Let's derive the **transmission-line equations** for the per-unit-length representations in Fig. P4.1.1 **in the limit as** \(\Delta z \rightarrow 0\). We'll use the circuit in (a) for the derivation, but the result will be the same for (b) and (c) as \(\Delta z \rightarrow 0\). --- ### **Step 1: Equivalent Circuit for a Small Segment** For a small segment \(\Delta z\) of the transmission line: - Series inductance: \(L = l \Delta z\) - Shunt capacitance: \(C = c \Delta z\) Where: - \(l\) = inductance per unit length (H/m) - \(c\) = capacitance per unit length (F/m) Let: - \(V(z, t)\) = voltage at position \(z\) - \(V(z + \Delta z, t)\) = voltage at position \(z + \Delta z\) - \(I(z, t)\) = current at position \(z\) - \(I(z + \Delta z, t)\) = current at position \(z + \Delta z\) --- ### **Step 2: Apply KVL and KCL** #### (a) **KVL (Kirchhoff's Voltage Law) for the Series Inductor** \[ V(z, t) - V(z + \Delta z, t) = L \frac{\partial I(z, t)}{\partial t} \] \[ V(z, t) - V(z + \Delta z, t) = l \Delta z \frac{\partial I(z, t)}{\partial t} \] Rearrange: \[ \frac{V(z, t) - V(z + \Delta z, t)}{\Delta z} = l \frac{\partial I(z, t)}{\partial t} \] Take the limit as \(\Delta z \to 0\): \[ - \frac{\partial V(z, t)}{\partial z} = l \frac{\partial I(z, t)}{\partial t} \] \[ \boxed{ \frac{\partial V(z, t)}{\partial z} = -l \frac{\partial I(z, t)}{\partial t} } \] --- #### (b) **KCL (Kirchhoff's Current Law) for the Shunt Capacitor** The current entering the capacitor: \[ I(z, t) - I(z + \Delta z, t) = C \frac{\partial V(z + \Delta z, t)}{\partial t} \] \[ I(z, t) - I(z + \Delta z, t) = c \Delta z \frac{\partial V(z + \Delta z, t)}{\partial t} \] Rearrange: \[ \frac{I(z, t) - I(z + \Delta z, t)}{\Delta z} = c \frac{\partial V(z + \Delta z, t)}{\partial t} \] Take the limit as \(\Delta z \to 0\): \[ - \frac{\partial I(z, t)}{\partial z} = c \frac{\partial V(z, t)}{\partial t} \] \[ \boxed{ \frac{\partial I(z, t)}{\partial z} = -c \frac{\partial V(z, t)}{\partial t} } \] --- ### **Step 3: Final Transmission Line Equations** The **lossless transmission line equations** are: \[ \boxed{ \begin{align*} \frac{\partial V(z, t)}{\partial z} &= -l \frac{\partial I(z, t)}{\partial t} \\ \frac{\partial I(z, t)}{\partial z} &= -c \frac{\partial V(z, t)}{\partial t} \end{align*} } \] These equations are **independent of the specific structure** of the per-unit-length circuit as \(\Delta z \to 0\), as long as the total per-unit-length \(l\) and \(c\) are the same. --- ## **Summary** - **Derived the transmission-line equations** using KVL and KCL for a small segment. - **Final Answer:** The equations are \[ \frac{\partial V}{\partial z} = -l \frac{\partial I}{\partial t}, \qquad \frac{\partial I}{\partial z} = -c \frac{\partial V}{\partial t} \] - **Structure of the circuit does not matter** in the limit \(\Delta z \to 0\), only the total per-unit-length values \(l\) and \(c\).