Copyright Paul Tobin 63

Transcription

1 DT, Kevin t. lectric Circuit Thery DT87/ Tw-Prt netwrk parameters ummary We have seen previusly that a tw-prt netwrk has a pair f input terminals and a pair f utput terminals figure. These circuits were analysed using several circuit techniques such as mesh, ndal, superpsitin, Thevenin and Nrtn t predict the behavir f the netwrk. A further technique is t use the input and utput current and vltages,, V i, V t generate sets f parameters. By cnsidering a cmbinatin f currents and vltages and taking the input current r the input vltage r a cmbinatin f current and vltage as the independent variable we generate six different types f parameters. Fr example if we cnsider the input and utput current as the independent variables, we generate the parameters. f we take the input and utput vltages as the independent variables, we generate the Y parameters. Anther set f parameters, the hybrid (mixed and Y r h-parameters uses the input current and utput vltage as the independent variables. The remaining three cmbinatins have specialized use and will nt be discussed here. The - Parameters These parameters are ften referred t as the pen-circuit parameters and have uses in analyzing passive netwrks and sme active netwrks. Here and are the independent variables. The equatins are: ( And in matrix frm ( Figure 3: A tw - prt netwrk. Cpyright Paul Tbin 63

2 DT, Kevin t. lectric Circuit Thery DT87/ The fur parameters have the dimensins f impedance and fr the general case can be cnsidered cmplex (R jx. We can define each f these parameters by letting certain currents t be zer. Open circuiting the input r the utput terminals prduces the name pen circuit parameters. We define the parameters as: 0 i.e. pen-circuit at the /p nput r driving pint impedance. imilarly i.e. pen circuit at the /P Reverse transfer impedance. 0 Als 0 utput impedance. 0 frward transfer impedance. Figure : Obtaining an expressin fr the utput impedance. The -parameters can be calculated by using the definitin fr each parameter and slve fr a particular current r vltage. Cpyright Paul Tbin 6

3 -equivalent circuit DT, Kevin t. lectric Circuit Thery DT87/ Figure 5: The - equivalent circuit. The next step in this technique is t define f btain expressins fr the current gain, vltage gain and pwer gain and als expressins fr input and utput impedance. We can write the circuit parameters in matrix frm as: (3 We can say that. The negative sign takes care f the directin f current and plarity f ubstituting fr in the abve matrix using ( (5 Als the input vltage is expressed as: (6 (7 0 ( 0 ( Cpyright Paul Tbin 65

4 DT, Kevin t. lectric Circuit Thery DT87/ Obtain an expressin fr the vltage gain V Av V ut in (8 0 s (9 0 (0 ( s ( ( ( ( Current Gain ut (3 The current gain fr a tw-prt netwrk is defined as: s in rder t find an expressin, fr the current gain in terms f the -parameters we must slve fr and separately. ( The matrix equatin is changed if a lad is present then 0 (5 T slve fr 0 0 (6 Fr the input current Cpyright Paul Tbin 66

5 DT, Kevin t. lectric Circuit Thery DT87/ ( (7 imilarly 0 (8 (9 Current Gain A i (0 A i ( (. Ai ( T find an expressin fr the input impedance, we use this matrix and slve fr. We write using Cramer's Rule. ( ( (3 ( ( in (5 Find an expressin fr ut Tw-Prt Netwrks A tw-prt netwrk, shwn in figure 5, has a pair f input terminals and a pair f utput terminals. xamples f such tw-prt netwrks are: Transfrmer Cpyright Paul Tbin 67

6 DT, Kevin t. lectric Circuit Thery DT87/ T netwrk, eries-tuned CR We can classify these as ymmetrical r asymmetrical netwrks and Balanced r unbalanced netwrks. A symmetrical netwrk is defined as a netwrk such that when the i/p and /p terminals are interchanged, the electrical prperties f the i/p and /p remain unaltered. A Tee netwrk with equal series arms is an example f such a symmetrical netwrk. f the series arms were nt equal then the netwrk is asymmetrical. nterchanging the i/p and /p wuld change the electrical prperties f the i/p and /p. Further classificatin is a balanced netwrk where the tw input arms cntain the same elements. A tee netwrk is an example f an unbalanced netwrk. The characteristic impedance is defined as the impedance lking int ne pair f terminals when the ther pair f terminals is terminated in the characteristic impedance. in characteristic impedance Alternatively it is defined as the input impedance f a netwrk terminated at infinity ymmetrical and Unbalanced Ttal series Arm Ttal shunt Arm Frm the definitin f characteristic impedance, the input impedance is the characteristic impedance when terminated in. in [ ] (6 ( ( x ( (7 (8 / c s / c 5Ω 5 5( Ω (9 Cpyright Paul Tbin 68

7 DT, Kevin t. lectric Circuit Thery DT87/ 50(5 75 Ω Asymmetrical Netwrks f a netwrk is asymmetrical we cannt interchange the input and /p terminals withut affecting the electrical prperties f the netwrk. n this situatin the characteristic impedance has a different value when lking at either i/p r /p. n this situatin we have t intrduce the cncept f the image impedance mage mpedance These are defined fr a tw-prt netwrk as the tw impedances, which are such, that when ne f them is cnnected acrss the apprpriate pair f terminals f the netwrk the ther is seen acrss the ppsite pair f terminals. Fr example, cnsider an asymmetrical T netwrk. T simplify the calculatins fr different netwrks we can shw that the characteristic impedance fr a tw-prt netwrk is equal t the square rt f the prduct f the shrt - circuit and pen circuit impedance. T prve this, cnsider the unbalanced symmetrical T netwrk. The input impedance, when the /p is pen circuited, is / c The input impedance when the /p is shrt circuit is // s / c (30 } (3 { / c s / c ( (3 (33 (3 Thus we can find a value fr the characteristic impedance in terms f the elements f the tw-prt netwrks. e.g. 50 Ω and 00 Ω. Cpyright Paul Tbin 69

8 DT, Kevin t. lectric Circuit Thery DT87/ Cpyright Paul Tbin 70 Ω 75 50(00 (50 Ω //00 5 N N Verify the π netwrk using π c s c / / / ( c / ( ( c ( π x π π π Or in terms f the Tee netwrk characteristic impedance: T π