Filters in Network Theory

In this tutorial, you will learn the basics of filters, including Nepers and attenuation, the relation between decibels and Nepers, and the types of filters: low pass, high pass, band pass, and band reject. You will also explore impedances and propagation constants in T- and π-networks, key terms in the propagation constant expression, and the differences between attenuation bands and transmission bands.

Contents:

1. Introduction to Filters
2. Nepers and Attenuation in Filters
3. Relation Between Decibels and Nepers
4. Low Pass and High Pass Filters
5. Band Pass and Band Reject Filters
6. Impedances in T-network
7. Propagation Constant in T-Network
8. Propagation Constant Forms in T-Networks
9. Impedances and Propagation Constant in π-Networks
10. Terms in the Propagation Constant Expression
11. Attenuation Constant with Identical Reactances
12. Attenuation and Phase Shift Constants with Varying Reactances
13. Graphical Expression for Change in Attenuation Constant
14. Graphical Expression for Change in Phase Shift Constant
15. Differences Between Attenuation Band and Transmission Band

Introduction to Filters

Filters are essential components in electronics, designed to allow certain frequencies to pass while blocking others. They are reactive networks made from purely reactive elements (inductors and capacitors), which allows them to have zero attenuation within the passband.

Features of Filters

• A filter enables us to freely pass the required bands of frequencies and suppress all other bands of frequencies.
• A filter is a reactive network and is constructed from purely reactive elements. This was necessary to make the attenuation zero in the pass band of the filter network.
• They provide constant transmission over the band they accept the signals to pass.

Applications of Filters

• They are used in communication systems to distinguish different voice channels in carrier frequency circuits.
• They are used in instrumentation.
• They are used in telemetering equipment.

Nepers and Attenuation in Filters

The attenuation of a filter is measured in Nepers. Nepers quantify the ratio of input to output voltage or current in terms of the natural logarithm, provided the system is terminated in its characteristic impedance.

• The input and output can be either be a voltage or current. The expression for Nepers can be given as follows:
  \(N = \log_e \left| \frac{V_1}{V_2} \right| = \log_e \left| \frac{I_1}{I_2} \right|\)
  Where
  N is the number of nepers
  V₁ is the input voltage
  V₂ is the output voltage
  I₁ is the input current
  I₂ is the output current
• The nepers can be expressed in terms of power using the following expression:
  \(N = \frac{1}{2} \log_e \left| \frac{P_1}{P_2} \right|\)
  Where
  N is the number of nepers
  P₁ is the input power
  P₂ is the output power

Relation Between Decibels and Nepers

Decibels (dB) are another measure of attenuation, widely used in filter design and analysis. The relationship between Decibels and Nepers is important to understand how these units correlate.

• A decibel is ten times the common logarithm of the ratio of input power to output power.
  \(D = 10 \log_{10} \left| \frac{P_1}{P_2} \right|\)
  Where
  D is the decibel
  P₁ is the input power
  P₂ is the output power
• A decibel can be expressed in terms of voltage and current using the following expressionl:
  \(D = 20 \log_{10} \left| \frac{V_1}{V_2} \right| = 20 \log_{10} \left| \frac{I_1}{I_2} \right|\)
  Where
  D is the decibel
  V₁ is the input voltage
  V₂ is the output voltage
  I₁ is the input current
  I₂ is the output current
• The relation between nepers and decibels can be given as follows:
  \(D = 10 \log_{10} \left| \frac{P_1}{P_2} \right| = \frac{10}{2.303} \log_e \left| \frac{P_1}{P_2} \right|\)
  \(D = 4.3421 \cdot \frac{2}{2} \log_e \left| \frac{P_1}{P_2} \right| = 8.685 \cdot \frac{1}{2} \log_e \left| \frac{P_1}{P_2} \right|\)
  \(D = 8.685 N\)
  \(N = 0.115 D\)
• Thus, we can convert decibel to neper and vice versa using the following expression:
  \(\text{Neper value = Decibel value}*0.115\)
  \(\text{Decibel value = Neper value}*8.685\)

Low Pass and High Pass Filters

Low Pass Filter

• A low pass filter allows all frequencies without attenuation upto the cut-off frequency and attenuates other frequencies greater than the cut-off frequency.
• The pass band or transmission band in a low pass filter is the frequency range between zero to the cut-off frequency.
• The stop band or attenuation band in a low pass filter is the frequency range beyond the cut-off frequency.

High Pass Filter

• A high pass filter allows all frequencies without attenuation beyond the cut-off frequency and attenuates other frequencies below the cut-off frequency.
• The pass band or transmission band in a high pass filter is the frequency range beyond the cut-off frequency.
• The stop band or attenuation band in a high pass filter is the frequency range between zero to the cut-off frequency.

Band Pass and Band Reject Filters

Band Pass Filter

• A band pass filter allows all frequencies between two frequencies and attenuates other frequencies.
• The pass band or transmission band in a band pass filter is the frequency range between the lower cut-off frequency and higher cut-off frequency.
• The stop band or attenuation band is the frequency range less than the lower cut-off frequency and beyond the higher cut-off frequency.

Band Elimination Filter

• A band elimination filter rejects all frequencies between two frequencies and attenuates other frequencies.
• The pass band or transmission band in a band elimination filter is the frequency range less than the lower cut-off frequency and beyond the higher cut-off frequency.
• The stop band or attenuation band is the frequency range between the lower cut-off frequency and higher cut-off frequency.

Impedances in T-network

The figure given below represents a symmetrical T-network.

Characteristic Impedance

• Image impedance is called the characteristic impedance if the impedance at port 11’ is equal to the impedance at port 22’.
• Image impedance also called as iterative impedance is represented by Z₀.
• The following gives the expression of characteristic impedance for a symmetrical T network, Z_0T.
  \(Z_{OT} = \sqrt{\frac{Z_1^2}{4} + Z_1 Z_2}\)

Open Circuit and Short Circuit Impedance

• The open-circuit impedance for a symmetrical T network can be obtained by finding the equivalent resistance when port 22’ is open-circuited.
  \(Z_{OCT}=\frac{Z_1}{2+Z_2}\)
• The short circuit impedance for a symmetrical T network can be obtained by finding the equivalent resistance when port 22’ is short-circuited.
  \(Z_{SCT} = \frac{Z_1}{2} + \frac{\left(\frac{Z_1}{2} \cdot Z_2\right)}{\left(\frac{Z_1}{2} + Z_2\right)}\)
• The characteristic impedance is the square root of the product of open-circuit impedance and short circuit impedance.
  \(Z_{OT} = \sqrt{Z_{OCT} \cdot Z_{SCT}}\)

Propagation Constant in T-Network

The following points give the propagation constant of a T-network.

• Consider a T network with two impedances Z₁/2 in the series branch and impedance Z₂ in the parallel branch. A voltage source V is placed in the first port and the second port is terminated with impedance, Z_o.
• The propagation constant, γ is given by the following expression:
  \(\gamma = \log_e \left( \frac{I_1}{I_2} \right)\)
  \(\frac{I_1}{I_2} = e^{\gamma}\)
• When mesh analysis is used in the second loop, the relation between I₁ and I₂ is obtained.
  \(I_1 Z_2 = I_2 \left( \frac{Z_1}{2} + Z_2 + Z_{0T} \right)\)
  \(\frac{I_1}{I_2} = \frac{1}{Z_2} \left( \frac{Z_1}{2} + Z_2 + Z_{0T} \right) = e^{\gamma}\)
  \(\left( \frac{Z_1}{2} + Z_2 + Z_{0T} \right) = Z_2 e^{\gamma}\)
  \(Z_{0T} = Z_2 (e^{\gamma} – 1) – \frac{Z_1}{2} \quad \text{(1)}\)
• The characteristic impedance for a symmetrical T network can be given as Z_0T.
  \(Z_{OT} = \sqrt{\frac{Z_1^2}{4} + Z_1 Z_2} \quad \text{(2)}\)
• Equating equations (1) and (2) the value for propagation constant is obtained.
  \(\gamma = \cosh^{-1} \left( 1 + \frac{Z_1}{2 Z_2} \right)\)

Propagation Constant Forms in T-Networks

The following points give the different forms in which the propagation constant of a T-network can be expressed.

• The image impedance (Z_0T), open circuit impedance (Z_OC), and short circuit impedance (Z_SC) for a T-network can be expressed using the following expressions:
  \(Z_{OC} = \frac{Z_1}{2} + Z_2\)
  \(Z_{OT} = \sqrt{\frac{Z_1^2}{4} + Z_1 Z_2}\)
  \(Z_{OT} = \sqrt{Z_{OC} \cdot Z_{SC}}\)
• The propagation constant for a T-network is given by the following expression:
  \(\gamma = \cosh^{-1} \left( 1 + \frac{Z_1}{2 Z_2} \right)\)
  \(\cosh \gamma = 1 + \frac{Z_1}{2 Z_2} \)
• The propagation constant in terms of hyperbolic sine function can be obtained as follows:
  \(\cosh \gamma = 1 + 2 \sinh^2 \left( \frac{\gamma}{2} \right)\)
  \(\sinh \left( \frac{\gamma}{2} \right) = \sqrt{\frac{\cosh \gamma – 1}{2}} = \sqrt{\frac{1 + \frac{Z_1}{2 Z_2} – 1}{2}}\)
  \(\sinh \left( \frac{\gamma}{2} \right) = \sqrt{\frac{Z_1}{4 Z_2}}\)
• The propagation constant in terms of the hyperbolic sine function and image impedance can be obtained as follows:
  \(\sinh \gamma = \sqrt{\cosh^2 \gamma – 1}\)
  \(\sinh \gamma = \sqrt{\left(1 + \frac{Z_1}{2 Z_2}\right)^2 – 1} = \sqrt{1 + \left(\frac{Z_1}{2 Z_2}\right)^2 + \frac{Z_1}{Z_2} – 1}\)
  \(\sinh \gamma = \sqrt{\left(\frac{Z_1}{2 Z_2}\right)^2 + \frac{Z_1}{Z_2}} = \frac{1}{Z_2} \sqrt{\left(\frac{Z_1}{2}\right)^2 + Z_1 Z_2}\)
  \(\sinh \gamma = \frac{Z_{0T}}{Z_2}\)
• The propagation constant in terms of hyperbolic tan function can be obtained as follows:
  \(\tanh \gamma = \frac{\sinh \gamma}{\cosh \gamma} = \frac{Z_{0T}}{Z_2 \left(1 + \frac{Z_1}{2 Z_2}\right)}\)
  \(\tanh \gamma = \frac{Z_{0T}}{Z_2 + \frac{Z_1}{2}}\)
  \(\tanh \gamma = \frac{Z_{0T}}{Z_{0C}} = \frac{\sqrt{Z_{0C} Z_{SC}}}{Z_{0C}}\)
  \(\tanh \gamma = \sqrt{\frac{Z_{SC}}{Z_{0C}}}\)

Impedances and Propagation Constant in π-Networks

The figure given below represents a symmetrical π-network.

Characteristic Impedance

• Image impedance becomes the characteristic impedance if the input impedance at port 11’ is equal to the impedance at port 22’.
• Image impedance also called as the iterative impedance is represented by Z₀.
• The following gives the expression of characteristic impedance for a symmetrical π network, Z_0π:
  \(Z_{O\pi} = \sqrt{\frac{Z_1 Z_2}{1 + \frac{Z_1}{4 Z_2}}}\)

Open Circuit and Short Circuit Impedance

• The open-circuit impedance for a symmetrical π network can be obtained by finding the equivalent resistance when port 22’ is open-circuited.
  \(Z_{OC\pi} = \frac{2Z_2 (Z_1 + 2Z_2)}{Z_1 + 4Z_2}\)
• The short circuit impedance for a symmetrical π network can be obtained by finding the equivalent resistance when port 22’ is short-circuited.
  \(Z_{SC\pi} = \frac{2Z_1 Z_2}{Z_1 + 2Z_2}\)
• The characteristic impedance is the square root of the product of open-circuit impedance and short circuit impedance.
  \(Z_{O\pi} = \sqrt{Z_{OC\pi} \cdot Z_{SC\pi}}\)

Propagation Constant

• The propagation constant for a π- network is the same as that for T-network and is given by the following expression:
  \(\gamma = \cosh^{-1} \left( 1 + \frac{Z_1}{2 Z_2} \right)\)
  \(\cosh \gamma = 1 + \frac{Z_1}{2 Z_2}\)

Terms in the Propagation Constant Expression

The following points give the different terms in the propagation constant expression.

• The propagation constant is a complex term and is represented by γ.
  \(γ=α+jβ\)
• The real part, α is the attenuation constant and is expressed in nepers.
• The attenuation constant helps in measuring the magnitude of current or voltage present in the network.
• The imaginary part, β is the phase shift constant and is expressed in radians.
• The phase shift constant helps in measuring the phase difference between the input and output of current or voltage present in the network.

Attenuation Constant with Identical Reactances

The following points give the attenuation constant when the same type of reactances are present in the network.

• The propagation constant, γ in terms of hyperbolic sine function can be written and split into its real and imaginary terms.
  \(\sinh\left(\frac{\gamma}{2}\right) = \sqrt{\frac{Z_1}{4Z_2}}\)
  \(\sinh\left(\frac{\alpha}{2} + j\frac{\beta}{2}\right) = \sinh\left(\frac{\alpha}{2}\right) \cos\left(\frac{\beta}{2}\right) + j \cosh\left(\frac{\alpha}{2}\right) \sin\left(\frac{\beta}{2}\right) = \sqrt{\frac{Z_1}{4Z_2}}\)
• When the same kind of reactances is present in the network \(\left| \frac{Z_1}{4Z_2} \right|\) will be real. Consider \(\sqrt{\frac{Z_1}{4Z_2}}\) be equal to x
  \(\left| \frac{Z_1}{4Z_2} \right| > 0\).
• The imaginary part of the propagation constant will be zero. Then the real part of the propagation constant is x.
  \(\sinh\left(\frac{\alpha}{2}\right) \cos\left(\frac{\beta}{2}\right) = x\)
  \(\cosh\left(\frac{\alpha}{2}\right) \sin\left(\frac{\beta}{2}\right) = 0\)
• The above two equations can be satisfied when \((\frac{β}{2}nπ)\) where n = 0,1,2,…. Substituting the value of \(\frac{β}{2}\) in the above equations the following are obtained. \(\sinh\left(\frac{\alpha}{2}\right) \cos\left(\frac{0}{2}\right) = x\)
  \(\sinh\left(\frac{\alpha}{2}\right) = x = \sqrt{\frac{Z_1}{4Z_2}}\)
  \(\alpha = 2 \sinh^{-1}\left(\sqrt{\frac{Z_1}{4Z_2}}\right)\)
• Since the attenuation constant is not equal to zero, attenuation exists in this case.

Attenuation and Phase Shift Constants with Varying Reactances

The following points give the expressions for attenuation constant and phase shift constant when different types of reactances are present in the network.

• The propagation constant, γ can be expressed in its real and imaginary terms using the following expression:
  \(\sinh\left(\frac{\gamma}{2}\right) = \sqrt{\frac{Z_1}{4Z_2}}\)
  \(\sinh\left(\frac{\alpha}{2} + j\frac{\beta}{2}\right) = \sinh\left(\frac{\alpha}{2}\right) \cos\left(\frac{\beta}{2}\right) + j \cosh\left(\frac{\alpha}{2}\right) \sin\left(\frac{\beta}{2}\right) = \sqrt{\frac{Z_1}{4Z_2}}\)
• When different type of reactances is present in the network \(\frac{Z_1}{4Z_2}\) is negative. Thus,\(\sqrt{\frac{Z_1}{4Z_2}}\) will be imaginary and equal to j_x.
• The real part of the propagation constant must be zero. Thus, the imaginary part of the propagation constant will be x. These two equations will be satisfied when (α=0 or β/2=π).
  \(sinh⁡(\frac{α}{2}) cos⁡(\frac{β}{2})=0\)
  \(cosh(\frac{α}{2})sin(\frac{β}{2})=x\)
• When α=0 is considered the value of β is obtained. Since the attenuation constant is zero and phase constant exists in this case. This gives the transmission band.
  \(-1 \leq \frac{Z_1}{4Z_2} \leq 0\)
  \(\cosh\left(\frac{0}{2}\right) \sin\left(\frac{\beta}{2}\right) = x\)
  \(\sin\left(\frac{\beta}{2}\right) = x = \sqrt{\frac{Z_1}{4Z_2}}\)
  \(\beta = 2 \sin^{-1}\left(\sqrt{\frac{Z_1}{4Z_2}}\right)\)
• When β=π is considered the value of α is obtained. The attenuation constant exists in this case and thus, the condition for stop band is obtained. \(\cosh\left(\frac{\alpha}{2}\right) \sin\left(\frac{\pi}{2}\right) = x\)
  \(\cosh\left(\frac{\alpha}{2}\right) = x = \sqrt{\frac{Z_1}{4Z_2}}\)
  \(\alpha = 2 \cosh^{-1}\left(\sqrt{\frac{Z_1}{4Z_2}}\right)\)

Graphical Expression for Change in Attenuation Constant

The following figure gives the graphical expression for the change in attenuation constant.

• The attenuation occurs in the attenuation band or the stop band.
• The attenuation bands are obtained at \((\frac{β}{2}=nπ)\) where n=0,1,2…. when Z₁ and Z₂ are of the same type. This expression is for positive values of \(\frac{Z_1}{4Z_2}\). The expression is obtained as follows:
  \(\alpha = 2 \sinh^{-1}\left(\sqrt{\frac{Z_1}{4Z_2}}\right)\)
• The attenuation bands are obtained at (β=π) when Z1 and Z2 are of different types. This expression is for negative values of \(\frac{Z_1}{4Z_2}\). The expression is obtained as follows:
  \(\alpha = 2 \cosh^{-1}\left(\sqrt{\frac{Z_1}{4Z_2}}\right)\)
• The cut-off frequencies are obtained when \(\frac{Z_1}{2Z_2}=0\) or -1.
  \(\frac{Z_1}{4Z_2} = 0\)
  \(Z_1 = 0\)
  \(\frac{Z_1}{4Z_2} = -1\)
  \(Z_1 + 4Z_2 = 0\)

Graphical Expression for Change in Phase Shift Constant

The following figure gives the graphical expression for the change in phase shift constant.

• The transmission occurs in the pass band or the transmission band.
• The transmission bands are obtained at (α=0) when Z₁ and Z₂ are of different types. This expression occurs for negative values of \(\frac{Z_1}{4Z_2}\).
  \(\beta = 2 \sin^{-1}\left(\sqrt{\frac{Z_1}{4Z_2}}\right)\)
• The cut-off frequencies are obtained at the following values.
  \(\frac{Z_1}{4Z_2} = 0 \quad \text{or} \quad Z_1 = 0\)
  \(\frac{Z_1}{4Z_2} = -1 \quad \text{or} \quad Z_1 + 4Z_2 = 0\)
• This cut-off frequency separates the attenuation band from the pass band.

Differences Between Attenuation Band and Transmission Band

The following table compares the differences between the attenuation band and the transmission band.