220-A1.2

• Activity: Simulating simple RLC circuits
• At the end of this activity, the student should be able to:

1. Run DC, AC, and transient simulations using ngspice.

A Wideband RC Voltage Divider

One way to build high-speed circuits with relatively large input impedances and capacitances is to use a simple RC voltage divider, as shown in the figure below. This RC divider is commonly found in oscilloscope 10X probes.

Let $Z_{1}=R_{1}\|{\frac {1}{sC_{1}}}={\frac {R_{1}}{1+sR_{1}C_{1}}}$ and similarly $Z_{2}={\frac {R_{2}}{1+sR_{2}C_{2}}}$. Thus, the output voltage can be expressed as:

$v_{out}={\frac {Z_{2}}{Z_{1}+Z_{2}}}\cdot v_{in}={\frac {R_{2}}{R_{1}+R_{2}}}\cdot {\frac {1+sR_{1}C_{1}}{1+s{\frac {R_{1}R_{2}}{R_{1}+R_{2}}}\left(C_{1}+C_{2}\right)}}\cdot v_{in}={\frac {R_{2}}{R_{1}+R_{2}}}\cdot {\frac {1+{\frac {s}{\omega _{z}}}}{1+{\frac {s}{\omega _{p}}}}}\cdot v_{in}$

(1)

Notice that we can cancel out the pole with the zero when we set $\omega _{z}=\omega _{p}$, or equivalently,

${\frac {C_{1}}{C_{1}+C_{2}}}={\frac {R_{2}}{R_{1}+R_{2}}}$

(2)

Intuitively, we can think if this as a resistive voltage divider at low frequencies, and a capacitive divider with the same ratio at high frequencies. Thus, the output voltage will simply be equal to:

$v_{out}={\frac {R_{2}}{R_{1}+R_{2}}}\cdot v_{in}$

(3)

We can then build a simple 10X oscilloscope probe circuit with an input impedance of $1\,\mathrm {M\Omega }$ and an input capacitance of $1\,\mathrm {pF}$ using $R_{1}=900\,\mathrm {k\Omega }$, $R_{2}=100\,\mathrm {k\Omega }$, $C_{1}=1.11\,\mathrm {pF}$, and $C_{2}=10\,\mathrm {pF}$. Since the pole and zero cancel each other out, the bandwidth of the probe circuit is not limited by its RC values.