on the other) fails to realize a single shunt admittance $Y_s=(G+sC)\Delta z$ at any node; the first-order $C$ entry of the ABCD matrix no longer equals $Y_s$. \(\Rightarrow\) Invalid.
\[ \boxed{\text{Valid infinitesimal models: (B), (C), and (D).}} \]