Tranmsission line theory
Transmission Line Equations: Derivation, Wave Equations and Lossless Line Analysis
The transmission line equations, also known as the Telegrapher’s equations, describe the relationship between voltage and current as electromagnetic waves propagate along a transmission line. These equations are fundamental to transmission-line theory because they connect the distributed electrical parameters of a practical transmission line with the voltage and current waves traveling along it. The equations can be derived from an infinitesimally small section of the transmission line using Kirchhoff’s Voltage Law (KVL) and Kirchhoff’s Current Law (KCL). They can also be related to Maxwell’s equations, showing how electromagnetic field behavior is represented by the voltage and current quantities used in circuit analysis.
From Electromagnetic Fields to Transmission Line Quantities
Fundamentally, a transmission-line problem is an electromagnetic problem because the transmission of electrical energy occurs through electric and magnetic fields surrounding the conductors. For a two-conductor transmission line supporting a Transverse Electromagnetic (TEM) wave, or a wave that can be closely approximated as TEM, the field analysis can be simplified considerably. An important property of a TEM wave is that the electric field is directly related to the voltage between the conductors, while the magnetic field is related to the current flowing along the conductors. This relationship allows the complex electromagnetic field problem to be represented using the scalar quantities voltage V and current I.
Fig: Voltage Relation
The voltage between the two conductors is related to the electric field by a line integral taken along a path from one conductor to the other. The voltage can therefore be expressed as:
\[ V = -\int_{L} \mathbf{E} \cdot d\mathbf{l} \]
The negative sign follows the conventional relationship between electric potential difference and electric field. This equation provides the electromagnetic interpretation of the voltage used in transmission-line circuit analysis.
Fig: Current Relation
Similarly, the total current flowing through a conductor is related to the magnetic field surrounding the conductor. According to Ampere’s law, the current can be obtained from the closed-loop integral of the magnetic field around the conductor:
\[ I = \oint_{L} \mathbf{H} \cdot d\mathbf{l} \]
These relationships provide the connection between the electromagnetic fields and the circuit quantities used in transmission-line analysis. The distributed resistance, inductance, capacitance, and conductance of the line can then be used to describe how voltage and current vary with position and time.
Transmission Line Equations and Maxwell’s Equations
The transmission line equations can be viewed as a circuit-level representation of the electromagnetic relationships described by Maxwell’s equations. For a transmission line, the distributed series resistance and inductance determine the voltage variation along the line, while the distributed shunt conductance and capacitance determine the current variation. The resulting equations are:
\[ -\frac{\partial V}{\partial z} = RI + L\frac{\partial I}{\partial t} \]
\[ -\frac{\partial I}{\partial z} = GV + C\frac{\partial V}{\partial t} \]
These are the fundamental time-domain transmission line equations. They are commonly called the Telegrapher’s equations. The first equation describes how voltage changes along the transmission-line direction because of the series resistance and inductance. The second equation describes how current changes along the line because of the shunt conductance and capacitance.
Derivation of the Transmission Line Equations
To derive the transmission line equations, consider an infinitesimally small section of a transmission line having a length of Δz. The distributed parameters of the line are represented by resistance R, inductance L, conductance G, and capacitance C per unit length. The equivalent circuit of this small section provides the basis for applying KVL and KCL.
Fig: Equivalent Transmission Line Circuit
Derivation Using KVL
Apply Kirchhoff’s Voltage Law around the infinitesimal transmission-line section. The voltage drop across the series resistance and inductance must be equal to the difference between the voltage at positions z and z + Δz. Therefore:
\[ -V(z,t) + I(z,t)R\Delta z + L\Delta z\frac{\partial I(z,t)}{\partial t} + V(z+\Delta z,t) =0 \]
Rearranging the equation gives:
\[ V(z+\Delta z,t)-V(z,t) = -RI(z,t)\Delta z - L\Delta z\frac{\partial I(z,t)}{\partial t} \]
Dividing both sides by Δz gives:
\[ \frac{V(z+\Delta z,t)-V(z,t)}{\Delta z} = -RI(z,t) - L\frac{\partial I(z,t)}{\partial t} \]
Taking the limit as Δz approaches zero, the finite difference becomes a partial derivative with respect to the transmission-line coordinate z:
\[ \frac{\partial V(z,t)}{\partial z} = -RI(z,t) - L\frac{\partial I(z,t)}{\partial t} \]
Therefore, the first transmission line equation can be written as:
\[ -\frac{\partial V(z,t)}{\partial z} = RI(z,t) + L\frac{\partial I(z,t)}{\partial t} \]
This equation shows that the spatial variation of voltage along the line is determined by the series resistance R and series inductance L per unit length.
Derivation Using KCL
Next, apply Kirchhoff’s Current Law to the same infinitesimal transmission-line section. The current entering the section is divided between the current continuing along the line and the current flowing through the shunt conductance and capacitance. Thus:
\[ I(z,t) = I(z+\Delta z,t) + G\Delta zV(z+\Delta z,t) + C\Delta z\frac{\partial V(z+\Delta z,t)}{\partial t} \]
Rearranging gives:
\[ I(z+\Delta z,t)-I(z,t) = -G\Delta zV(z+\Delta z,t) - C\Delta z\frac{\partial V(z+\Delta z,t)}{\partial t} \]
Dividing by Δz:
\[ \frac{I(z+\Delta z,t)-I(z,t)}{\Delta z} = -GV(z+\Delta z,t) - C\frac{\partial V(z+\Delta z,t)}{\partial t} \]
Taking the limit as Δz approaches zero:
\[ \frac{\partial I(z,t)}{\partial z} = -GV(z,t) - C\frac{\partial V(z,t)}{\partial t} \]
Therefore, the second transmission line equation is:
\[ -\frac{\partial I(z,t)}{\partial z} = GV(z,t) + C\frac{\partial V(z,t)}{\partial t} \]
This equation shows that the spatial variation of current depends on the shunt conductance G and shunt capacitance C per unit length.
Frequency-Domain Transmission Line Equations
For sinusoidal steady-state analysis, voltage and current can be represented using phasors. The time derivative can then be replaced by multiplication by jω. The time-domain Telegrapher’s equations consequently become the frequency-domain transmission line equations:
\[ -\frac{dV(z)}{dz} = (R+j\omega L)I(z) \]
\[ -\frac{dI(z)}{dz} = (G+j\omega C)V(z) \]
These equations are particularly useful in RF and microwave engineering because they directly describe the voltage and current behavior of a sinusoidal signal propagating along a transmission line.
Transmission Line Wave Equations
The first-order transmission line equations can be combined to obtain second-order differential equations for voltage and current. These are known as the transmission line wave equations. They describe the propagation of voltage and current waves along the transmission line.
Wave Equation for Voltage
Starting with the first frequency-domain transmission line equation:
\[ -\frac{dV(z)}{dz} = (R+j\omega L)I(z) \]
Differentiate both sides with respect to z:
\[ -\frac{d^2V(z)}{dz^2} = (R+j\omega L)\frac{dI(z)}{dz} \]
From the second transmission line equation:
\[ \frac{dI(z)}{dz} = -(G+j\omega C)V(z) \]
Substituting this into the differentiated voltage equation gives:
\[ -\frac{d^2V(z)}{dz^2} = (R+j\omega L) \left[-(G+j\omega C)V(z)\right] \]
Therefore:
\[ \frac{d^2V(z)}{dz^2} - (R+j\omega L)(G+j\omega C)V(z) = 0 \]
This is the voltage wave equation for a general transmission line.
Wave Equation for Current
Starting with the second transmission line equation:
\[ -\frac{dI(z)}{dz} = (G+j\omega C)V(z) \]
Differentiating with respect to z gives:
\[ -\frac{d^2I(z)}{dz^2} = (G+j\omega C)\frac{dV(z)}{dz} \]
From the first transmission line equation:
\[ \frac{dV(z)}{dz} = -(R+j\omega L)I(z) \]
Substituting gives:
\[ -\frac{d^2I(z)}{dz^2} = (G+j\omega C) \left[-(R+j\omega L)I(z)\right] \]
Therefore:
\[ \frac{d^2I(z)}{dz^2} - (G+j\omega C)(R+j\omega L)I(z) = 0 \]
This is the current wave equation for a general transmission line.
Propagation Constant of a Transmission Line
The quantity multiplying the voltage and current terms in the wave equations is defined in terms of the propagation constant. The propagation constant describes how a wave changes in amplitude and phase as it travels along the transmission line.
\[ \gamma = \alpha+j\beta = \sqrt{(R+j\omega L)(G+j\omega C)} \]
Here, α is the attenuation constant, which represents the reduction in signal amplitude as the wave propagates, while β is the phase constant, which represents the rate at which the phase changes with distance.
Using the propagation constant, the wave equations can be written in a simpler form:
\[ \frac{d^2V(z)}{dz^2} - \gamma^2V(z) = 0 \]
\[ \frac{d^2I(z)}{dz^2} - \gamma^2I(z) = 0 \]
General Voltage and Current Wave Solutions
The general solution of the voltage wave equation consists of two traveling-wave components. One component travels in the forward direction and the other travels in the reverse direction:
\[ V(z) = V_0^+e^{-\gamma z} + V_0^-e^{+\gamma z} \]
The corresponding current is:
\[ I(z) = \frac{V_0^+}{Z_0}e^{-\gamma z} - \frac{V_0^-}{Z_0}e^{+\gamma z} \]
Here, V0+ represents the amplitude of the forward-traveling voltage wave, while V0− represents the amplitude of the backward-traveling voltage wave. The negative sign associated with the reflected current wave results from the opposite direction of current associated with the backward-traveling wave.
Characteristic Impedance of a Transmission Line
The characteristic impedance is an important transmission-line parameter that represents the voltage-to-current ratio of a single traveling wave. From the transmission line equations, it is given by:
\[ Z_0 = \frac{R+j\omega L}{\gamma} \]
Substituting the expression for the propagation constant gives:
\[ Z_0 = \sqrt{\frac{R+j\omega L}{G+j\omega C}} \]
The characteristic impedance depends on the distributed resistance, inductance, conductance, and capacitance of the transmission line. Proper matching between the source, transmission line, and load is important because impedance mismatch causes reflections and standing waves.
Lossless Transmission Line
A lossless transmission line is an ideal transmission line in which the series resistance and shunt conductance are zero. Therefore, the conditions for a lossless line are:
\[ R=0,\qquad G=0 \]
Under these conditions, the transmission line equations become considerably simpler. There is no attenuation due to the distributed resistance or conductance, although the voltage and current still undergo phase changes as they propagate along the line.
Propagation Constant of a Lossless Line
The general propagation constant is:
\[ \gamma = \sqrt{(R+j\omega L)(G+j\omega C)} \]
For a lossless line, substituting R = 0 and G = 0 gives:
\[ \gamma = \sqrt{(j\omega L)(j\omega C)} \]
Since j2 = -1:
\[ \gamma = \sqrt{-\omega^2LC} \]
Therefore:
\[ \gamma = j\omega\sqrt{LC} \]
Since the general propagation constant is:
\[ \gamma=\alpha+j\beta \]
comparison gives:
\[ \alpha=0,\qquad \beta=\omega\sqrt{LC} \]
The attenuation constant is therefore zero for an ideal lossless line, while the phase constant remains nonzero.
Characteristic Impedance of a Lossless Line
The general characteristic impedance is:
\[ Z_0 = \sqrt{\frac{R+j\omega L}{G+j\omega C}} \]
For a lossless transmission line:
\[ Z_0 = \sqrt{\frac{j\omega L}{j\omega C}} \]
Therefore:
\[ Z_0 = \sqrt{\frac{L}{C}} \]
Thus, the characteristic impedance of an ideal lossless transmission line is purely real and depends only on the distributed inductance and capacitance per unit length.
Voltage and Current Waves on a Lossless Line
For a lossless line, the propagation constant is:
\[ \gamma=j\beta \]
Therefore, the general voltage and current solutions become:
\[ V(z) = V_0^+e^{-j\beta z} + V_0^-e^{j\beta z} \]
\[ I(z) = \frac{V_0^+}{Z_0}e^{-j\beta z} - \frac{V_0^-}{Z_0}e^{j\beta z} \]
The term e−jβz represents the forward-traveling wave, while e+jβz represents the backward-traveling wave under the selected propagation convention.
Phase Velocity of a Lossless Line
The phase velocity is the velocity at which a constant phase point of the wave propagates along the transmission line. It is given by:
\[ v_p = \frac{\omega}{\beta} \]
For a lossless transmission line:
\[ \beta=\omega\sqrt{LC} \]
Therefore:
\[ v_p = \frac{\omega}{\omega\sqrt{LC}} \]
Hence:
\[ v_p = \frac{1}{\sqrt{LC}} \]
This result shows that the phase velocity of a lossless transmission line is determined by its distributed inductance and capacitance per unit length.
Wavelength of a Lossless Transmission Line
The wavelength is related to the phase constant by:
\[ \lambda = \frac{2\pi}{\beta} \]
Substituting the lossless-line expression for the phase constant gives:
\[ \lambda = \frac{2\pi}{\omega\sqrt{LC}} \]
Since ω = 2πf, the wavelength can also be related to the operating frequency and phase velocity. The wavelength is an important quantity when determining the electrical length of a transmission line and analyzing standing waves, impedance transformation, and transmission-line matching networks.
Effective Complex Permittivity
In a lossy dielectric, the permittivity can be represented as a complex quantity. Complex permittivity provides a convenient way to represent dielectric polarization and loss within a frequency-domain electromagnetic model. It is commonly written as:
\[ \varepsilon = \varepsilon' - j\varepsilon'' \]
Here, ε′ represents the real part of the permittivity and is associated primarily with energy storage in the dielectric, while ε″ represents the dielectric-loss component under the stated sign convention.
Derivation of Effective Complex Permittivity
Starting with Ampere’s law in phasor form, the total current density consists of both conduction current and displacement current:
\[ \nabla\times\mathbf{H} = \mathbf{J}_{\text{total}} \]
The total current density is:
\[ \mathbf{J}_{\text{total}} = \mathbf{J}_c+\mathbf{J}_d \]
The conduction current density is:
\[ \mathbf{J}_c = \sigma\mathbf{E} \]
For sinusoidal fields, the displacement current density is:
\[ \mathbf{J}_d = j\omega\varepsilon\mathbf{E} \]
Using the complex permittivity:
\[ \varepsilon = \varepsilon' - j\varepsilon'' \]
the total current density becomes:
\[ \mathbf{J}_{\text{total}} = \sigma\mathbf{E} + j\omega(\varepsilon'-j\varepsilon'')\mathbf{E} \]
Expanding:
\[ \mathbf{J}_{\text{total}} = \sigma\mathbf{E} + j\omega\varepsilon'\mathbf{E} - j^2\omega\varepsilon''\mathbf{E} \]
Since j2 = -1:
\[ \mathbf{J}_{\text{total}} = \sigma\mathbf{E} + j\omega\varepsilon'\mathbf{E} + \omega\varepsilon''\mathbf{E} \]
Grouping the real loss terms gives:
\[ \mathbf{J}_{\text{total}} = (\sigma+\omega\varepsilon'')\mathbf{E} + j\omega\varepsilon'\mathbf{E} \]
Now define an effective complex permittivity such that the total current density can be represented as:
\[ \mathbf{J}_{\text{total}} = j\omega\varepsilon_{\text{eff}}\mathbf{E} \]
Therefore:
\[ j\omega\varepsilon_{\text{eff}}\mathbf{E} = (\sigma+\omega\varepsilon'')\mathbf{E} + j\omega\varepsilon'\mathbf{E} \]
Dividing by jωE:
\[ \varepsilon_{\text{eff}} = \varepsilon' + \frac{\sigma+\omega\varepsilon''}{j\omega} \]
Using:
\[ \frac{1}{j}=-j \]
gives:
\[ \varepsilon_{\text{eff}} = \varepsilon' - j\frac{\sigma+\omega\varepsilon''}{\omega} \]
Therefore, the effective complex permittivity is:
\[ \varepsilon_{\text{eff}} = \varepsilon' - j\left( \varepsilon'' + \frac{\sigma}{\omega} \right) \]
This expression shows that both dielectric loss and conduction loss contribute to the imaginary part of the effective complex permittivity. The term σ/ω represents the contribution of finite conductivity to the effective dielectric loss.
Loss Tangent
The loss tangent describes the ratio of the loss component of a dielectric to its energy-storage component. With the convention used above, the dielectric loss tangent is:
\[ \tan\delta = \frac{\varepsilon''}{\varepsilon'} \]
When conductor or finite-conductivity loss is incorporated into the effective complex permittivity, the corresponding effective loss tangent becomes:
\[ \tan\delta_{\text{eff}} = \frac{\varepsilon''+\frac{\sigma}{\omega}} {\varepsilon'} \]
A larger loss tangent indicates greater dielectric loss relative to the energy stored in the electric field. This concept is important when selecting dielectric materials for high-frequency and microwave transmission-line applications because dielectric loss contributes to signal attenuation.