KIRCHHOFF'S LAWS IN AC CIRCUITS

As we have already seen, circuits with sinusoidal excitation can be solved using complex impedances for the elements and complex peak or complex rms values for the currents and voltages. Using the complex values version of Kirchhoff's laws, nodal and mesh analysis techniques can be employed to solve AC circuits in a manner similar to DC circuits. In this chapter we will show this through examples of Kirchhoff's laws.

Example 1

Find the amplitude and phase angle of the current ivs(t)  if
vS(t) = VSM cos 2pft; i(t) =ISM cos 2pft;  VSM = 10 V; ISM = 1 A; f = 10 kHz; 

R = 5 ohm; L =0.2 mH;  C1 = 10 mF;  C2 = 5 mF

 
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Altogether we have 10 unknown voltages and currents, namely: i, iC1, iR, iL, iC2, vC1, vR, vL, vC2 and vIS. (If we use complex peak or rms values for the voltages and currents, we have altogether 20 real equations!)

The equations:

Loop or mesh equations: for  M1                    - VSM +VC1M+VRM = 0

                                                    M2              - VRM + VLM = 0

                                                    M3              - VLM + VC2M = 0

                                                    M4              - VC2M + VIsM = 0

Ohm's laws                                                   VRM = R*IRM

                                                                       VLM = j*w*L*ILM

                                                                       IC1M = j*w*C1*VC1M

                                                                       IC2M = j*w*C2*VC2M

Nodal equation for N1                                  - IC1M - ISM + IRM + ILM +IC2M = 0

for series elements                                      I = IC1M

Solving the system of equations you can find the unknown current:

ivs (t) = 1.81 cos (wt + 79.96°) A

Solving such a large system of complex equations is very complicated, so we haven't shown it in detail. Each complex equation leads to two real equations, so we show the solution only by the values calculated with TINA's Interpreter.

The solution using TINA's Interpreter:

{Solution by TINA's Interpreter}
om:=20000*pi;
Vs:=10;
Is:=1;
Sys Ic1,Ir,IL,Ic2,Vc1,Vr,VL,Vc2,Vis,Ivs
Vs=Vc1+Vr    {M1}
Vr=VL        {M2}
Vr=Vc2       {M3}
Vc2=Vis      {M4}
Ivs=Ir+IL+Ic2-Is   {N1}
{Ohm's rules}
Ic1=j*om*C1*Vc1
Vr=R*Ir
VL=j*om*L*IL
Ic2=j*om*C2*Vc2
Ivs=Ic1
end;
Ivs=[3.1531E-1+1.7812E0*j]
abs(Ivs)=[1.8089]
fiIvs:=180*arc(Ivs)/pi
fiIvs=[79.9613]

 The solution using TINA:

 
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To solve this problem by hand, work with the complex impedances. For example, R, L and C2 are connected in parallel, so you can simplify the circuit by computing their parallel equivalent. || means the parallel equivalent of the impedances:

Numerically:


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The simplified circuit using the impedance:

The equations in ordered form :                 I + IG1 = IZ

                                                          VS = VC1 +VZ

                                                                       VZ = Z · IZ

                                   I = j w C1· VC1

There are four unknowns' I; IZ; VC1; VZ ' and we have four equations, so a solution is possible.

Express I after substituting the other unknowns from the equations:

 

Numerically


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According to TINA's Interpreter's result.

{Solution using the impedance Z}
om:=20000*pi;
Vs:=10;
Is:=1;
Z:=replus(R,replus(j*om*L,1/j/om/C2));
Z=[2.1046E0-2.4685E0*j]
sys I
I=j*om*C1*(Vs-Z*(I+Is))
end;
I=[3.1531E-1+1.7812E0*j]
abs(I)=[1.8089]
180*arc(I)/pi=[79.9613]  

The time function of the current, then, is:

i(t) = 1.81 cos (wt + 80°) A


You can check Kirchhoff's current rule using phasor diagrams. The picture below was developed by checking the node equation in iZ = i + iG1 form. The first diagram shows the phasors added by parallelogram rule, the second one illustrates the triangular rule of the phasor addition.

Now let's demonstrate KVR using TINA's phasor diagram feature. Since the source voltage is negative in the equation, we connected the voltmeter 'backwards.' The phasor diagram illustrates the original form of the Kirchhoff's voltage rule.


 
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The first phasor diagram uses the parallelogram rule, while the second uses the triangular rule.


 


To illustrate KVR in the form VC1 + VZ ' VS = 0, we again connected the voltmeter to the voltage source backwards. You can see that the phasor triangle is closed.

Note that TINA lets you use either sine or cosine function as a base function. Depending on the function chosen, the complex amplitudes seen in phasor diagrams may differ by 90º. You can set the base function under 'View' 'Options' 'Base function for AC'. In our examples we always used cosine function as a base.

Example 2

Find the voltages and currents of all the components if:

vS(t) = 10 cos wt V,  iS(t) = 5 cos (w t + 30°) mA;

C1 = 100 nF,  C2 = 50 nF,  R1 = R2 = 4 k; L = 0.2 H,  f = 10 kHz.


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Let the unknowns be the complex peak values of the voltages and currents of 'passive' elements, as well as the current of the voltage source ( iVS ) and the voltage of the current source ( vIS ). Altogether, there are twelve complex unknowns. We have three independent nodes, four independent loops ( marked as MI), and five passive elements which can be characterized by five 'Ohm's laws' ' altogether there are 3+4+5 = 12 equations:

 Nodal equations      for N1              IVsM = IR1M + IC2M

                                               for N2               IR1M = ILM + IC1M

                                               for N3              IC2M + ILM + IC1M +IsM = IR2M

 Loop equations                   for M1              VSM = VC2M + VR2M

                                               for M2             VSM = VC1M + VR1M+ VR2M

                                               for M3             VLM = VC1M

                                               for M4             VR2M = VIsM

 Ohm's laws   VR1M = R1*IR1M

                                   VR2M = R2*IR2M

                                   IC1m = j*w*C1*VC1M

                                   IC2m = j*w*C2*VC2M

                                   VLM = j*w*L*ILM

Don't forget that any complex equation might lead to two real equations, so Kirchhoff's method requires many calculations. It's much simpler to solve for the time functions of the voltages and currents using a system of differential equations (not discussed here). First we show the results calculated by TINA's Interpreter:

{Solution by TINA's Interpreter}
f:=10000;
Vs:=10;
s:=0.005*exp(j*pi/6);
om:=2*pi*f;
sys ir1,ir2,ic1,ic2,iL,vr1,vr2,vc1,vc2,vL,vis,ivs
ivs=ir1+ic2         {1}
ir1=iL+ic1           {2}
ic2+iL+ic1+Is=ir2    {3}
Vs=vc2+vr2           {4}
Vs=vr1+vr2+vc1       {5}
vc1=vL               {6}
vr2=vis              {7}
vr1=ir1*R1           {8}
vr2=ir2*R2           {9}
ic1=j*om*C1*vc1      {10}
ic2=j*om*C2*vc2      {11}
vL=j*om*L*iL         {12}
end;
abs(vr1)=[970.1563m]
abs(vr2)=[10.8726]
abs(ic1)=[245.6503u]
abs(ic2)=[3.0503m]
abs(vc1)=[39.0965m]
abs(vc2)=[970.9437m]
abs(iL)=[3.1112u]
abs(vL)=[39.0965m]
abs(ivs)=[3.0697m]
180+radtodeg(arc(ivs))=[58.2734]
abs(vis)=[10.8726]
radtodeg(arc(vis))=[-2.3393]
radtodeg(arc(vr1))=[155.1092]
radtodeg(arc(vr2))=[-2.3393]
radtodeg(arc(ic1))=[155.1092]
radtodeg(arc(ic2))=[-117.1985]
radtodeg(arc(vc2))=[152.8015]
radtodeg(arc(vc1))=[65.1092]
radtodeg(arc(iL))=[-24.8908]
radtodeg(arc(vL))=[65.1092]  

Now try to simplify the equations by hand using substitution. First substitute eq.9. into eq 5. 

                        VS = VC2 + R2 IR2                                         a.)

then eq.8 and eq.9. into eq 5.

                        VS = VC1 + R2 IR2 + R1 IR1                 b.)

then eq 12., eq. 10. and IL from eq. 2 into eq.6.                   

                        VC1 = VL = jwL IL = jwL(IR1 ' IC1) = jwL IR1 - jwL jwC1 VC1

Express VC1             

                    c.)

Express VC2 from eq.4. and eq.5. and substitute eq.8., eq.11. and VC1:

                d.)

Substitute eq.2., 10., 11. and d.) into eq.3. and express IR2

IR2 = IC2 + IR1 + IS = jwC2 VC2 + IR1 + IS

          e.)

Now substitute d.) and e.) into eq.4 and express IR1

Numerically:


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 According to TINA's results.

The time function of iR1 is the following:

iR1(t) = 0.242 cos (wt+155.5°) mA

The measured voltages:


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