Low-Pass Prototype Element Values: What g1 to gN Mean
Low-pass prototype g-values are element values normalized to a 1 Ω source and ωc = 1 rad/s, with g0 and g(N+1) as the source and load terms. For Butterworth, g_k = 2·sin((2k − 1)π/(2N)), so N = 3 gives 1, 2, 1.
Engineer’s note: Divide by 2π·fc, not fc. Miss the 2π and every inductor and capacitor comes out more than six times too large, yet the values still look plausible enough to slip past a review. One part worked by hand catches it.
For practitioners
A filter design table gives you a row of plain numbers: g0, g1, g2 and so on up to g(N+1). There are no henries, no farads and no ohms in it. That row is a low-pass prototype, and once you know what each number stands for, it works for any impedance level and any cut-off frequency.
What a prototype row actually is
A low-pass prototype is a lossless LC ladder between a source resistor and a load resistor. It is normalized in two ways:
- The source impedance is 1 Ω.
- The cut-off frequency is ωc = 1 rad/s.
Within that normalized world, g1 to gN are the reactive element values. In a ladder, each one is either a series inductance or a shunt capacitance, and the two types alternate along the ladder. If g1 is a shunt capacitor, g2 is a series inductor, g3 is a shunt capacitor again, and so on. If g1 is a series inductor, the pattern flips.
So a g-value is a number of henries or farads, but only in a circuit that runs from a 1 Ω source with a cut-off at 1 rad/s. It is not tied to any real frequency or impedance until you scale it. That is why one row serves every design: the response shape lives in the ratios between the g-values, and scaling only moves that shape to a new impedance level and frequency.
g0 and g(N+1): the terminations
The two end values are not reactive elements. g0 is the source and g(N+1) is the load. Each one is either a resistance or a conductance, and which one it is depends on its neighbour:
- g0 is a resistance if g1 is a shunt capacitor, and a conductance if g1 is a series inductor.
- g(N+1) is a resistance if gN is a shunt capacitor, and a conductance if gN is a series inductor.
The rule follows from duality. A ladder that starts with a shunt C is naturally described with a source resistance. Its dual, which starts with a series L, swaps every impedance for an admittance, so the termination becomes a conductance. When g0 = 1 the difference disappears, because 1 Ω and 1 S are the same termination. When a termination value is not 1, getting this wrong inverts the load.
A table always lists g(N+1) in its own column. Read it from the table; do not assume it is 1 for every response family.
Computing Butterworth values directly
For the maximally flat (Butterworth) response you do not need a table. The element values have a closed form:
with g0 = g(N+1) = 1. At ω = 1 rad/s the normalized prototype is 3 dB down, which is the standard Butterworth cut-off.
Worked example, N = 3:
- g1 = 2·sin(π/6) = 2·sin 30° = 1
- g2 = 2·sin(3π/6) = 2·sin 90° = 2
- g3 = 2·sin(5π/6) = 2·sin 150° = 1
- g0 = g4 = 1
Worked example, N = 5 (angles in steps of 36°, starting at 18°):
- g1 = 2·sin 18° = 0.618
- g2 = 2·sin 54° = 1.618
- g3 = 2·sin 90° = 2
- g4 = 2·sin 126° = 1.618
- g5 = 2·sin 162° = 0.618
- g0 = g6 = 1
The values are symmetric about the middle element. That comes straight from the formula: replacing k with N + 1 − k changes the angle to π − (2k − 1)π/(2N), and sin(π − x) = sin x. For odd N the middle element sits at k = (N + 1)/2, where the angle is π/2, so it is always exactly 2. For even N there is no middle element and no value reaches 2. The outer elements are the smallest because the angle is closest to 0 or π there.
From one row to any impedance and cut-off
To move the prototype to a system impedance R0 and a cut-off ωc = 2π·fc, scale each element by its type:
Terminations scale as R = g·R0 when g is a resistance, and R = R0/g when g is a conductance.
Inductors scale up with impedance and capacitors scale down. Both scale down with frequency. That is the full content of impedance and frequency scaling for a low-pass filter.
Take the N = 3 Butterworth row in a 50 Ω system, starting with a shunt capacitor:
- C1 = C3 = 1/(50·ωc)
- L2 = 2·50/ωc = 100/ωc
- Source and load: 50 Ω each
The dual form, starting with a series inductor, uses the same row:
- L1 = L3 = 50/ωc
- C2 = 2/(50·ωc) = 1/(25·ωc)
Both ladders give the same response. Pick a value for ωc and you have real component values. The response itself is easier to check from S21 and S11 than from the element list; see Low-Pass Filter S-Parameters Explained.
Common reading errors
Other mistakes come up often:
- Treating g-values as picofarads or nanohenries. They have no unit prefix. They are henries and farads in a 1 Ω, 1 rad/s circuit.
- Using f instead of ω. The prototype is normalized to 1 rad/s. Divide by 2π·fc, not by fc.
- Assuming g(N+1) = 1. That holds for Butterworth. For other families, read the table column.
- Mixing up the terminations. With a series-L start, g0 is a conductance. If it is not 1, then R = R0/g0, not R0·g0.
In plain words · for students and beginners
Low-pass prototype values in plain words
A recipe written for one person
A prototype table is like a recipe written for exactly one serving. Every number in it assumes one standard situation. Here that situation is a 1 Ω source and a cut-off of 1 rad/s. A cut-off is the frequency where the filter starts blocking signals. rad/s (radians per second) is a way of measuring frequency, and 1 rad/s is about 0.16 Hz. To cook for a crowd, you scale the recipe. To build a real filter, you scale the g-values to your real impedance (often 50 Ω) and your real cut-off frequency.
Where the analogy breaks down: in a recipe every ingredient grows the same way. In a filter, inductors (coils) get bigger when you raise the impedance, but capacitors get smaller. Both get smaller when you raise the frequency. Each type has its own scaling rule.
The three-element example, step by step
A ladder is a chain of parts: one part goes in series (in line with the signal path), the next goes in shunt (from the signal path down to ground), and so on, taking turns.
- Pick the order: N = 3, meaning three reactive parts (parts that store energy, here coils and capacitors).
- Use the Butterworth formula, 2 times the sine of an angle. The angles for N = 3 are 30°, 90° and 150°.
- 2·sin 30° = 1, so g1 = 1. 2·sin 90° = 2, so g2 = 2. 2·sin 150° = 1, so g3 = 1.
- The ends are g0 = 1 (the source) and g4 = 1 (the load). In the drawing they are resistors, because the parts next to them are shunt capacitors.
- The result: shunt capacitor 1, series coil 2, shunt capacitor 1, between two 1 Ω resistors.
To make it real in a 50 Ω system: each capacitor becomes 1/(50·ωc), the coil becomes 100/ωc, and both resistors become 50 Ω. Here ωc is 2π times your cut-off frequency in hertz.
Mistakes beginners make
- Reading g-values as real component values. A "2" is not 2 nH or 2 pF. It means nothing in hardware until you scale it.
- Forgetting the 2π. The formulas use ωc = 2π·fc. If you leave out the 2π, every part comes out about six times too large.
- Forgetting that the row works both ways. The same numbers can start with a capacitor or with a coil. Decide which form you are building first, then scale each part by its own type.
- Treating g0 and g(N+1) like coils or capacitors. They are the source and load ends of the filter, not filter parts.
Remember
- g-values are normalized: 1 Ω source, cut-off 1 rad/s, no units until you scale.
- g0 and g(N+1) are the terminations. g1 to gN take turns between series coil and shunt capacitor.
- Butterworth values follow 2·sin((2k − 1)π/(2N)): 1, 2, 1 for N = 3, and 0.618, 1.618, 2, 1.618, 0.618 for N = 5.
Related pages on RF View
Frequently asked questions
What do g0 and g(N+1) mean in a low-pass prototype table?
g0 is the source termination and g(N+1) is the load termination. Each one is a resistance when the element next to it is a shunt capacitor and a conductance when the element next to it is a series inductor.
How do I calculate Butterworth g-values without a table?
Use g_k = 2·sin((2k − 1)π/(2N)) for k = 1 to N, with g0 = g(N+1) = 1. For N = 3 this gives 1, 2, 1.
What units do low-pass prototype g-values have?
None until you scale them. They are normalized to a 1 Ω source and a cut-off of 1 rad/s, so a g-value turns into henries, farads or ohms only after you apply impedance and frequency scaling.
Can the same g-values start with either a series inductor or a shunt capacitor?
Yes. The two ladder forms are duals and give the same response. Only the meaning of each g changes: inductance or capacitance, resistance or conductance.