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Solving (4) and (5) simultaneously results in I1 2 A and I2 1 A. The current in the center branch, shown dotted, is I1 I2 1 A. In Example 4.1 this was branch current I3 .

(11.1)

The currents do not have to be restricted to the windows in order to result in a valid set of simultaneous equations, although that is the usual case with the mesh current method. For example, see Problem 4.6, where each of the currents passes through the source. In that problem they are called loop currents. The applicable rule is that each element in the network must have a current or a combination of currents and no two elements in di erent branches can be assigned the same current or the same combination of currents.

" " Through use of the .PRINT statement, both magnitudes and phase angles of V1 , V5 , I"Ci , and I"Co are written to <Ex1_10.OUT> and can be retrieved by viewing of the le.

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In Section A1 complex numbers were de ned in rectangular form. In Fig. A-3, x r cos , y r sin , and the complex number z can be written in trigonometric form as z x jy r cos j sin where r is the modulus or absolute value (the notation r jzj is common), given by r angle tan 1 y=x is the argument of z. p x2 y2 , and the

Fig. 11-15

Ex1_10.CIR - h-parameter evaluation .PARAM I1value=0mA V5value=1mV I1 0 1 AC {I1value} R10 1 0 1Tohm ; Large resistor to avoid floating node Ci 1 2 100uF RB 2 3 10kohm VB 0 3 DC 10V R1 2 4 1kohm R2 4 0 5kohm C2 4 0 0.05uF Co 5 4 100uF V5 5 0 AC {V5value} .AC LIN 1 10kHz 10kHz .PRINT AC Vm(1) Vp(1) Im(Ci) Ip(Ci) ; Mag & phase of inputs .PRINT AC Vm(5) Vp(5) Im(Co) Ip(Co) ; Mag & phase of outputs .END

Fig. A-3

Fig. 11-16

Euler s formula, e j cos j sin , permits another representation of a complex number, called the exponential form: z r cos jr sin rej A third form, widely used in circuit analysis, is the polar or Steinmetz form, z r , where is usually in degrees.

" Set V5value=0 (deactivates V5 ) and I1value=1mA. Execute <Ex1_10.CIR> and retrieve the necessary " values of V1 ; I"Ci ; and I"Co to calculate h11 and h21 by use of (1.16) and (1.18). h11 h21 Vm 1 0:9091 Vp 1 Ip Ci 0:028 08 909:1 0:028 Im Ci 0:001 Im Co 9:08 10 4 Ip Co Ip Ci 1808 08 0:908 1808 Im Ci 1 10 3

11.5. Find the locus of points in rhombus ABCD in Fig. 11-16, which are equidistant from (a) AB and AD; (b) AB and (11.1) BC; (c) A and C; (d) B and D; (e) each of the four sides. 11.6. In Fig. 11-17, find the locus of points which are on or inside circle C and (a) 5 units from O (b) 15 units from O (c) Equidistant from circles A and C (d) 10 units from circle C (e) 10 units from circle A (f) 5 units from circle B (g) The center of a circle tangent to circles A and C (11.1 and 11.2)

To add two complex numbers, add the real parts and the imaginary parts separately. To subtract two complex numbers, subtract the real parts and the imaginary parts separately. From the practical standpoint, addition and subtraction of complex numbers can be performed conveniently only when both numbers are in the rectangular form.

Fig. 11-17

Set V5value=1mV and I1value=0 (deactivates I"1 ). Execute <Exl_10.CIR> and retrieve the needed values of " " V1 ; V5 ; and I"Co to evaluate h12 and h22 by use of (1.17) and (1.19). h21 h22 Vm 1 9:08 10 4 08 08 0:908 08 Vp 1 Vp 5 Vm 5 1 10 3 Im Co 3:15 10 6 Ip Co Vp 5 84:78 08 3:15 10 3 84:78 Vm 5 1 10 3

EXAMPLE A1 Given z1 5 j2 and z2 3 j8, z1 z2 5 3 j 2 8 2 j10 z2 z1 3 5 j 8 2 8 j6

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