NT Unit V Part II - Network Functions PDF

## SOLVED QUESTION BANK **Q. 5.1.** Explain the following terms: * Driving point function * Transfer function * Voltage gain * Current gain * Poles & zeroes of Network function **Ans:** * **Driving point function:** In a two-port network (see figure below), a driving point function relat...

## SOLVED QUESTION BANK **Q. 5.1.** Explain the following terms: * Driving point function * Transfer function * Voltage gain * Current gain * Poles & zeroes of Network function **Ans:** * **Driving point function:** In a two-port network (see figure below), a driving point function relates the voltages and currents at the same port. For example, $Z_{11} = \frac{V_1}{I_1}$ is the driving point impedance at port 1. * $Z_{22} = \frac{V_2}{I_2}$ is the driving point impedance at port 2 * $Y_{11} = \frac{I_1}{V_1}$ is the driving point admittance at port 1 * $Y_{22} = \frac{I_2}{V_2}$ is the driving point admittance at port 2 * **Transfer Function:** The transfer function is the ratio of the Laplace transform of the output voltage to the Laplace transform of the input voltage. For a two-port network, the transfer function is: > $T(S) = \frac{V_2(S)}{V_1(S)}$ * **Voltage Gain:** The voltage gain is the ratio of the output voltage to the input voltage. For a two-port network, the voltage gain is: > $\frac{V_2}{V_1}$ * **Current Gain:** The current gain is the ratio of the output current to the input current. For a two-port network, the current gain is: > $\frac{I_2}{I_1}$ * **Poles & Zeroes of Network Function:** If a network function or transfer function is given as: > $N(S) = T(S) = K \frac{(S+Z_1)(S+Z_2)---(S+Z_n)}{(S+P_1)(S+P_2)---(S+P_n)}$ where K is constant. * **Poles:** The values of 'S' at which the transfer function T(S) becomes infinite are called as poles of T(S). > $S=-P_1, S=-P_2, ... , S=-P_n$ * **Zeroes:** Values of 'S' at which the transfer function T(S) becomes zero are called as zeroes of T(S). > $S=-Z_1, S=-Z_2, ... , S=-Z_n$ **Figure:** Two-port network ``` ____ ____ | | | | _ | 1 |______| 2 | _ | |___|______|___| | _ | 1' |______| 2' | _ | | | | |____|______|____| V1 V2 ``` ## Network Functions **Q. 5.2.** Define the following with respect to Network functions: * Poles * Zeroes * Driving point impedance * Transfer admittance **Ans:** * **Driving Point Impedance:** The ratio of transform voltage to current. > $Z_{11}=\frac{V_1}{I_1}$ and $Z_{22}=\frac{V_2}{I_2}$ * **Transfer Admittance:** The ratio of transform current at the output port to the transform voltage at the input port. > $Y_{12} = \frac{I_2}{V_1}$ and $Y_{21} = \frac{I_1}{V_2}$ **Q. 5.3.** For the network in the figure (see below), find the transfer admittance $Y_{12}(S) = \frac{I_2(S)}{V_1(S)}$ and plot the poles and zeroes of $Y_{12}(S)$. **Ans:** <start_of_image> схемы в области S: **Figure:** Two-port network for Q. 5.3 ``` ____ ____ | | | | _ | 1 |______| 2 | _ | |___|______|___| | _ | 1' |______| 2' | _ | | | | |____|______|____| V1 V2 ``` **Figure:** Two-port network in S-domain ``` ____ ____ | | | | _ | 1 |______| 2 | _ | |___|______|___| | _ | 1' |______| 2' | _ | | | | |____|______|____| I1(s) I2(s) V1(s) V2(s) ``` **Q. 5.4.** For a given network function $I(S)$, draw the pole-zero diagram and hence obtain the time domain response. Verify the result analytically. > $I(S) = \frac{3S}{(S+1)(S+3)}$ **Ans:** **Pole-zero diagram:** **Q. 5.5.** Find the voltage ratio and transfer admittance function of the ladder network shown (see below). **Figure:** Ladder Network ``` + 1H | | ------- | | | | 1F | | | | | +---- | |-----+ | | | | 1F | | 1F | | | | +----- | |-----+ | | | | 1H | | + V1 V2 ``` **Figure:** Ladder Network in S-domain ``` + 1/S | | ------- | | | | 1/S | | | | | +---- | |-----+ | | | | 1/S | | 1/S | | | | +----- | |-----+ | | | | 1/S | | + I1(s) I4(s) V1(s) V4(s) ``` **Ans:** **Q. 5.6.** Explain how a time response can be obtained from a pole-zero plot. Consider a transform of current: > $I(S) = K \frac{(S+Z_1)(S+Z_2)---(S+Z_n)}{(S+P_1)(S+P_2)---(S+P_n)}$ **Ans:** **Q. 5.7.** Find the time domain response of the current transform, given by $I(s) = \frac{3S}{(S+2)(S^2+2S+2)} $ by locating the poles and zeroes in the S-plane. **Ans:** **Q. 5.8.** * Let $f_1(t) = 2u(t)$ and $f_2(t) = e^{-3t}u(t)$. Determine the convolution between $f_1(t)$ and $f_2(t)$. * A function in the Laplace domain is given by $F(S) = \frac{2(S+4)}{(S+3)(S+8)}$. Find the initial and the final value theorem and verify the same by finding $f(t)$. **Ans:** **Q. 5.9.** Obtain the current transfer ratio $\frac{I_2(S)}{I_1(S)}$ for the network in the figure (see below): **Figure:** Network for Q. 5.9 ``` + | | ------- | | | C_1 | | | +---- | |-----+ | | | | R | | ------- | | | C_2 | | | +---- |----|-----+ | | + I1 I2 ``` **Ans:** **Q. 5.10.** The driving point admittance of a network is given by: > $Y(S) = \frac{10S}{s^2 + 10S + 250}$ If a unit impulse voltage is applied to the network, find the current $i(t)$ entering the network using a pole-zero diagram, or the pole-zero plot. **Ans:** **Q. 5.11.** Plot the pole and zeroes in the S-plane for the following network function $F(S) = \frac{S^2+4S+3}{S^2+2S}$. Also, find $f(t)$. **Ans:** **Q. 5.12.** The network in the figure (see below) is terminated at port 2 by a resistance of 1 Ω. Find the transform admittance $Y_{12} = \frac{I_2(S)}{V_1(S)}$ for this terminated network. **Figure:** Two-port network for Q. 5.12 ``` + | | ------- | | | 2H | | | +---- | |-----+ | | | | 1Ω | | | ------- | | | 1H | | | +---- |----|-----+ | | + I1 V1 I2 V2 ``` **Ans:** **Q. 5.13.** Find the transform impedance of the network shown (see below) **Figure:** Network for Q. 5.13 ``` + | | ------- |-----+ | | | 2 Ω | | | | | +------ | |-----+ | | | | | | ------- |-----+ | | | 1/S | | | +------- | |------+ | | | | | | ------- |-----+ | | | 1 Ω | | | | | +------- | |------+ | | | | | + Z ``` **Ans:** **Q. 5.14.** A network function N(S) has the following data: * Zero at origin * A simple pole at -1 * Pairs of complex conjugate zero at -4+j1 & poles at -1+j1 * N(S) = 20 at infinite frequency. Determine network function N(S) **Ans:** **Q. 5.15.** For the given network function, plot poles and zeroes in the S-plane and, hence, obtain the time domain response. > $I(S) = \frac{3S}{(S+1)(S+4)}$ **Ans:** **Q. 5.16.** Show that the voltage transfer function of the following network (see below) has imaginary zeroes given by $S = \pm \frac{j}{CR}$. **Figure:** Network for Q. 5.16 ``` + | | ------- |-----+ | | | R | | | | | +------ | |-----+ | | | | | | ------- |-----+ | | | C | | | | | +------- | |------+ | | | | | | ------- |-----+ | | | R | | | | | +------- | |------+ | | | | | | ------- |-----+ | | | C | | | | | +------- | |------+ | | | | | + V1 V2 ``` **Ans:** **Q. 5.17.** Draw the pole-zero diagram for the given network function and, hence, obtain $V(t)$. > $V(S) = \frac{4(S+2)}{(S+1)(S+3)}$ **Ans:** **Q. 5.18.** For the ladder network shown in the figure (see below), find the: * Driving point input impedance $Z_{11}$ * Transfer impedance $Z_{12}$ * Voltage-ratio transfer function $G_{12}$ Assume $I_2 = 0$ **Figure:** Network for Q. 5.18 ``` + | | ------- | | | 1H | | | +---- | |-----+ | | | | 1F | | | ------- | | | 1H | | | +---- |----|-----+ | | + I1 V1 I2 = 0 V2 ``` **Ans:** **Q. 5.19.** Draw the pole-zero diagram for the given network function $I(S)$ and, hence, obtain i(t) using the pole-zero diagram. > $I(S) = \frac{20S}{(S+5)(S+2)}$ **Ans:** **Q. 5.20.** Obtain the pole-zero diagram of the given function. > $I(S) = \frac{2S}{(S+1)(S^2+2S+4)}$ **Ans:** **Q. 5.21.** Find the time domain response of the current transform, given by $I(s) = \frac{5S}{(S+1)(s^2+2S+2)} $ by locating the poles and zeroes in the S-plane. **Ans:** **Q. 5.22.** The given network is terminated at port 2 by a resistance of 1 Ω. Find the transform admittance $Y_{12} = \frac{I_2(S)}{V_1(S)}$ for this terminated network. **Figure:** Network for Q. 5.22 ``` + | | ------- | | | | 2 Ω | | | | | +---- | |-----+ | | | | 1F | | 1 Ω | | | | +----- | |-----+ | | | | 1 | | + I1(s) I2(s) V1(s) V2(s) ``` **Ans:** **Q. 5.23.** State the necessary condition for driving point functions and transfer functions. **Ans:** **Q. 5.24.** Find the voltage transfer function $G_{12}(S)$ for the network shown (see below). **Figure:** Network for Q. 5.24 ``` + | | ------- | | | | 2 Ω | | | | | +---- | |-----+ | | | | 1 Ω | | 1 Ω | | | | +----- | |-----+ | | | | 1 | | + V1 V2 ``` **Ans:** **Q. 5.25.** For a network function given as > $I(S) = \frac{10(S+1)}{(S+5)S^2+6S+ 13}$ Place the poles and zeroes in the S-plane and hence find i(t). **Ans:** **Q. 5.26.** Show that the transfer function of the following function (see below) is: > $F(S) = \frac{1}{S^2+1}coth(\frac{πS}{2})$ **Figure:** Function For Q. 5.26 ``` 1 | | 1 | | | | | | | | +------+ | | | | | | | +------+ 0 P t ``` **Ans:** **Q. 5.27.** For the network shown (see below) find the voltage transfer function $G_{12} = \frac{V_2(S)}{V_1(S)}$. **Figure:** Network for Q. 5.27 ``` + | | ------- | | | | 1 Ω | | | | | +---- | |-----+ | | | | 1F | | 2 Ω | | | | +----- | |-----+ | | | | 2F | | + V1 V2 ``` **Ans:** **Q. 5.28.** Select the function which represents the driving point impedance of a passive network, and justify. > i). $\frac{S^3+5S^2+S+5}{S(S^2+1)} $ > ii). $\frac{15(S^3+2S^2+3S+5)}{4(S^2+6S^3+8S^2+5)}$ **Ans:** **Q. 5.29.** Determine the driving point impedance of the given ladder network. **Figure:** Network for Q. 5.29 ``` + | | ------- | | | | 5F | | | | | +---- | |-----+ | | | | 5F | | 5H | | | | +----- | |-----+ | | | | 5H | | + V1 V2 ``` **Ans:** **Q. 5.30.** For the network shown (see below), find the current transfer ratio $o_{12} = \frac{I_2}{I_1}$. **Figure:** Network for Q. 5.30 ``` + | | ------- | | | 1 Ω | | | +------- | |--------+ | | | | 2 Ω | | | | | +------- | |--------+ | | | | 21 b | | | | | +------- | |--------+ | | | | 2 Ω | | | | | +------- | |--------+ | | | | 1 Ω | | | | | ------- | | | + I1 I2 V1 V2 ``` **Ans:** **Q. 5.31.** Find the voltage transfer function $G_{12}(S) = \frac{V_2(S)}{V_1(S)}$ for the network (see below) **Figure:** Network for Q. 5.31 ``` + | | ------- | | | | 1F | | | | | +---- | |-----+ | | | | 1 Ω | | 1F | | | | +----- | |-----+ | | | | 1 Ω | | + V1 V2 ``` **Ans:** **Q. 5.32.** Find the time domain response by locating the poles and zeroes in the S-plane for the following function: > $I(S) = \frac{3S}{S^2+2S+2}$ **Ans:** **Q. 5.33.** Explain the concept of complex frequency. **Ans:** **Q. 5.34** Find the voltage transfer function $G_{12}(S)$ for the network shown in the figure (see below). **Figure:** Network for Q. 5.34 ``` + | | ------- | | | | 2 Ω | | | | | +---- | |-----+ | | | | 1 Ω | | 1 Ω | | | | +----- | |-----+ | | | | 1 | | + V1 V2 ``` **Ans:** **Q. 5.35.** Draw the pole-zero diagram for the given network function and hence obtain $V(t)$. > $V(S) = \frac{4(S+2)}{(S+1)(S+3)}$ **Ans:** **Q. 5.36.** For the ladder network shown in the figure (see below), find the: * Driving point input impedance $Z_{11}$ * Transfer impedance $Z_{12}$ * Voltage-ratio transfer function $G_{12}$ Assume $I_2 = 0$. **Figure:** Network for Q. 5.36 ``` + | | ------- | | | 1H | | | +---- | |-----+ | | | | 1F | | | ------- | | | 1H | | | +---- |----|-----+ | | + I1 V1 I2 = 0 V2 ``` **Ans:** **Q. 5.37.** Find the voltage transfer ratio $G_{12}(S) = \frac{V_2(S)}{V_1(S)}$ for the network shown (see below). **Figure:** Network for Q. 5.37 ``` + | | ------- | | | | 1F | | | | | +---- | |-----+ | | | | 1 Ω | | 1F | | | | +----- | |-----+ | | | | 1 Ω | | + V1 V2 ``` **Ans:** **Q. 5.38.** Obtain the pole-zero diagram of the given function. > $I(S) = \frac{2S}{(S+1)(S^2+2S+4)}$ **Ans:** **Q. 5.39.** Find the time domain response of the current transform, given by $I(s) = \frac{5S}{(S+1)(s^2+2S+2)} $ by locating the poles and zeroes in the S-plane. **Ans:** **Q. 5.40.** Find the voltage transfer function $G_{12}(S)$ for the network shown in the figure (see below). **Figure:** Network for Q. 5.40 ``` + | | ------- | | | | 2 Ω | | | | | +---- | |-----+ | | | | 1 Ω | | 1 Ω | | | | +----- | |-----+ | | | | 1 | | + V1 V2 ``` **Ans:** **Q. 5.41.** Find the voltage transfer ratio $G_{12}(S) = \frac{V_2(S)}{V_1(S)}$ for the network shown (see below). **Figure:** Network for Q. 5.41 ``` + | | ------- | | | | 1F | | | | | +---- | |-----+ | | | | 1 Ω | | 1F | | | | +----- | |-----+ | | | | 1 Ω | | + V1 V2 ``` **Ans:** **Q. 5.42.** The given network is terminated at port 2 by a resistance of 1 Ω. Find the transform admittance $Y_{12} = \frac{I_2(S)}{V_1(S)}$ for this terminated network. **Figure:** Network for Q. 5.42 ``` + | | ------- | | | | 2 Ω | | | | | +---- | |-----+ | | | | 1F | | 1 Ω | | | | +----- | |-----+ | | | | 1 | | + I1(s) I2(s) V1(s) V2(s) ``` **Ans:** **Q. 5.43.** Explain the concept of complex frequency. **Ans:** **Q. 5.44** Find the voltage transfer function $G_{12}(S)$ for the network shown (see below). **Figure:** Network for Q. 5.44 ``` + | | ------- | | | | 2 Ω | | | | | +---- | |-----+ | | | | 1 Ω | | 1 Ω | | | | +----- | |-----+ | | | | 1 | | + V1 V2 ``` **Ans:** **Q. 5.45** For a network function given as > $I(S) = \frac{10(S+1)}{(S+5)S^2+6S+13}$ Place the poles and zeroes in the S-plane and hence find $i(t)$. **Ans:** **Q. 5.46.** Show that the transfer function of the following function (see below) is: > $F(S) = \frac{1}{S^2+1}coth(\frac{πS}{2})$ **Figure:** Function For Q 5.46 ``` 1 | | 1 | | | | | | | | +------+ | | | | | | | +------+ 0 P t ``` **Ans:** **Q. 5.47.** For the network shown in the figure (see below), find the voltage transfer function $G_{12} = \frac{V_2(S)}{V_1(S)}$. **Figure:** Network for Q. 5.47 ``` + | | ------- | | | | 1 Ω | | | | | +---- | |-----+ | | | | 1F | | 2 Ω | | | | +----- | |-----+ | | | | 2F | | + V1 V2 ``` **Ans:** **Q. 5.48.** Select the function which represents the driving point impedance of a passive network, and justify. > i). $\frac{S^3+5S^2+S+5}{S(S^2+1)} $ > ii). $\frac{15(S^3+2S^2+3S+5)}{4(S^2+6S^3+8S^2+5)}$ **Ans:** **Q. 5.49.** Determine the driving point impedance of the given ladder network. **Figure:** Network for Q 5.49 ``` + | | ------- | | | | 5F | | | | | +---- | |-----+ | | | | 5F | | 5H | | | | +----- | |-----+ | | | | 5H | | + V1 V2 ``` **Ans:** **Q. 5.50.** For the network shown (see below), find the current transfer ratio $o_{12} = \frac{I_2}{I_1}$. **Figure:** Network for Q. 5.50 ``` + | | ------- | | | 2H | | | +---- | |-----+ | | | | 1Ω | | | ------- | | | 1H | | | +---- |----|-----+ | | + I1 V1 I2 V2 ``` **Ans:** **Q. 5.51.** Find the voltage transfer ratio $G_{12}(S) = \frac{V_2(S)}{V_1(S)}$ for the network shown (see below). **Figure:** Network for Q. 5.51 ``` + | | ------- | | | | 1F | | | | | +---- | |-----+ | | | | 1 Ω | | 1F | | | | +----- | |-----+ | | | | 1 Ω | | + V1 V2 ``` **Ans:** **Q. 5.52.** The given network is terminated at port 2 by a resistance of 1 Ω. Find the transform admittance $Y_{12} = \frac{I_2(S)}{V_1(S)}$ for this terminated network. **Figure:** Network for Q. 5.52 ``` + | | ------- | | | | 2 Ω | | | | | +---- | |-----+ | | | | 1F | | 1 Ω | | | | +----- | |-----+ | | | | 1 | | + I1(s) I2(s) V1(s) V2(s) ``` **Ans:** **Q. 5.53.** Find the voltage transfer ratio $G_{12}(S) = \frac{V_2(S)}{V_1(S)}$ for the network shown (see below). **Figure:** Network for Q. 5.53 ``` + | | ------- | | | | 1F | | | | | +---- | |-----+ | | | | 1 Ω | | 1F | | | | +----- | |-----+ | | | | 1 Ω | | + V1 V2 ``` **Ans:** **Q. 5.54.** Find the time domain response by locating the poles and zeroes in the S-plane for the following function: > $I(S) = \frac{3S}{S^2+2S+2}$ **Ans:**