Let's derive the transmission-line equations from the per-unit-length representations shown in Fig. P4.1.1. We will use the segment in (a) for clarity, but the result will be the same for all, as Δz → 0. ## 1. **Transmission Line Segment (Fig. a)** ### Per-unit-length parameters: - Inductance per unit length: \( l \) - Capacitance per unit length: \( c \) ### For a small segment \( \Delta z \): - Series inductance: \( l \Delta z \) - Shunt capacitance: \( c \Delta z \) Let \( v(z, t) \) be the voltage and \( i(z, t) \) the current at position \( z \). At \( z + \Delta z \), they are \( v(z + \Delta z, t) \) and \( i(z + \Delta z, t) \). --- ## 2. **Applying KVL and KCL** ### **KVL (Voltage drop across inductor):** \[ v(z, t) - v(z + \Delta z, t) = l \Delta z \frac{\partial i(z, t)}{\partial t} \] Rearrange: \[ \frac{v(z + \Delta z, t) - v(z, t)}{\Delta z} = -l \frac{\partial i(z, t)}{\partial t} \] Take the limit as \( \Delta z \to 0 \): \[ \frac{\partial v(z, t)}{\partial z} = -l \frac{\partial i(z, t)}{\partial t} \] --- ### **KCL (Current into capacitor):** \[ i(z, t) - i(z + \Delta z, t) = c \Delta z \frac{\partial v(z + \Delta z, t)}{\partial t} \] Since \( \Delta z \) is small, \( v(z + \Delta z, t) \approx v(z, t) \): \[ i(z, t) - i(z + \Delta z, t) = c \Delta z \frac{\partial v(z, t)}{\partial t} \] \[ \frac{i(z, t) - i(z + \Delta z, t)}{\Delta z} = c \frac{\partial v(z, t)}{\partial t} \] or \[ -\frac{i(z + \Delta z, t) - i(z, t)}{\Delta z} = c \frac{\partial v(z, t)}{\partial t} \] Take the limit as \( \Delta z \to 0 \): \[ \frac{\partial i(z, t)}{\partial z} = -c \frac{\partial v(z, t)}{\partial t} \] --- ## 3. **Final Transmission-Line Equations** \[ \boxed{ \begin{align*} \frac{\partial v(z, t)}{\partial z} &= -l \frac{\partial i(z, t)}{\partial t} \\ \frac{\partial i(z, t)}{\partial z} &= -c \frac{\partial v(z, t)}{\partial t} \end{align*} } \] These are the **lossless transmission-line (telegrapher's) equations**. --- ### **Explanation:** - These equations describe how voltage and current vary along a transmission line. - They are derived directly from the lumped-element circuit model, and the form is independent of the specific arrangement of L and C as \( \Delta z \to 0 \). - The key is to apply KVL and KCL to a differential segment and then take the limit as the segment size approaches zero. --- ### **Conclusion:** No matter the detailed structure (as in Fig. (a), (b), or (c)), as long as the total per-unit-length inductance and capacitance are \( l \) and \( c \), the differential equations governing the transmission line are: \[ \boxed{ \frac{\partial v}{\partial z} = -l \frac{\partial i}{\partial t}, \quad \frac{\partial i}{\partial z} = -c \frac{\partial v}{\partial t} } \]