# Solve Parallel RC Circuit Diff Eq | Current Flow

• cpuwildman
In summary, the conversation discussed an RC circuit driven by a voltage source with a resistor and capacitor in parallel. The problem was figuring out the governing differential equation for the current, which was easier in a series circuit. Impedances were also discussed, with the reminder that they only have meaning with time varying voltages and currents. The expression for a capacitor's impedance was given, and for a step excitation, a different concept should be used. The governing differential equation for the current in the circuit was also derived and it was noted that it would be different if an inductor was substituted for the capacitor.
cpuwildman
I have an RC circuit driven by a voltage source. The resister and capacitor are in parallel. I'm having a problem figuring out the governing differential equation for the current of the circuit. I could figure it out easily if it were a series circuit as the current is the same for each branch, but since the current splits between the resister and capacitor branches, I cannot figure it out. I would appreciate any help.

For any two impedances z1 and z2 in parallel, the total impedance is (z1-1 + z2-1)-1.

- Warren

Do impedances apply for both AC and DC voltage sources?

The two currents are independent. You have:
iR = V/R
iC = C dV/dt

If The initial voltage of the capacitor is different from the initial value of the voltage source you will have an impulse of current in your capacitor.

Impedances only have meaning with time varying voltages and currents. Notice that a step is a time varying function. So if you have a constant voltage source that is not initially connected to the circuit and you switch it on, this corresponds to a step of voltage. The impedance of the capacitor will be 1/sC, where s is the Laplace transform variable, but you really don't need it.

I know that a capacitor's impedance is given by $$Z_c=\frac{1}{\omega C}$$. If the circuit in question is driven by a DC voltage and a switch closes at t=0 to deliver the voltage, what should I use for $$\omega$$?

cpuwildman said:
I know that a capacitor's impedance is given by $$Z_c=\frac{1}{\omega C}$$. If the circuit in question is driven by a DC voltage and a switch closes at t=0 to deliver the voltage, what should I use for $$\omega$$?

No, this expression is valid only for a senoidal wave of frequency ω
For a step excitation (DC source switched at t = 0, you should use $$Z_c=\frac{1}{s C}$$, where s = σ + jω is a complex frequency associated with the Laplace transform variable.
In your case you don't need to use this concept. Use directly $$i_c=\frac{C dv}{dt}$$, where v(t) = V.u(t).
The derivative of the step function u(t) is δ(t), the unit impulse.
For the current in the resistor, it is simply $$i_r=\frac{v(t)}{R}$$.
The total current driven from the source is $$i_s=i_c + i_r$$

So then the governing differential equation would be $$i(t)=\frac{v(t)}{R}+C\frac{dv(t)}{dt}$$?

cpuwildman said:
So then the governing differential equation would be $$i(t)=\frac{v(t)}{R}+C\frac{dv(t)}{dt}$$?

You are right.

Thanks for the help.

Now, if I had the same circuit substituting an inductor for the capacitor, I would get $$i(t)=\frac{v(t)}{R}+\frac{1}{L}\int^{t}_{0}v(\tau)d\tau$$. Is this the governing differential equation for $$i(t)$$ in the new circuit? I'm not sure because of the integral.

cpuwildman said:
Thanks for the help.

Now, if I had the same circuit substituting an inductor for the capacitor, I would get $$i(t)=\frac{v(t)}{R}+\frac{1}{L}\int^{t}_{0}v(\tau)d\tau$$. Is this the governing differential equation for $$i(t)$$ in the new circuit? I'm not sure because of the integral.
Yes, this would be an integral equation. The solution for a step of amplitude V is:
$$i(t)=\frac{V.u(t)}{R}+\frac{V}{L}t.u(t)$$.

## What is a parallel RC circuit?

A parallel RC circuit is an electrical circuit that has a combination of a resistor (R) and a capacitor (C) connected in parallel. This means that the two components share the same voltage source, but have different current paths.

## How do you solve a parallel RC circuit?

To solve a parallel RC circuit, you need to use the differential equation that describes the relationship between the voltage and current in the circuit. This equation is called the RC circuit differential equation and can be solved using various techniques such as separation of variables or Laplace transforms.

## What is the current flow in a parallel RC circuit?

The current flow in a parallel RC circuit is determined by Ohm's law, which states that the current (I) is equal to the voltage (V) divided by the resistance (R). In addition, the current also changes over time due to the charging and discharging of the capacitor in the circuit.

## How does a parallel RC circuit affect the frequency of the current?

The presence of a capacitor in a parallel RC circuit affects the frequency of the current by creating a phase shift between the voltage and current. This means that the current will reach its maximum value at a different time than the voltage, resulting in a change in the frequency of the current.

## What are some real-world applications of parallel RC circuits?

Parallel RC circuits are commonly used in electronic devices such as filters, oscillators, and timer circuits. They are also used in power supplies to provide a stable and constant output voltage. Additionally, parallel RC circuits are used in radio frequency (RF) circuits for signal processing and amplification.

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