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Theoretical foundations of electrical engineering

# Calculation of circuits with sinusoidal current (“Electric circuits of the single-phase sinusoidal current”), construction of vector and topographical diagrams

It is necessary to define in the electric circuit of the single-phase sinusoidal current which elements and parameters are specified for each case in the table:
1) resistance of the circuit and its nature;
2) RMS currents in branches;
3) voltmeter and wattmeter indications;
4) construct vector diagram of currents and topographical diagram of voltages for entire circuit.

Given
E = 130 V;
f = 50 Hz;
R1 = 9 ohms;
L1 = 15,9 mH;
C2 = 318 μF;
L2 = 9,4 mH;
R3 = 8 ohms;
C3 = 500 μF;

## Solution

Figure 1. Original scheme of the circuit. Chose arbitrary directions of currents in branches. Replace meters with its internal resistance:
Figure 2. Design circuit

Cyclic frequency of the circuit:

Resistance of reactive components:

XL1=ωL1=314•15,9•10^(-3)=4,99 ohms;

XC2=1/(ωC2)=1/(314•318•10^(-6))=10,01 ohms;

XL2=ωL2=314•9,4•10^(-3)=2,95 ohms;

XC3=1/(ωC1 )=1/(314•500•10^(-6))=6,37 ohms;

Total resistance of the circuit:

The input resistance has active-inductive nature; it is indicated by the presence of a real part and positive imaginary part of the resistance.

Complex of the effective value of the input voltage:

E=E•(cosφ+jsinφ)=130•(cos0o+jsin0o)=130+0j V;

Rms complex currents in the circuit:

I1=E/Zвх =130/(10,632+0,669j)=12,179-0,767j=12,203•e(-4oj) A;

I2=I1•((R3-jXC3))/(R3+j(XL2-XC2-XC3))=(12,179-0,767j)•(8-6,37j)/(8+j•(2,95-10,01-6,37))=

=7,631+2,346j=7,983•e(17oj) A;

I3=I1-I2=12,179-0,767j-7,631-2,346j=4,548-3,113j=5,511•e(-35oj) A;

Voltage drops on elements:

UL1=I1•jXL1=(12,179-0,767j)•4,99j=3,83+60,77j V;

UR1=I1•R1=(12,179-0,767j)•9=109,61-6,90j V;

UL2=I2•jXL2=(7,631+2,346j)•2,95j=-6,92+22,51j V;

UC2=I2•(-jXC2)=(7,631+2,346j)•(-10,01j)=23,48-76,39j V;

UR3=I3•R3=(4,548-3,113j)•8=36,38-24,90j V;

UC3=I4•(-jXC3)=(4,548-3,113j)•(-6,37j)=-19,83-28,97j=35,11•e(-124oj) V;

Voltmeter indications (which measures Rms voltage):

UV=UC3=35,11 V;

Wattmeter indications (which measures active power):

Pw=Re(E•I1*)=Re(130•(12,179+0,767j))=1583,3 W;

I* — conjugate current. For example, if I=a+jb, then I*=a-jb;

Figure 3. Topographical diagram of voltages
Figure 4. Vector diagram of currents
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