Строительный блокнот  Introduction to electronics 

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 [ 282 ] 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300

Transferfunction

Transfer function

Gis)

Linear circuit

-0 [--

Linear circuit

Input Output

Inpul Output

Port

Port

Open-circuit

Fig. C.l How an added element changes a transfer function Gis); (a) (irlglnal ciinditions, before addition of the flew element; (b) addition uf element having impedanee Z(jf).

Transfer function

Transfer function

Linear circuit Input Output Port

Linear circuit

Input Output

Port

Short-circuit

Fig. C.2 The dual form of the Extra Element Theorem, iir which the extra eleutetit replaces a short circuit: (a) original conditions, (b) additioti of element having impedance Z(.v),

ад Z[s)

Z(s) j

(C,2)

The right-hand side terms involving Z{s) account for the influence of Z{s) on G(s), and are known as the correction factor.

The Extra Element Theorem also apphes to the dual form illustrated in Fig. C.l. In this form, the transfer functitm is initially known under the conditions that the port is short-circuited. In Fig. C.2(b), the short-circuit is replaced by the impedance Z(.t). In this case, the addition of the impedance Z(.!) causes the transfer function to become

vsl

z(.v) - 0

ZM j

(C,3)



Short-circuit

Linear circuit

input Output

Port

+ vis) ~

Linear circuit Input Ouiput

Port

Fig. С J Determination of the quantities Zt) and Zfj): (a) Z,(i) is the Thevenin-equivalent iinpedanue at the poit, and is measured with the input Vj {s) set to zero; (b) Zi) is the impedance seen at the port under the condition that the output is nulled.

The Zy{j) and Z(j) terms in Eqs. (C.2) and (C.3) are identical. By equating the G(s) expressions of Eqs. (C.2) and (C.3), one can show that

HA - Zj)(i)

(C,4)

This is known as the reciprocity relationship.

The quantities ZJ,s) and Zq(,s) can be found by measuring impediinces at the port. The term Zg(s) is the Thevenin equivalent impedance seen looking into the port, also known as the driving-point impedance. As illustrated in Fig. C.3(a), this impedance is found by setting the independent source vO?) to zero, and then measuring the impedance between the terminals of the port:

Zy(j) =

(C.5)

Thus, Z{.v) is the impedance between the port terminals when the input v(i) is set to zero.

Determination of the impedance Zs) is illustrated in Fig. C.3(b). The term 2(s) is found under the conditions that the output v ,(.t) is nulled to zero. A current source s{s) is connected to the terminals ofthe port. In the presence of the input signal Vf (.r), the current i{s) is adjusted so that the output

nulled to zero. Under these conditions, the quantity Zfs) is given by



(.CA)

Note that mtlling the output is not the same as .horlin tlie output. If one simply shorted the output, then a euiient would flow through the short, which would induce voltage drops and currents in other elements of fhe network. These voltage drops and currents ;ue not present when the output is nulled. The null condition of Fig. C.3(b) does not employ any connections to the output of the circuit. Rather, the null condition employs the adjustment of the independent .sources ) t<) and i(s) in a special way that causes the output V ,(.t) to be zero. By superposition, can be expressed as a linear combinatitm of У,- -) and

i(j); therefore, for a given v-,Jis), it is always possible to choose an i(i) that will cause v (.v) to be zero. Under these null conditions, Zfs) is measured as the ratio of vj) to i(s). In practice, the circuit analysis to find Zfjs) is simpler than analysis of Zp(s), because the null condition cau.ses many of the signals within the circuit to be zero. Several examples ate given in Section C.4.

The input and output quantities need not be voltages, but could also be currents or other signals that can be set or nulled to zero. The next section contains a derivation of the Extra Element Tlieoreni with a general input u[s) and output y(s).

C.2 DERIVATION

Figure C.4(a) illustrates a general linear system having an input ii(.s) and an output y(.0- In addition, the system contains an electrical port having voltage v(.t) and current i(.i), widi the polarities illustiated. Initially, the port is open-circuited; ((.t) = 0. Tlie transfer function of this system, with the port open-circuited, is

(C.7)

The objective of the extra element theorem is to determine the new transfer function G(s) that is obtained when an impedance Z{s) is connected to the pott:

<C,8)

The situation is illustrated in Fig. C.4(b). It can be seen that the conditions at the port are now given by

Inptit

Linear network

Port

Open-circuit

Oittput

Linear network

uis)

y{s)

Input

Port

Output

Ф) -j

Fig. C.4 Modifioatiou of a hnear network by addinon of an extra element: (a) origiitEil .system, (b) modified system, with impedance Z(,s) connected at an eltscUical port.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 [ 282 ] 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300