# AC Currents and Voltages Solver and Calculator

  

A calculator to calculate the voltages and currencts of an AC circuit with a load $Z_L$ given $Z_1$, $Z_2$, $Z_3$ and $Z_L$. The calculator computes all currents and voltages in polar form.

## Formulae for Currents and Voltages Used in the Calculator

Use Kirchhoff's law of current to write
$I_1 = I_2 + I_3$
and Kirchhoff's law of voltage
$V_i - V_{Z_1} - V_{Z_2} = 0$
$V_{Z_2} - V_{Z_3} - V_{Z_L} = 0$
Use Ohm's law to rewrite the above equations as
$I_1 = I_2 + I_3$
$V_i - Z_1 I_1 - Z_2 I_2 = 0$
$Z_2 I_2 - Z_3 I_3 - Z_L I_3 = 0$
Rewrite the above system in standard form
$I_1 - I_2 - I_3 = 0$
$Z_1 I_1 + Z_2 I_2 = V_i$
$Z_2 I_2 - (Z_3 + Z_L) I_3 = 0$
Solve the above system to obtain currents
$I_3 = \dfrac{Z_2 V_i}{(Z_1+Z_2)(Z_3+Z_L)+Z_1Z_2}$

$I_2 = \dfrac{(Z_3+Z_L) V_i}{(Z_1+Z_2)(Z_3+Z_L)+Z_1Z_2}$
$I_1 = I_2 + I_3$
Use Ohm's law to claculate voltages as follows
$V_{Z_1} = Z_1 I_1$
$V_{Z_2} = Z_2 I_2$
$V_{Z_3} = Z_3 I_3$
$V_o = Z_L I_3$

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## Example Using the Calculator

In the AC circuit below, we are given $v_i = 10 \angle 0^{\circ}$ , $R_1 = 100 \; \Omega$, $C = 0.47 \; \mu F$, $R_2 = 120 \; \Omega$, $R_3 = 200 \; \Omega$, $R_4 = 400 \; \Omega$, $L = 20 \; mH$ , frequency $f = 2$ kHz.
Find the currents $I_1$, $I_2$ , $I_3$ and the voltages across each resistor.

Let
$z_1 = R_1 = 100 \; \Omega \angle 0$

$\dfrac{1}{Z_2} = \dfrac{1}{R_2} + j 2 \pi f C$ , resisitor $R_2$ and capacitor $C$ are in parallel
Use Parallel RC circuit Impedance Calculator to calculate $Z_2$ and obtain
$Z_2 = 97.9040 \; \Omega \angle -35.3269^{\circ}$

$Z_3 = R_3 = 200 \; \Omega \angle 0$
$\dfrac{1}{Z_L} = \dfrac{1}{R_4} + \dfrac{1}{j 2 \pi f L }$ , resisitor $R_4$ and inductor $L$ are in parallel
Use Parallel RL circuit Impedance Calculator to calculate $Z_L$ and obtain
$Z_L = 212.8072 \; \Omega \angle 57.8581^{\circ}$
The above values for $Z_1$, $Z_2$, $Z_3$ and $Z_L$ are the default values for the calculator but of course you may change these values.

## Use of the calculator

Enter the impedances $Z_1$, $Z_2$, $Z_3$ and $Z_L$ as complex numbers in polar form (modulus and argument in degress) then press "calculate".
The calculator presented may be used to calculate the ac currens and voltages in any circuit that may be reduced to the basic circuit shown above.
The currents and voltages are in polar form.

 Peak Source Voltage V = 10 V   Impedance $Z_1$ = 100   $\angle$ 0 $^{\circ}$ Impedance $Z_2$ = 97.9040   $\angle$ -35.3269 $^{\circ}$ Impedance $Z_3$ = 200   $\angle$ 0 $^{\circ}$ Load Impedance $Z_L$ = 212.8072   $\angle$ 57.8581 $^{\circ}$