Unit 4 - Notes

ELE205

Unit 4: Network Functions for two Port Networks

1. Singularity Functions

Singularity functions are discontinuous functions or functions with discontinuous derivatives, widely used to model switching operations, impulsive sources, and transient responses in network analysis.

1.1 Unit Step Function, $u(t)$

The unit step function represents a signal that switches on at $t=0$ and stays constant. It models the application of a DC source.

Definition:
$u(t) = \begin{cases} 0 & \text{for } t < 0 \\ 1 & \text{for } t > 0 \end{cases}$
Laplace Transform:
$\mathcal{L}[u(t)] = \frac{1}{s}$

1.2 Unit Impulse Function, $\delta(t)$

The unit impulse function (Dirac Delta function) is the derivative of the unit step function. It represents a pulse of infinite height and zero width with an area of unity.

Definition:
$\delta(t) = 0 \text{ for } t \neq 0$
$\int_{-\infty}^{\infty} \delta(t) dt = 1$
Relation to Step: $\delta(t) = \frac{d}{dt}u(t)$
Laplace Transform:
$\mathcal{L}[\delta(t)] = 1$
Sampling Property: $\int_{-\infty}^{\infty} f(t)\delta(t-a) dt = f(a)$

1.3 Unit Ramp Function, $r(t)$

The unit ramp function is the integral of the unit step function. It increases linearly with time.

Definition:
$r(t) = \begin{cases} 0 & \text{for } t < 0 \\ t & \text{for } t \ge 0 \end{cases}$
Relation to Step: $r(t) = \int_{-\infty}^{t} u(\tau) d\tau$
Laplace Transform:
$\mathcal{L}[r(t)] = \frac{1}{s^2}$

1.4 Shifted Functions

Time-shifting allows singularity functions to represent events occurring at $t = a$ rather than $t = 0$ .

Shifted Step: $u(t-a)$ is 0 for $t<a$ and 1 for $t>a$ .
Shifted Impulse: $\delta(t-a)$ exists only at $t=a$ .
General Property: If $\mathcal{L}[f(t)] = F(s)$ , then $\mathcal{L}[f(t-a)u(t-a)] = e^{-as}F(s)$ .

1.5 Gate Function (Pulse Function)

A gate function represents a rectangular pulse of finite duration. It is constructed by subtracting two step functions.

Mathematical Representation: A pulse of magnitude $A$ starting at $t=a$ and ending at $t=b$ :
$g(t) = A[u(t-a) - u(t-b)]$
Laplace Transform:
$G(s) = A\left[ \frac{e^{-as}}{s} - \frac{e^{-bs}}{s} \right]$

2. Network Functions and Transfer Functions

The network function concept generalizes the relationship between input (excitation) and output (response) in the $s$ -domain (frequency domain).

2.1 Definition

For a linear, time-invariant (LTI) passive network with zero initial conditions:

H(s) = \frac{R(s)}{E(s)} = \frac{\text{Laplace Transform of Response}}{\text{Laplace Transform of Excitation}}

2.2 Classification of Network Functions

Driving Point Functions: Relate voltage and current at the same port.
- Driving Point Impedance: $Z(s) = V(s)/I(s)$
- Driving Point Admittance: $Y(s) = I(s)/V(s)$
Transfer Functions: Relate a quantity at one port to a quantity at a different port.
- Voltage Transfer Ratio: $G_{21}(s) = V_2(s) / V_1(s)$
- Current Transfer Ratio: $\alpha_{21}(s) = I_2(s) / I_1(s)$
- Transfer Impedance: $Z_{21}(s) = V_2(s) / I_1(s)$
- Transfer Admittance: $Y_{21}(s) = I_2(s) / V_1(s)$

3. Poles and Zeros

A network function $H(s)$ is typically a rational function:

H(s) = \frac{N(s)}{D(s)} = K \frac{(s-z_1)(s-z_2)\dots(s-z_m)}{(s-p_1)(s-p_2)\dots(s-p_n)}

3.1 Definitions

Zeros ( $z_i$ ): Values of complex frequency $s$ for which $N(s) = 0$ , causing $H(s) = 0$ . These are frequencies where transmission is blocked.
Poles ( $p_i$ ): Values of complex frequency $s$ for which $D(s) = 0$ , causing $H(s) = \infty$ . These determine the natural response and stability of the system.
Scale Factor ( $K$ ): A constant multiplier.

3.2 Significance

Stability: For a stable active network or any passive network, all poles must lie in the left half of the $s$ -plane (Real part $< 0$ ).
Frequency Response: The proximity of poles and zeros to the $j\omega$ axis determines the frequency response (magnitude and phase) of the network.

4. Necessary Conditions for Network Functions

For a function to be realizable as a physical passive network, it must satisfy specific mathematical properties.

4.1 Necessary Conditions for Driving Point Functions

Driving point functions ( $Z(s)$ or $Y(s)$ ) for passive networks must be Positive Real Functions (PRF).

Polynomial Properties: Coefficients of numerator $N(s)$ and denominator $D(s)$ must be real and positive.
Degree Constraint: The highest degrees of $N(s)$ and $D(s)$ may differ by at most 1. The lowest degrees may also differ by at most 1.
No Missing Terms: No terms of intermediate powers of $s$ can be missing in $N(s)$ or $D(s)$ unless all odd or all even terms are missing.
Pole/Zero Locations: Poles and zeros must lie in the left half-plane or on the $j\omega$ axis. Those on the $j\omega$ axis must be simple (non-repeated).
Real Part Condition: $\text{Re}[H(j\omega)] \ge 0$ for all $\omega$ .

4.2 Necessary Conditions for Transfer Functions

Stability: All poles must lie in the left-half $s$ -plane.
Zeros: Zeros may lie anywhere in the $s$ -plane.
Degree Constraint: The maximum degree of $N(s)$ must be less than or equal to the degree of $D(s)$ (usually).
Coefficients: Polynomial coefficients must be real.

5. Two-Port Networks

A two-port network is a "black box" circuit with two pairs of terminals:

Port 1 (Input): Voltage $V_1$ , Current $I_1$ .
Port 2 (Output): Voltage $V_2$ , Current $I_2$ .

Standard Convention: Current is usually assumed to flow into the positive terminal of both ports.

5.1 Z-Parameters (Open-Circuit Impedance Parameters)

Express voltages in terms of currents.

V_1 = z_{11}I_1 + z_{12}I_2

V_2 = z_{21}I_1 + z_{22}I_2

Parameter Definitions:
- $z_{11} = \frac{V_1}{I_1} \bigg|_{I_2=0}$ : Input impedance with output open.
- $z_{12} = \frac{V_1}{I_2} \bigg|_{I_1=0}$ : Reverse transfer impedance with input open.
- $z_{21} = \frac{V_2}{I_1} \bigg|_{I_2=0}$ : Forward transfer impedance with output open.
- $z_{22} = \frac{V_2}{I_2} \bigg|_{I_1=0}$ : Output impedance with input open.

Conditions:

Reciprocity: $z_{12} = z_{21}$
Symmetry: $z_{11} = z_{22}$

5.2 Y-Parameters (Short-Circuit Admittance Parameters)

Express currents in terms of voltages.

I_1 = y_{11}V_1 + y_{12}V_2

I_2 = y_{21}V_1 + y_{22}V_2

Parameter Definitions:
- $y_{11} = \frac{I_1}{V_1} \bigg|_{V_2=0}$ : Input admittance with output shorted.
- $y_{12} = \frac{I_1}{V_2} \bigg|_{V_1=0}$ : Reverse transfer admittance.
- $y_{21} = \frac{I_2}{V_1} \bigg|_{V_2=0}$ : Forward transfer admittance.
- $y_{22} = \frac{I_2}{V_2} \bigg|_{V_1=0}$ : Output admittance with input shorted.

Conditions:

Reciprocity: $y_{12} = y_{21}$
Symmetry: $y_{11} = y_{22}$

5.3 ABCD Parameters (Transmission/T-Parameters)

Used for cascaded networks. Relates input quantities to output quantities. Note: The convention for $I_2$ here is often negative (leaving the port), so equations reflect standard source-to-load direction. Assuming standard sign convention (current entering):

V_1 = A V_2 - B I_2

I_1 = C V_2 - D I_2

(Note: The negative sign exists because $I_2$ is defined entering the network, but transmission moves current to the load).

Parameter Definitions:
- $A = \frac{V_1}{V_2} \bigg|_{I_2=0}$ : Reverse voltage gain (Open circuit).
- $B = -\frac{V_1}{I_2} \bigg|_{V_2=0}$ : Transfer impedance (Short circuit).
- $C = \frac{I_1}{V_2} \bigg|_{I_2=0}$ : Transfer admittance (Open circuit).
- $D = -\frac{I_1}{I_2} \bigg|_{V_2=0}$ : Reverse current gain (Short circuit).

Conditions:

Reciprocity: $AD - BC = 1$
Symmetry: $A = D$

5.4 Hybrid (h) Parameters

Used extensively in transistor modeling (BJT). Mixes voltage and current.

V_1 = h_{11}I_1 + h_{12}V_2

I_2 = h_{21}I_1 + h_{22}V_2

Parameter Definitions:
- $h_{11} = \frac{V_1}{I_1} \bigg|_{V_2=0}$ : Input Impedance.
- $h_{12} = \frac{V_1}{V_2} \bigg|_{I_1=0}$ : Reverse Voltage Gain.
- $h_{21} = \frac{I_2}{I_1} \bigg|_{V_2=0}$ : Forward Current Gain.
- $h_{22} = \frac{I_2}{V_2} \bigg|_{I_1=0}$ : Output Admittance.

Conditions:

Reciprocity: $h_{12} = -h_{21}$
Symmetry: $\Delta h = h_{11}h_{22} - h_{12}h_{21} = 1$

6. Interrelationship Between Various Network Parameters

It is often necessary to convert one set of parameters to another.

6.1 Relation between Z and Y

The Y-parameter matrix is the inverse of the Z-parameter matrix.

[Y] = [Z]^{-1} \quad \text{and} \quad [Z] = [Y]^{-1}

y_{11} = \frac{z_{22}}{\Delta z}, \quad y_{12} = \frac{-z_{12}}{\Delta z}

y_{21} = \frac{-z_{21}}{\Delta z}, \quad y_{22} = \frac{z_{11}}{\Delta z}

Where

\Delta z = z_{11}z_{22} - z_{12}z_{21}

6.2 Relation between Z and ABCD (T)

z_{11} = \frac{A}{C}, \quad z_{12} = \frac{AD-BC}{C}

z_{21} = \frac{1}{C}, \quad z_{22} = \frac{D}{C}

6.3 Relation between h and Z

h_{11} = \frac{\Delta z}{z_{22}}, \quad h_{12} = \frac{z_{12}}{z_{22}}

h_{21} = \frac{-z_{21}}{z_{22}}, \quad h_{22} = \frac{1}{z_{22}}

6.4 Summary of Reciprocity and Symmetry

Parameter	Condition for Reciprocity	Condition for Symmetry
Z	$z_{12} = z_{21}$	$z_{11} = z_{22}$
Y	$y_{12} = y_{21}$	$y_{11} = y_{22}$
ABCD	$AD - BC = 1$	$A = D$
h	$h_{12} = -h_{21}$	$\Delta h = 1$

Unit 4 - Notes

Table of Contents

Unit 4: Network Functions for two Port Networks

1. Singularity Functions

1.1 Unit Step Function,

1.2 Unit Impulse Function,

1.3 Unit Ramp Function,

1.4 Shifted Functions

1.5 Gate Function (Pulse Function)

2. Network Functions and Transfer Functions

2.1 Definition

2.2 Classification of Network Functions

3. Poles and Zeros

3.1 Definitions

3.2 Significance

4. Necessary Conditions for Network Functions

4.1 Necessary Conditions for Driving Point Functions

4.2 Necessary Conditions for Transfer Functions

5. Two-Port Networks

5.1 Z-Parameters (Open-Circuit Impedance Parameters)

5.2 Y-Parameters (Short-Circuit Admittance Parameters)

5.3 ABCD Parameters (Transmission/T-Parameters)

5.4 Hybrid (h) Parameters

6. Interrelationship Between Various Network Parameters

6.1 Relation between Z and Y

6.2 Relation between Z and ABCD (T)

6.3 Relation between h and Z

6.4 Summary of Reciprocity and Symmetry

1.1 Unit Step Function, $u(t)$

1.2 Unit Impulse Function, $\delta(t)$

1.3 Unit Ramp Function, $r(t)$